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بسم اهلل الرحمن الرحیم

نام کتاب: چکیده مبسوط مقاالت پنجمین سمینارآنالیزهارمونیک تدوین کننده: حمیدرضاابراهیمی ویشکی صفحه آرا: فاطمه عزیزی، رامین فرشچیان باهمکاری: زهرا امیری هفشجانی، فاطمه خسروی، معصومه زارع و ماهره مهماندوست. ناشر: کمیته برگزاری سمینار سال انتشار: دی ماه 1395 شمارگان: 150 امورفنی و چاپ: مؤسسه چاپ وانتشارات دانشگاه فردوسی مشهد

فهرست مندرجات

پیش گفتار ...... آ

هیات موسس ...... ت

کمیته علمی ...... ث

کمیته برگزارکننده ...... ج

کادر اجرایی ...... ح

حمایت کنندگان ...... ح

سخنرانان مدعو ...... خ

شرکت کنندگان ...... د

پوسترهای فارسی ...... 2

مقاالت و پوسترهای انگلیسی ...... 1

پیش گفتار

به نام خداوند لوح و قلم حقیقت نگار وجود وعدم

با تائیدات خداوند متعال، به همت دانشگاه فردوسی مشهد و انجمن ریاضی ایران افتخار میزبانی پنجمین سمینار آنالیز هارمونیک و کاربردها نصیب گروه ریاضی محض دانشگاه فردوسی مشهد گردیده است. امیدواریم که تالش کمیته های مختلف برگزاری این سمینار بتواند رضایت خاطر میهمانان گرامی را فراهم نموده باشد.

در بازه زمانی تعیین شده تعداد 90 مقاله برای بررسی به دبیر خانه سمینار ارسال شد که پس از بررسی وانجام داوری توسط کمیته علمی سمینار، تعداد 51 مقاله بصورت ارائه شفاهی و 15 مقاله بصورت ارائه پوستر پذیرفته شدند. برای هر سخنرانی تخصصی 20 دقیقه برای ارائه و 5 دقیقه برای پرسش و پاسخ در نظر گرفته شده است. 4 سخنرانی عمومی )مدعو( 40 دقیقه ای نیز توسط 4 نفر از متخصصین شاخه آنالیز هارمونیک ارائه می گردد که تلفیقی مناسب ازمباحث مجرد و کاربردی در آن گنجانده شده است. . به منظور مشارکت هر چه بیشتر متخصصین این شاخه به ویژه تشویق ریاضیدانان جوان، درتعیین اعضای کمیته علمی سعی شده است تا از هر یک از دانشگاههای کشور که در این شاخه فعالیت دارند، الاقل یک عضو درکمیته علمی سمینار داشته باشیم.

ضمنا برای عالقه مندان به مباحث کاربردی، کارگاه آموزشی و پژوهشی نرم افزار متلب در روزهای 27 و 28 دی ماه تدارک دیده شده است که در آن به جنبه های عملی و کاربردی مباحث آنالیز هارمونیک پرداخته شده است، که امیدواریم مورد توجه شرکت کنندگان محترم قرار گیرد.

برگزارکنندگان این سمینار و دانشگاه فردوسی مشهد مفتخر هستند تا از دو تن از پیش کسوتان آنالیز هارمونیک کشور که از بنیانگذاران این شاخه از ریاضیات در کشور محسوب می گردند، تجلیل به عمل آورند.

استاد دکتر عبدالحمید ریاضی، استاد بازنشسته دانشگاه صتعتی امیر کبیر تهران و استاد دکتر محمد علی پورعبداهلل استاد بازنشسته دانشگاه فردوسی مشهد. مراسم تجلیل از این بزرگواران همزمان با جلسه افتتاحیه برگزار خواهد گردید.

برگزاری این سمینار مرهون هم فکری و حمایتهای همه جانبه افراد زیادی بوده است. حمایت های هیات رئیسه دانشگاه فردوسی مشهد، دانشکده های علوم ریاضی و مهندسی، انجمن ریاضی ایران، اعضای کمیته علمی و اجرایی سمینار، مرکز آثارومفاخردانشگاه، گروه ریاضی محض، کارکنان و دانشجویان ارجمند دانشکده علوم ریاضی و سایر حمایت کنندگانی که نام آنها در این کتابچه ذکر گردیده است، شایسته تقدیر فراوان وسپاس گزاری می باشند.

در خاتمه ضمن آرزوی اقامتی خوش در جوار بارگاه مقدس امام هشتم، امیدواریم که از برنامه های علمی سمینار بهره الزم را برده و کم و کاستی های موجود را با دیده اغماض بنگرید.

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با آرزوی سالمتی و شادکامی برای همه شرکت کنندگان

رجبعلی کامیابی گل، محمد جانفدا، معصومه فشندی، ریحانه رئیسی طوسی و

حمیدرضا ابراهیمی ویشکی

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‌ هیئت موسس سمینار

دکتر حمیدرضا ابراهیمی ویشکی دانشگاه فردوسی مشهد

دکتر غالمحسین اسالم زاده دانشگاه شیراز

دکتر سید مسعود امینی دانشگاه تربیت مدرس

دکتر عبدالرسول پورعباس دانشگاه صنعتی امیرکبیر

دکتر محمد علی پورعبداهلل دانشگاه فردوسی مشهد

دکتر سید علیرضا حسینیون دانشگاه شهید بهشتی

دکتر محمدعلی دهقان دانشگاه ولی عصر )عج( رفسنجان

دکترعلی رجالی دانشگاه اصفهان

دکتر مهدی رجبعلیپور دانشگاه شهید باهنر کرمان

دکتر عبدالحمید ریاضی دانشگاه صنعتی امیرکبیر

دکتر طاهر قاسمی هنری دانشگاه خوارزمی

دکتر رجبعلی کامیابی گل دانشگاه فردوسی مشهد

دکتر محمود لشکری زاده بمی دانشگاه اصفهان

دکتر علیرضا مدقالچی دانشگاه خوارزمی

دکتر رسول نصراصفهانی دانشگاه صنعتی اصفهان

‌ کمیته علمی سمینار

دکتر رجبعلی کامیابی گل )دبیرسمینار( دانشگاه فردوسی مشهد دکتر حمیدرضا ابراهیمی ویشکی )دبیر علمی( دانشگاه فردوسی مشهد دکتر مجید اسحاقی گرجی دانشگاه سمنان دکتر غالمحسین اسالم زاده )نماینده انجمن ریاضی( دانشگاه شیراز دکتر فاطمه اسماعیل زاده دانشگاه آزاد بجنورد دکتر مرتضی اسمعیلی دانشگاه خوارزمی دکتر محمد اکبری تتکابنی دانشگاه شاهد دکتر سیدمسعود امینی )نماینده انجمن ریاضی( دانشگاه تربیت مدرس دکتر صدیقه باروط کوب دانشگاه بجنورد دکتر عبدالرسول پور عباس دانشگاه صنعتی امیرکبیر دکترنرگس توالیی دانشگاه دامغان دکترمحمد جانفدا دانشگاه فردوسی مشهد دکتر علی جباری شاهزاده محمدی دانشگاه شهید باهنرکرمان دکترحسین جوانشیری دانشگاه یزد دکترشیرین حجازیان دانشگاه فردوسی مشهد دکترکاظم حق نژاد آذر دانشگاه محقق اردبیلی دکتر علی اکبرخادم معبودی دانشگاه علوم پزشکی شهید بهشتی دکتر علیرضا خدامی دانشگاه صنعتی شاهرود دکترمحمدعلی دهقان دانشگاه ولی عصر )عج( رفسنجان دکتراصغررحیمی دانشگاه مراغه دکترحمیدرضا رحیمی دانشگاه آزاد تهران دکترمهدی رستمی دانشگاه صنعتی امیرکبیر دکترمحمدرمضان پور دانشگاه دامغان

دکترریحانه رئیسی طوسی دانشگاه فردوسی مشهد دکتر عباس سهله دانشگاه گیالن دکترمحمدصال مصلحیان دانشگاه فردوسی مشهد دکتراحمدصفاپور دانشگاه ولی عصر )عج( رفسنجان دکترعلی اکبرعارفی جمال دانشگاه حکیم سبزواری دکترعبدالعزیز عبداللهی دانشگاه شیراز دکترعطااهلل عسکری همت دانشگاه شهید باهنرکرمان دکتر علی غفاری دانشگاه سمنان دکترمعصومه فشندی دانشگاه فردوسی مشهد دکتر سیدعلی رضا کامل میرمصطفایی دانشگاه فردوسی مشهد دکتر علیرضا مدقالچی دانشگاه خوارزمی دکتر سیدمحمدصادق مصدق مدرس دانشگاه یزد دکتر سعید مقصودی دانشگاه زنجان دکترمجید میرزاوزیری دانشگاه فردوسی مشهد دکترحامد نجفی دانشگاه فردوسی مشهد دکترعبدالرسول نصراصفهانی دانشگاه صنعتی اصفهان دکترمهدی نعمتی دانشگاه صنعتی اصفهان دکتراسداهلل نیکنام دانشگاه فردوسی مشهد دکتراحدهراتی دانشگاه فردوسی مشهد

کمیته برگزارکننده سمینار

محمد جانفدا، ریحانه رئیسی طوسی، احمد عرفانیان مشیری نژاد، معصومه فشندی، فاطمه هلن قانع استادقاسمی، رجبعلی کامیابی گل، حمیدرضا ابراهیمی ویشکی

کادراجرایی سمینار

رضا احمدئی، زهرا امیری هفشجانی ، امین اهلل خسروی، فاطمه خسروی، معصومه زارع، عاطفه رحیمی، فاطمه عزیزی، رامین فرشچیان، ماهره مهماندوست.

حمایت کنندگان سمینار

دانشگاه فردوسی مشهد انجمن ریاضی ایران قطب علمی آنالیز روی ساختارهای جبری پارک علم وفناوری خراسان دانشگاه مهندسی فناوریهای نوین قوچان دانشگاه بجنورد قطب علمی مدل سازی ومحاسبات درسیستم های خطی وغیرخطی مرکزپژوهشی موجکها درسیستم های خطی وغیرخطی

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سخنرانان مدعو سمینار

دکتر نرگس توالیی )دانشگاه دامغان(

دکتر حسین جوانشیری )دانشگاه یزد(

دکتر رجبعلی کامیابی گل )دانشگاه فردوسی مشهد(

‌ شرکت کنندگان سمینار

ردیف نام خانوادگی نام دانشگاه محل خدمت – تحصیل 1 ابراهیمی ویشکی حمیدرضا دانشگاه فردوسی مشهد 2 احمدی جعفر دانشگاه فردوسی مشهد 3 احمدی گندمانی محمدحسین دانشگاه صنعتی شیراز 4 اختری فاطمه دانشگاه صنعتی اصفهان 5 اسحاقی گرجی مجید دانشگاه سمنان 6 اسفندانی مریم دانشگاه صنعتی اصفهان 7 اسالم زاده غالمحسین دانشگاه شیراز 8 اسماعیل زاده فاطمه دانشگاه آزاد بحنورد 9 اسمعیلی مرتضی دانشگاه خوارزمی 10 افخمی فرشاد دانشگاه فردوسی مشهد 11 اکبری تتکابنی محمد دانشگاه شاهد 12 امیری راما دانشگاه محقق اردبیلی 13 امیری هفشجانی زهرا دانشگاه فردوسی مشهد 14 امین خواه مژگان دانشگاه فردوسی مشهد 15 امینی سیدمسعود دانشگاه تربیت مدرس 16 ایلخانی زاده منش اسما دانشگاه ولی عصر )عج( رفسنجان 17 باروط کوب صدیقه دانشگاه بجنورد 18 باغانی امید دانشگاه حکیم سبزواری 19 باقری ثالث علی رضا دانشگاه قم 20 بدیع عابد مهناز دانشگاه سیستان وبلوچستان 21 بهارلویی یانچشمه سارا دانشگاه صنعتی اصفهان

22 پورعباس عبدالرسول دانشگاه صنعتی امیرکبیر 23 پورعبداهلل محمدعلی دانشگاه فردوسی مشهد 24 تخته فرخنده دانشگاه خوارزمی 25 توالیی نرگس دانشگاه دامغان 26 جانفدا محمد دانشگاه فردوسی مشهد 27 جباری علی دانشگاه شهیدباهنرکزمان 28 جعفری سیده سمیه دانشگاه صنعتی اصفهان 29 جالئی خادمی سیدیونس دانشگاه پیام نور مشهد 30 جمال زاده جواد دانشگاه سیستان بلوچستان 31 جوانشیری حسین دانشگاه یزد 32 جوکار سهیال دانشگاه قم 33 حجازیان شیرین دانشگاه فردوسی مشهد 34 حسینی شریف سیده زهرا دانشگاه صنعتی اصفهان 35 حسینیون سیدعلیرضا دانشگاه شهیدبهشتی 36 حق نژاد آذر کاظم دانشگاه محقق اردبیلی 37 خادم معبودی علی اکبر دانشگاه علوم پزشکی شهید بهشتی 38 خدامی علیرضا دانشگاه صنعتی شاهرود 39 خسروی امین اهلل دانشگاه فردوسی مشهد 40 خسروی فاطمه دانشگاه فردوسی مشهد 41 دریکوند تاج الدین دانشگاه فردوسی مشهد 42 دهقان محمدعلی دانشگاه ولی عصر )عج( رفسنجان 43 دهمرده سمیرا دانشگاه سیستان وبلوچستان 44 دیالی شریفه دانشگاه سیستان وبلوچستان 45 رابعی اردشیر دانشگاه رازی کرمانشاه 46 رازقندی عاطفه دانشگاه حکیم سبزواری 47 رجالی علی دانشگاه اصفهان 48 رجبعلی پور مهدی دانشگاه شهیدباهنرکرمان

49 رحمتی نصرآباد زهرا دانشگاه فردوسی مشهد 50 رحیم خانی پریسا دانشگاه الزهرا 51 رحیمی حمیدرضا دانشگاه آزاداسالمی واحد تهران مرکزی 52 رحیمی عاطفه دانشگاه فردوسی مشهد 53 رحیمی اصغر دانشگاه مراغه 54 رستمی علی دانشگاه آزاد اسالمی واحد نیشابور 55 رستمی مهدی دانشگاه امیرکبیر 56 رضایی علی اصغر دانشگاه کاشان 57 رمضان پور محمد دانشگاه دامغان 58 روحی افرا پلی فاطمه دانشگاه فردوسی مشهد 59 ریاضی عبدالحمید دانشگاه صنعتی امیرکبیر 60 رئیسی طوسی ریحانه دانشگاه فردوسی مشهد 61 زارع معصومه دانشگاه فردوسی مشهد 62 زبرجد ساناز دانشگاه یزد 63 سعادتمندان جواد دانشگاه قم 64 سلطانی سروستانی فاطمه دانشگاه فردوسی مشهد 65 سهله عباس دانشگاه گیالن 66 سهیلی علیرضا دانشگاه فردوسی مشهد 67 سیالنی سنا دانشگاه آزاد اسالمی واحد تهران شمال 68 سیمی مرضیه دانشگاه یزد 69 شاکری چناری منانه دانشگاه آزاد اسالمی واحد تهران مرکز 70 شمس آبادی میترا دانشگاه حکیم سبزواری 71 شیردل فهیمه دانشگاه فردوسی مشهد 72 شیواک صدیقه دانشگاه گلستان 73 صادقی قدیر دانشگاه حکیم سبزواری 74 صادقی محمود دانشگاه هرمزگان 75 صال مصلحیان محمد دانشگاه فردوسی مشهد

76 صفاپور احمد دانشگاه ولی عصر )عج( رفسنجان 77 صالح زهی نسیمه دانشگاه والیت ایرانشهر 78 ضارب نیا محمد دانشگاه محقق اردبیلی 79 طالعی یونس دانشگاه تبریز 80 طباطبایی سیدمحمد دانشگاه قم 81 عارفی جمال علی اکبر دانشگاه حکیم سبزواری 82 عالی پوراحمدی نیلوفر دانشگاه هرمزگان 83 عبدالهی عبدالعزیز دانشگاه شیراز 84 عربیانی نیشابوری فهیمه دانشگاه نیشابور 85 عرفانیان مشیری نژاد احمد دانشگاه فردوسی مشهد 86 عسکری همت عطااهلل دانشگاه شهیدباهنرکرمان 87 علی نژاد احمد دانشگاه تهران 88 علیزاده نظرکندی حسین دانشگاه آزاد مرند 89 غفاری علی دانشگاه سمنان 90 غالمی نفیسه دانشگاه فردوسی مشهد 91 غالمی چهکند حسین دانشگاه فردوسی مشهد 92 فرشچیان رامین دانشگاه فردوسی مشهد 93 فشندی معصومه دانشگاه فردوسی مشهد 94 فعال رامین دانشگاه فردوسی مشهد 95 قاسمی احمد شرکت بهره برداری قطارشهری 96 قاسمی هنری طاهر دانشگاه خوارزمی 97 قانع استادقاسمی فاطمه هلن دانشگاه فردوسی مشهد 98 قنبری رضا دانشگاه فردوسی مشهد 99 قومنجانی فاطمه مرکز آموزش عالی کاشمر 100 کامل میرمصطفایی علیرضا دانشگاه فردوسی مشهد 101 کامیابی گل رجبعلی دانشگاه فردوسی مشهد 102 کشاورزیان جاثیه دانشگاه شاهد

103 کالته بجدی زهرا دانشگاه کرمان 104 گل مرادی مسعود خانه ریاضیات اردبیل 105 لشکری زاده بمی محمود دانشگاه اصفهان 106 لکزیان حسین دانشگاه فردوسی مشهد 107 لنگری ملیحه دانشگاه آزاد بجنورد 108 محمدزاده سمیه دانشگاه بجنورد 109 محمدیان نرگس دانشگاه دریانوردی وعلوم دریایی چابهار 110 مختاری امیرحسین دانشگاه فنی مهندسی فردوس 111 مدقالچی علیرضا دانشگاه خوارزمی 112 مرشدی وحیدرضا دانشگاه فردوسی مشهد 113 مصدق مدرس سیدمحمدصادق دانشگاه یزد 114 مصطفوی سمانه مرکزآموزش عالی اقلید 115 مصطفی زاده بهناز دانشگاه گلستان 116 معیری زهرا دانشگاه سیستان و بلوچستان 117 مقصودی سعید دانشگاه زنجان 118 منصوری شریفه دانشگاه سیستان وبلوچستان 119 مهماندوست ماهره دانشگاه فردوسی مشهد 120 موحد سیما دانشگاه والیت 121 موسویان خطیر سیدربیع دانشگاه مازندران 122 میرزاوزیری مجید دانشگاه فردوسی مشهد 123 میرزایی رئوف دانشگاه رازی کرمانشاه 124 ناصحی مهری دانشگاه صنعتی اصفهان 125 نجارپیشه لیال دانشگاه گیالن 126 نجفی حامد دانشگاه فردوسی مشهد 127 نصراصفهانی رسول دانشگاه صنعتی اصفهان 128 نظریان پور مهدی دانشگاه حکیم سبزواری 129 نعمتی مهدی دانشگاه صنعتی اصفهان

برنامه کارگاه متلب

130 نیازی محسن دانشگاه بیرجند 131 نیازی مطلق ابولفضل دانشگاه بجنورد 132 نیکبخت نغمه دانشگاه آزادبجنورد 133 نیکنام اسداهلل دانشگاه فردوسی مشهد 134 هراتی احد دانشگاه فردوسی مشهد 135 ولی زاده فهیمه دانشگاه آزادتهران مرکزی ‌

ﻧﻤﺎﯾﻪ ﻧﻮﯾﺴﻨﺪﮔﺎن

ج. ﺟﻤﺎل زاده، ٧، ٢٢ ز. ﻣﻌﯿﺮی، ٢٢ ش. دﯾﺎﻟﯽ، ١١ ش. ﻣﻨﺼﻮری، ١۴، ١٨ غ. رﺿﺎﯾﯽ، ١١ م. ﺑﺪﯾﻊ ﻋﺎﺑﺪ، ٢

س. دﻫﻤﺮده، ٧

١ ﭘﻮﺳﺘﺮﻫﺎ آﻧﺘﺮوﭘﯽ روی ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژی و ﯾﮑﻨﻮاﺧﺖ

ﻣﻬﻨﺎز ﺑﺪﯾﻊ ﻋﺎﺑﺪ

[email protected] ﮔﺮوه رﯾﺎﺿﯽ، داﻧﺸﮑﺪه ﻋﻠﻮم رﯾﺎﺿﯽ، داﻧﺸﮕﺎه ﺳﯿﺴﺘﺎن و ﺑﻠﻮﭼﺴﺘﺎن

ﭼﮑﯿﺪه. در ﺳﺎل ١٩۶۵ آدﻟﺮ و ﻫﻤﮑﺎران آﻧﺘﺮوﭘﯽ ﺗﻮﭘﻮﻟﻮژی را روی ﻧﮕﺎﺷﺖ ﺧﻮدرﯾﺨﺘﯽ ﭘﯿﻮﺳﺘﻪ از ﯾﮏ ﻓﻀﺎی ﻓﺸﺮده ﺗﻌﺮﯾﻒﮐﺮده اﻧﺪ. در ﺳﺎل ١٩٧١ اﯾﻦ ﻧﻈﺮﯾﻪ رویﻧﮕﺎﺷﺖ ﻫﺎی ﭘﯿﻮﺳﺘﻪ ﯾﮑﻨﻮاﺧﺖ ﮔﺴﺘﺮش داده ﺷﺪ. اﯾﻦ دﯾﺪﮔﺎه را روی ﻓﻀﺎﻫﺎی ﯾﮑﻨﻮاﺧﺖ وﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ ﻧﯿﺰﻣﯽ ﺗﻮان ﺗﻌﺮﯾﻒ ﮐﺮد. در ﺳﺎل ١٩٨١ ﭘﺘﺮز ﭘﯿﺸﻨﻬﺎد ﮐﺎﻣﻼ ﻣﺘﻔﺎوﺗﯽ از ﺗﺎﺑﻊ آﻧﺘﺮوﭘﯽ(ﺟﺒﺮی) ﺑﺮای ﺧﻮد رﯾﺨﺘﯽ ﻫﺎی ﭘﯿﻮﺳﺘﻪ روی ﻓﻀﺎﻫﺎی ﻣﻮﺿﻌﺎ ﻓﺸﺮده را اراﺋﻪ داد. ﻫﺪف ﮐﻠﯽ از اﯾﻦ ﻣﻘﺎﻟﻪ ﻣﻄﺮح ﮐﺮدن ﺑﺮﺧﯽ از ﻣﻔﺎﻫﯿﻢ و ﺧﻮاص آﻧﻬﺎ در ﺗﻮﺻﯿﻒ رواﺑﻂ ﻣﯿﺎن آﻧﺘﺮوﭘﯽ ﻫﺎی ﻣﺨﺘﻠﻒ و ﺗﺼﺤﯿﺢ ﺑﺮﺧﯽ از ﺧﻄﺎﻫﺎی ﻇﺎﻫﺮ ﺷﺪه درﻧﻮﺷﺘﻪ ﻫﺎ ﻣﯽ ﺑﺎﺷﺪ.

١. ﻣﻘﺪﻣﻪ در ﻧﮕﺎه اول اﯾﻦ ﻣﻔﻬﻮم ﻣﺘﻌﻠﻖ ﺑﻪ ﻧﻈﺮﯾﻪ ﺗﺮﻣﻮدﯾﻨﺎﻣﯿﮏ در ﻧﯿﻤﻪ اول ﻗﺮن XIX ﺑﻮده اﺳﺖ. در ﺳﺎل ١٩٣٠ ﺷﺎﻧﻮون ﻧﻈﺮﯾﻪ را ﺑﻬﺒﻮد ﺑﺨﺸﯿﺪ و ﯾﮏ ﻧﻈﺮﯾﻪ از آﻧﺘﺮوﭘﯽ را ﺑﺮای اﯾﻨﮑﻪ ﺗﻐﯿﯿﺮات اﯾﺠﺎد ﺷﺪه ﯾﮏ اﺑﺰار ﻣﻔﯿﺪ در ﻧﻈﺮﯾﻪ اﻃﻼﻋﺎت ﺗﻌﺮﯾﻒ ﮐﺮد، آﻧﺘﺮوﭘﯽ اﻧﺪازه ﺗﻮﺳﻂ ﮐﻮﻟﻤﻮﮔﺮو و ﺳﯿﻨﺎﯾﯽ ﺗﻌﺮﯾﻒ ﺷﺪ ﮐﻪ ﻧﻘﺶ اﺳﺎﺳﯽ در ﻓﺮﺿﯿﻪ ارﮔﻮدﯾﮏ درﺳﯿﺴﺘﻢ ﻫﺎی دﯾﻨﺎﻣﯿﮑﯽ ﺑﻌﺪ از ١٩۵٠ داﺷﺘﻪ اﺳﺖ. در ﺳﺎل ١٩۵٠ آدﻟﺮ و ﻫﻤﮑﺎران آﻧﺘﺮوﭘﯽ ﺗﻮﭘﻮﻟﻮژی روی ﻧﮕﺎﺷﺖ ﺧﻮد رﯾﺨﺘﯽ ﭘﯿﻮﺳﺘﻪ از ﯾﮏ ﻓﻀﺎی ﻓﺸﺮده ﺗﻌﺮﯾﻒﮐﺮده اﻧﺪ. در اﯾﻦ ﻧﻮﺷﺘﻪ، ﻣﺎ آﻧﺘﺮوﭘﯽ ﺗﻮاﺑﻊ ﭘﯿﻮﺳﺘﻪ را ﺑﺮای ﺣﺎﻟﺖ ﻓﺸﺮده و ﻓﻀﺎی ﯾﮑﻨﻮاﺧﺖ ﺗﻌﺮﯾﻒ ﮐﺮده و ﻣﻮرد ﺑﺮرﺳﯽ ﻗﺮارﻣﯽ دﻫﯿﻢ.

2010 Mathematics Subject Classification. Primary 99X99, Secondary 99Y99. واژﮔﺎن ﮐﻠﯿﺪی. آﻧﺘﺮوﭘﯽ ﺗﻮﭘﻮﻟﻮژی، آﻧﺘﺮوﭘﯽ ﯾﮑﻨﻮاﺧﺖ، ﺑﻮون آﻧﺘﺮوﭘﯽ. .

٣ م. ﺑﺪﯾﻊ ﻋﺎﺑﺪ

٢. آﻧﺘﺮوﭘﯽ ﺗﻮﭘﻮﻟﻮژی در ﺗﻘﺎﺑﻞ ﺑﺎ آﻧﺘﺮوﭘﯽ ﯾﮑﻨﻮاﺧﺖ ٢ . ١. آﻧﺘﺮوﭘﯽ ﺗﻮﭘﻮﻟﻮژی. آﻧﺘﺮوﭘﯽ ﺗﻮﭘﻮﻟﻮژی را در زﯾﺮ ﺗﻌﺮﯾﻒﻣﯽ ﮐﻨﯿﻢ. ﻓﺮض ﮐﻨﯿﺪ X ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﻓﺸﺮده ﺑﺎﺷﺪ { } (٢ . ١) U ﯾﮏ ﭘﻮﺷﺶ ﺑﺎزی از X اﺳﺖ : O(X) = U ﻗﺮاردادﻣﯽ ﮐﻨﯿﻢ ﮐﻪﭘﻮﺷﺶ ﻫﺎی ﺑﺎز ﺷﺎﻣﻞ ﺗﻬﯽ ﻧﯿﺰ ﺑﺎﺷﺪ. ﺑﺮای (U, V ∈ O(X در ﻧﻈﺮﻣﯽ ﮔﯿﺮﯾﻢ:

(٢ . ٢) {U ∨ V = {U ∩ V : U ∈ U,V ∈ V اﮔﺮ (U ∈ O(X، ﻓﺮض ﮐﻨﯿﺪ (N(U ﺑﺎ ﺗﻌﺪادی از ﻋﻨﺎﺻﺮ زﯾﺮﭘﻮﺷﺶ U ﺑﺎ ﺣﺪاﻗﻞ ﻋﺪد اﺻﻠﯽ ﻣﺸﺨﺺ ﺷﺪه ﺑﺎﺷﺪ. اﮔﺮ ϕ : X −→ X ﯾﮏ ﺧﻮد رﯾﺨﺘﯽ ﭘﯿﻮﺳﺘﻪ ﺑﺎﺷﺪ آﻧﮕﺎه آﻧﺘﺮوﭘﯽ ﺗﻮﭘﻮﻟﻮژی ϕ ﻧﺴﺒﺖ ﺑﻪ ﭘﻮﺷﺶ U را ﺗﻌﺮﯾﻒﻣﯽ ﮐﻨﯿﻢ

((U)١+U) · · · ∨ ϕ−n)٢−U) ∨ ϕ)١−logN(U ∨ ϕ h (ϕ, U) = lim (٢ . ٣) ∞→ T n { }n − − ﺑﻄﻮرﯾﮑﻪ (U)١ ϕ ﭘﺎﯾﻪ ای ﺑﺮای ﭘﻮﺷﺶ ﺑﺎز U): U ∈ U)١ ϕ ﺑﺎﺷﺪ. آﻧﺘﺮوﭘﯽ ﺗﻮﭘﻮﻟﻮژی ϕ ﺑﺼﻮرت

{ U U ∈ } (٢ . hT (ϕ) = sup hT (ϕ, ): O(x) (۴ ﺗﻌﺮﯾﻒﻣﯽ ﮐﻨﯿﻢ. ﻧﺘﺎﯾﺞ زﯾﺮ ﺑﺎ ﺗﻌﺮﯾﻒ ﺗﻮﭘﻮﻟﻮژی آﻧﺘﺮوﭘﯽ ﺑﺮای ﺗﻮاﺑﻊ ﺑﺪﺳﺖﻣﯽ آﯾﺪ. ﺑﺮای ﻫﺮ α : x −→ x ١− (٢ . hT (ϕ) = hT (α ϕα) (۵(١ ﺑﺮای ﯾﮏ ﻧﮕﺎﺷﺖ ﺧﻮد رﯾﺨﺘﯽ ﭘﯿﻮﺳﺘﻪ ϕ : X −→ X روی ﻓﻀﺎی ﻓﺸﺮده X، ﺑﺮای ﻫﺮ n ∈ N

n (hT (ϕ ) = n.hT (ϕ(٢ ١− (hT (ϕ ) = hT (ϕ(٣ ﺗﻌﺮﯾﻒﻣﯽ ﮐﻨﯿﻢ ∩ n (Eϕ(x) := ϕ (u(۴ دارﯾﻢ | hT (ϕ) = hT (ϕ Eϕ(X) )

٢ . ٢. آﻧﺘﺮوﭘﯽ ﯾﮑﻨﻮاﺧﺖ. آﻧﺘﺮوﭘﯽ ﺗﻮﭘﻮﻟﻮژی روی ﻓﻀﺎﻫﺎی ﻓﺸﺮده ﺗﻮﺳﻂ ﺑﻮون [٢] ﺑﺎ ﻧﮕﺎﺷﺖ ﺧﻮد رﯾﺨﺘﯽ ﭘﯿﻮﺳﺘﻪ ﯾﮑﻨﻮاﺧﺖ از ﯾﮏ ﻓﻀﺎی ﻣﺘﺮﯾﮏ (X, d) ﺑﺎ اﺳﺘﻔﺎده از ﯾﮑﻨﻮاﺧﺘﯽ ﻣﺘﺮی ﮔﺴﺘﺮش ﯾﺎﻓﺖ. ﻓﺮض ﮐﻨﯿﺪ (X, U) ﯾﮏ ﻓﻀﺎی ﯾﮑﻨﻮاﺧﺖ ﺑﺎﺷﺪ و ϕ : X −→ X ﯾﮏ ﻧﮕﺎﺷﺖ ﺧﻮد رﯾﺨﺘﯽ ﭘﯿﻮﺳﺘﻪ ﯾﮑﻨﻮاﺧﺖ ﺑﺎﺷﺪ ﻣﺠﻤﻮﻋﻪ

(٢ . K} (۶زﯾﺮﻣﺠﻤﻮﻋﻪ ﻓﺸﺮده از X اﺳﺖ. : K(X) = {K ﺑﻪ زﯾﺮ ﻣﺠﻤﻮﻋﻪ F از n, V ) ،X)- ﺗﻮﻟﯿﺪ ﮐﻨﻨﺪه ﺑﺮای ﯾﮏ ﻣﺠﻤﻮﻋﻪ ﻓﺸﺮده (K ∈ K(X ﻧﺎﻣﯿﺪهﻣﯽ ﺷﻮد، اﮔﺮ ﺑﺮای ﻫﺮ x ∈ K وﺟﻮد داﺷﺘﻪ ﺑﺎﺷﺪ y ﺑﻄﻮرﯾﮑﻪ ﺑﺮای ﻫﺮ j ⩽ n ⩾ ٠

j j (٢ . ٧) ϕ (x), ϕ (y)) ∈ V) ﺑﻪ زﯾﺮ ﻣﺠﻤﻮﻋﻪ F از r, V ) ،X)-ﻣﺠﺰا ﺑﺮای ϕ ﮔﻔﺘﻪﻣﯽ ﺷﻮد، اﮔﺮ ﺑﺮای ﻫﺮ زوج ﻧﻘﺎط ﻣﺘﻤﺎﯾﺰ x, y ∈ F وﺟﻮد داﺷﺘﻪ ﺑﺎﺷﺪ jای ﺑﻄﻮرﯾﮑﻪ j < n ⩾ ٠ و

۴ آﺗﺮوﭘﯽ روی ﻓﻀﺎﻫﺎی ﯾﮑﻨﻮاﺧﺖ

j j (٢ . ٨) ϕ (x), ϕ (y)) ̸∈ V) ﺑﺮای ﻫﺮ k ∈ K(x) ،V ∈ U و n ∈ N ﺗﻌﺮﯾﻒﻣﯽ ﮐﻨﯿﻢ

{| | } n, V )،F) -ﺗﻮﻟﯿﺪ ﮐﻨﻨﺪه K ﺑﺎ راﺑﻄﻪ rn(V, K, ϕ) = min F ; ϕ {| | ⊆ } F K و F ﯾﮏ ( n, V) -ﻣﺠﺰا ﺑﺎ راﺑﻄﮥ sn(V, K, ϕ) = max F ; ϕ V V اﻋﺪاد (rn( , K, ϕو (sn( , K, ϕ ﺗﻌﺮﯾﻒ ﺷﺪه و ﻣﺘﻨﺎﻫﯽ اﻧﺪ ﭼﻨﺎﻧﮑﻪ K ﻓﺸﺮده اﺳﺖ.

ﮔﺰاره ٢ . ١. ﻓﺮض ﮐﻨﯿﺪ ϕ ﯾﮏ ﻧﮕﺎﺷﺖ ﺧﻮد رﯾﺨﺘﯽ ﭘﯿﻮﺳﺘﻪ ﯾﮑﻨﻮاﺧﺖ روی ﻓﻀﺎی ﯾﮑﻨﻮاﺧﺖ (X, U) ﺑﺎﺷﺪ . اﮔﺮ (K ∈ K(X و W, V ∈ U ﺑﺎ W ⊆ V آﻧﮕﺎه

⩽ ⩾ ⩽ ⩾ ٠ ii)rn(V, K, ϕ) sn(W, K, ϕ), n ٠ i)σn(V, K, ϕ) σn(W, K, ϕ), n ∈ N ∈ K ﺑﻄﻮرﯾﮑﻪ σn ﯾﮑﯽ از sn و rn ﻣﯽ ﺑﺎﺷﺪ. (K (X را در ﻧﻈﺮ ﺑﮕﯿﺮﯾﺪ n و ﻗﺮار دﻫﯿﺪ σ = r, s، ﺗﻌﺮﯾﻒ ﮐﻨﯿﺪ

logσn(V, K, ϕ) σ (V, K, ϕ) = lim →∞sup (٢ . ٩) n n n ﺑﺮای ﻫﺮ V ∈ U. ﺑﻌﻼوه ﻓﺮض ﮐﻨﯿﺪ

{ ∈ U} (٢ . ١٠) hσ(k, ϕ) = sup σn(V, K, ϕ): V ﮔﺰاره ٢ . ٢. ﻓﺮض ﮐﻨﯿﺪ (X, U) ﯾﮏ ﻓﻀﺎی ﯾﮑﻨﻮاﺧﺖ و ϕ : X −→ X ﯾﮏ ﻧﮕﺎﺷﺖ ﺧﻮد رﯾﺨﺘﯽ ﭘﯿﻮﺳﺘﻪ ﯾﮑﻨﻮاﺧﺖ ﺑﺎﺷﺪ . اﮔﺮ (K ∈ K(x،

(٢ . ١١) (hr(K, ϕ) = hs(K, ϕ ﺣﺎل، ﺗﻌﺮﯾﻒ ﮐﻨﯿﺪ

{ ∈ K } { ∈ K } (٢ . ١٢) (hU (ϕ) = sup hr(X, ϕ): K (X) = sup hs(X, ϕ): K (X ٢ . ٣. آﻧﺘﺮوﭘﯽ ﯾﮑﻨﻮاﺧﺖ در ﺗﻘﺎﺑﻞ آﻧﺘﺮوﭘﯽ ﺗﻮﭘﻮﻟﻮژی. ﻓﻀﺎی ﯾﮑﻨﻮاﺧﺖ (X, U) را روی ﭘﻮﺷﺶ ﻫﺎی ﯾﮑﻨﻮاﺧﺖ {CU = {C(V ): V ∈ U در ﻧﻈﺮ ﺑﮕﯿﺮﯾﺪ ﭼﻨﺎﻧﮑﻪ

C(V ) = {V (x): x ∈ X} ,V ∈ U; ﺑﻄﻮرﯾﮑﻪ

V (x) = {y :(x, y) ∈ V }

ﺗﻌﺮﯾﻒ٢ . ٣. ﻓﺮض ﮐﻨﯿﺪ (X, U) ﯾﮏ ﻓﻀﺎی ﯾﮑﻨﻮاﺧﺖ ﺑﺎﺷﺪ. اﮔﺮ A ﯾﮏ ﭘﻮﺷﺶ ﯾﮑﻨﻮاﺧﺖ از X ﺑﺎﺷﺪ و X ﯾﮏ زﯾﺮﻣﺠﻤﻮﻋﻪ ﻓﺸﺮده، ﺗﻌﺮﯾﻒ ﮐﻨﯿﺪ

{| | ⊂ ⊆ ∪ } (٢ . ١٣) N(K,A) = min BK : BK A, K BK ﻓﺸﺮدﮔﯽ K ﺗﻀﻤﯿﻦﻣﯽ ﮐﻨﺪ ﮐﻪ ﻋﺪد (N(K,A ﺗﻌﺮﯾﻒ ﺷﺪه و ﻣﺘﻨﺎﻫﯽ اﺳﺖ.

۵ م. ﺑﺪﯾﻊ ﻋﺎﺑﺪ

ﺗﻌﺮﯾﻒ٢ . ۴. ﻓﺮض ﮐﻨﯿﺪ (X, U) ﯾﮏ ﻓﻀﺎی ﯾﮑﻨﻮاﺧﺖ و ϕ : X −→ X ﻧﮕﺎﺷﺖ ﺧﻮد رﯾﺨﺘﯽ ﭘﯿﻮﺳﺘﻪ ﯾﮑﻨﻮاﺧﺖ ﺑﺎﺷﺪ ﺑﺮای ﻫﺮ زﯾﺮ ﻣﺠﻤﻮﻋﻪ ﻓﺸﺮده K از X و ﻫﺮ V ∈ U، ﺗﻌﺮﯾﻒ ﮐﻨﯿﺪ ( ) (( ١ϕ−i(C(V−C (V, K, ϕ) = N K,V n (٢ . ١۴) ٠=n i ﻟﻢ٢ . ۵. ﻓﺮض ﮐﻨﯿﺪ (X, U) ﯾﮏ ﻓﻀﺎی ﯾﮑﻨﻮاﺧﺖ و ϕ : X −→ X ﯾﮏ ﺧﻮد رﯾﺨﺘﯽ ﭘﯿﻮﺳﺘﻪ ﯾﮑﻨﻮاﺧﺖ ﺑﺎﺷﺪ. اﮔﺮ K ∈ K(X) ،V ∈ U و n ∈ N ﺑﺮای ﻫﺮ W ∈ U و W oW ⊆ V دارﯾﻢ

i)sn(V, K, ϕ) ⩽ cn(W, K, ϕ) ii)cn(V, K, ϕ) ⩽ rn(W, K, ϕ)

ﺗﻌﺮﯾﻒ٢ . ۶. ﻓﺮض ﮐﻨﯿﺪ (X, U) ﯾﮏ ﻓﻀﺎی ﯾﮑﻨﻮاﺧﺖ و ϕ : X −→ X ﯾﮏ ﺧﻮد رﯾﺨﺘﯽ ﭘﯿﻮﺳﺘﻪ ﯾﮑﻨﻮاﺧﺖ ﺑﺎﺷﺪ. ﺑﺮای ﻫﺮ (V ∈ U ، K ∈ K(X و ﺑﺮای ﻫﺮ ٠ ⩽ n ﺗﻌﺮﯾﻒ ﮐﻨﯿﺪ

logCn(V, K, ϕ) C (V, K, ϕ) = lim →∞sup (٢ . ١۵) n n n ﮔﺰاره٢ . ٧. ﻓﺮض ﮐﻨﯿﺪ (X, U) ﯾﮏ ﻓﻀﺎی ﯾﮑﻨﻮاﺧﺖ ﺑﺎﺷﺪ و ϕ : X −→ X ﯾﮏ ﺧﻮد رﯾﺨﺘﯽ ﭘﯿﻮﺳﺘﻪ ﯾﮑﻨﻮاﺧﺖ ﺑﺎﺷﺪ ﺑﺮای ﻫﺮ (V,W ∈ U ،K ∈ K(K و W oW ⊆ V آﻧﮕﺎه

(٢ . ١۶) (i)s(V, K, ϕ) ⩽ c(W, K, ϕ و ﺑﺮای ﻫﺮ V ∈ U

(٢ . ١٧) (ii)c(V, K, ϕ) ⩽ r(W, K, ϕ ∈ R ∪ {∞} ﮔﺰاره ﻗﺒﻞ ﺑﻪ ﻣﺎ اﺟﺎزهﻣﯽ دﻫﺪ ﺗﺎ ﻣﻔﻬﻮم ٠⩽ (hUC (ϕ ﺗﻌﺮﯾﻒ ﺷﺪه ﺑﻪ ﺻﻮرت

hUC (ϕ) = sup {hUC (K, ϕ): K ∈ K(X)} ﺑﺮرﺳﯽ ﮐﻨﯿﻢ ﺑﻪ ﻃﻮری ﮐﻪ

hUC (K, ϕ) = sup {c(V, K, ϕ): V ∈ U}

ﻗﻀﯿﻪ٢ . ٨. اﮔﺮ ϕ : X −→ Xﯾﮏ ﺧﻮد رﯾﺨﺘﯽ ﭘﯿﻮﺳﺘﻪ ﯾﮑﻨﻮاﺧﺖ از ﻓﻀﺎی (X, U) ﺑﺎﺷﺪ، آﻧﮕﺎه

(٢ . ١٨) (hU (ϕ) = hUC (ϕ

ﻣﺮاﺟﻊ 1. R.L. Adler, A.G. Konheim, M.H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965) 309–319.

2. R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971) 401–414.

3. D. Dikranjan, A. Giordano Bruno, Entropy in a module category, Appl. Categ. Structures (2011), in press, doi:10.1007/s10485-011-9256-1.

4. R. Engelking, General Topology, PWN – Polish Sci. Publ., Warszawa, 1977.

۶ ﻧﻮع ﺟﺪﯾﺪ ﮐﺎﻣﻞ ﺳﺎزی در ﻓﻀﺎی ﻣﺘﺮﯾﮏ

∗ ﺳﻤﯿﺮا دﻫﻤﺮده ١ و ﺟﻮاد ﺟﻤﺎﻟﺰاده٢

١ آدرس ١ [email protected] ٢ آدرس ٢ [email protected] ٣ ﮔﺮوه رﯾﺎﺿﯽ، داﻧﺸﮑﺪه ﻋﻠﻮم رﯾﺎﺿﯽ، داﻧﺸﮕﺎه ﺳﯿﺴﺘﺎن و ﺑﻠﻮﭼﺴﺘﺎن

ﭼﮑﯿﺪه. اﯾﻦ ﻣﻘﺎﻟﻪ ﺑﻪ ﻣﻌﺮﻓﯽ و ﻣﻄﺎﻟﻌﻪ دو ﻧﻤﻮﻧﻪ از ﺧﻮاص ﺟﺪﯾﺪ ﮐﺎﻣﻞ ﺑﻮدن در ﻓﻀﺎی ﻣﺘﺮﯾﮏﻣﯽ ﭘﺮدازد ﮐﻪ ﻣﺎ آﻧﻬﺎ را ﺑﻮرﺑﺎﮐﯽ-ﮐﺎﻣﻞ و ﺑﻮرﺑﺎﮐﯽ-ﮐﺎﻣﻞﻫﻢ ﭘﺎﯾﺎنﻣﯽ ﻧﺎﻣﯿﻢ. ﻫﻢ ﭼﻨﯿﻦ ﺑﻪ ﻣﻄﺎﻟﻌﻪ ﻣﺴﺎﺋﻞ ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﻣﺘﺮﯾﮏ ﭘﺬﯾﺮ ﺑﺎ اﺳﺘﻔﺎده از ﺑﻮرﺑﺎﮐﯽ-ﮐﺎﻣﻞ ﯾﺎ ﺑﻮرﺑﺎﮐﯽ-ﮐﺎﻣﻞﻫﻢ ﭘﺎﯾﺎنﻣﯽ ﭘﺮدازﯾﻢ و ﺷﺮاﯾﻂ ﻣﻌﺎدل ﺑﻮرﺑﺎﮐﯽ-ﮐﻮﺷﯽ ﺑﻮدن را ﻣﺎﻧﻨﺪ ﮐﻮﺷﯽ ﺑﻮدن ﺑﺮایدﻧﺒﺎﻟﻪ ﻫﺎ ﺑﺮرﺳﯽﻣﯽ ﮐﻨﯿﻢ.

١. ﻣﻘﺪﻣﻪ ﺑﺮﺧﯽ از ﮐﻼﺳﻬﺎی ﻓﻀﺎﻫﺎی ﻣﺘﺮﯾﮏ ﺧﻮاﺻﯽ ﻗﻮﯾﺘﺮ از ﮐﺎﻣﻞ ﺑﻮدن و ﺿﻌﯿﻔﺘﺮ از ﻓﺸﺮدﮔﯽ دارﻧﺪ ﮐﻪ اﺧﯿﺮا ﺗﻮﺳﻂ ﺑﺴﯿﺎری از ﻧﻮﯾﺴﻨﺪﮔﺎن ﻣﻮرد ﻣﻄﺎﻟﻌﻪ ﻗﺮار ﮔﺮﻓﺘﻪ، ﯾﮏ ﻣﺮﺟﻊ ﺧﻮب ﺑﺮای اﯾﻦ ﻧﻮع ﻣﻘﺎﻟﻪ ﺑﯿﯿﺮ [١] ﺗﺤﺖ ﻋﻨﻮان راﺑﻂ ﺑﯿﻦ ﻓﺸﺮدﮔﯽ و ﮐﺎﻣﻞ ﺑﻮدنﻣﯽ ﺑﺎﺷﺪ. ﻣﺜﺎﻟﯽ از اﯾﻦ ﺧﻮاص ﻓﺸﺮدﮔﯽ ﮐﺮاﻧﺪار، ﻓﺸﺮدﮔﯽ ﻣﻮﺿﻌﺎ ﯾﮑﻨﻮاﺧﺖﮐﺎﻣﻞ ﺳﺎزیﻫﻢ ﭘﺎﯾﺎن، ﮐﺎﻣﻞ ﺑﻮدنﻫﻢ ﭘﺎﯾﺎن ﻗﻮی ﮐﻪ اﺧﯿﺮا ﺗﻮﺳﻂ ﺑﯿﯿﺮ در [٣] ﻣﻌﺮﻓﯽ ﺷﺪه ﻫﻤﭽﻨﯿﻦ ﺑﺮای ﻓﻀﺎی ﻣﺘﺮﯾﮏ UC-ﭘﺬﯾﺮ ﻧﺎﻣﯿﺪهﻣﯽ ﺷﻮد. ﻣﻄﺎﻟﻌﻪ ﻫﻤﻪ اﯾﻦ ﻓﻀﺎﻫﺎ ﺗﻤﺎم آن ﭼﯿﺰی را ﮐﻪﻣﯽ ﺧﻮاﻫﯿﻢ ﺑﻪ ﻣﺎﻧﻤﯽ دﻫﺪ اﻣﺎ ارﺗﺒﺎﻃﯽ ﺑﺎ ﺑﻌﻀﯽ ﻣﺴﺎﺋﻞ در آﻧﺎﻟﯿﺰ ﻣﺤﺪب، ﻗﻀﺎﯾﺎیﺑﻬﯿﻨﻪ ﺳﺎزی و در زﻣﯿﻨﻪ ﺳﺎﺧﺘﺎر ﻫﻤﮕﺮاﯾﯽ روی اﺑﺮﻓﻀﺎﻫﺎ راﺑﻪ وﺟﻮدﻣﯽ آورد. ﻫﺪف ﮐﻠﯽ اﯾﻦ ﻣﻘﺎﻟﻪ ﻣﻌﺮﻓﯽدوﺗﺎﯾﯽ ﻫﺎی ﺟﺪﯾﺪ اﯾﻦ ﺧﻮاص ﻣﺘﻮﺳﻂ در زﻣﯿﻨﻪ ﻓﻀﺎی ﻣﺘﺮﯾﮏ اﺳﺖ. اول از ﻫﻤﻪ ﺗﻮﺟﻪ داﺷﺘﻪ ﺑﺎﺷﯿﺪ ﮐﻪ ﯾﮏ راه ﺑﺮای رﺳﯿﺪن ﺑﻪ ﯾﮏ وﯾﮋﮔﯽ ﻗﻮی ﺗﺮ از ﮐﺎﻣﻞ ﺑﻮدن ﺑﺮای ﻓﻀﺎی ﻣﺘﺮﯾﮏ ﺧﻮﺷﻪ ای از ﻫﻤﻪ دﻧﺒﺎﻟﻪ ﻫﺎی ﻣﺘﻌﻠﻖ ﺑﻪ ﺑﺮﺧﯽ ﮐﻼﺳﻬﺎی ﺑﺰرﮔﺘﺮ از ﮐﻼﺳﻬﺎی دﻧﺒﺎﻟﻪ ﻫﺎی ﮐﻮﺷﯽ ﻣﯽ ﺑﺎﺷﺪ.ﺑﺪﯾﻦ ﺗﺮﺗﯿﺐ در ﺑﺨﺶ ٣ﮐﻼس ﻫﺎﯾﯽ از دﻧﺒﺎﻟﻪ ﺑﻮرﺑﺎﮐﯽ-ﮐﻮﺷﯽ و دﻧﺒﺎﻟﻪ ﺑﻮرﺑﺎﮐﯽ ﮐﻮﺷﯽ ﻫﻢ ﭘﺎﯾﺎن ﺗﻌﺮﯾﻒﻣﯽ ﮐﻨﯿﻢ. اﯾﻦ دﻧﺒﺎﻟﻪ در ﻓﻀﺎی ﻣﺘﺮﯾﮏ ﻇﺎﻫﺮﻣﯽ ﺷﻮد و ﺑﻪ اﺻﻄﻼح ﻣﺠﻤﻮﻋﻪ ﺑﻮرﺑﺎﮐﯽ-ﮐﺮاﻧﺪار ﻧﺎﻣﯿﺪهﻣﯽ ﺷﻮد. اﯾﻦ ﻧﻈﺮﯾﻪ ﮐﺮاﻧﺪاری ﺗﻮﺳﻂ آﺗﺴﻮﺟﯽ در [١] ﺑﻪ ﻣﻨﻈﻮر ﻧﺸﺎن دادن ﻓﻀﺎی ﻣﺘﺮﯾﮏ ﺑﻄﻮرﯾﮑﻪ ﻫﺮ ﺗﺎﺑﻊ ﭘﯿﻮﺳﺘﻪ ﯾﮑﻨﻮاﺧﺖ ﺣﻘﯿﻘﯽ ﮐﺮاﻧﺪار اﺳﺖ. ﻣﻌﺮﻓﯽ ﺷﺪه، اﻣﺎ ﻟﺰوﻣﺎ ﮐﻼ ﮐﺮاﻧﺪار ﻧﯿﺴﺖ. ﻧﺎم ﺑﻮرﺑﺎﮐﯽ-ﮐﺮاﻧﺪار از ﮐﺘﺎب ﺑﻮرﺑﺎﮐﯽ آﻣﺪه ﮐﻪ اﯾﻦ زﯾﺮﻣﺠﻤﻮﻋﻪ در ﻓﻀﺎی ﯾﮑﻨﻮاﺧﺖ ﻣﻄﺮح ﺷﺪه اﺳﺖ. در اﯾﻨﺠﺎ ﺛﺎﺑﺖﻣﯽ ﮐﻨﯿﻢ ﮐﻪزﯾﺮﻣﺠﻤﻮﻋﻪ ﻫﺎی ﺑﻮرﺑﺎﮐﯽ-ﮐﺮاﻧﺪار از ﻓﻀﺎی ﻣﺘﺮﯾﮏﻣﯽ ﺗﻮاﻧﺪ ﻣﺸﺨﺺ ﺷﻮد ﺑﻪ ﻓﺮمدﻧﺒﺎﻟﻪ ﻫﺎﯾﯽ ﮐﻪدﻧﺒﺎﻟﻪ ﻫﺎی ﮐﻮﺷﯽ را ﺑﺮای ﮐﻼ ﮐﺮاﻧﺪاری ﻣﺸﺨﺺﻣﯽ ﮐﻨﺪ.

2010 Mathematics Subject Classification. Primary 99X99, Secondary 99Y99. واژﮔﺎن ﮐﻠﯿﺪی. ﺑﻮرﺑﺎﮐﯽ-ﮐﻮﺷﯽ؛ ﺑﻮرﺑﺎﮐﯽ-ﮐﺮاﻧﺪاری؛ ﺑﻮرﺑﺎﮐﯽ-ﮐﺎﻣﻞ . ∗ ﺳﺨﻨﺮان ٧ س. دﻫﻤﺮده و ج. ﺟﻤﺎﻟﺰاده

ﺑﻨﺎﺑﺮاﯾﻦ ﻧﻮع ﺟﺪﯾﺪی ازدﻧﺒﺎﻟﻪ ﻫﺎ ﻇﺎﻫﺮﻣﯽ ﺷﻮد ﮐﻪ آﻧﺮا ﺑﻮرﺑﺎﮐﯽ-ﮐﻮﺷﯽﻣﯽ ﻧﺎﻣﯿﻢ. دﺳﺘﻪ ای دﯾﮕﺮ ازدﻧﺒﺎﻟﻪ ﻫﺎ را ﮐﻪﻫﻢ ﭘﺎﯾﺎن ﻫﺴﺘﻨﺪ را ﺗﻮﻟﯿﺪﻣﯽ ﮐﻨﯿﻢ. ﺑﻪ اﯾﻦ ﻣﻌﻨﺎ ﮐﻪﺑﺎﻗﯿﻤﺎﻧﺪه ای از ﺷﺎﺧﺺ ﺟﺎﯾﮕﺰﯾﻦ ﺗﻮﺳﻂﻫﻢ ﭘﺎﯾﺎﻧﯽ اﺳﺖ. ﺑﻨﺎﺑﺮاﯾﻦ آﻧﭽﻪ را ﮐﻪ دﻧﺒﺎﻟﻪ ﺑﻮرﺑﺎﮐﯽ-ﮐﺸﯽ ﻣﯽ ﻧﺎﻣﯿﻢ ﺑﺪﺳﺖ آوردﯾﻢ.

٢. ﻧﺘﺎﯾﺞ ﻣﻘﺪﻣﺎﺗﯽ ∈ ﺑﺮای ﻓﻀﺎی ﻣﺘﺮی (X, d)، ﻣﻌﻨﯽﻣﯽ ﮐﻨﯿﻢ (Bε(X، ﮔﻮیﺑﺎزی ﺳﺖ ﺑﻪ ﻣﺮﮐﺰ x X و ﺷﻌﺎع ٠ < ε. و ﺑﺮای ﻫﺮ زﯾﺮﻣﺠﻤﻮﻋﻪ A از X و ٠ < ε ﻣﯽ ﻧﻮﯾﺴﯿﻢ، ε-ﺑﺰرﮔﯽ از A ﺗﻮﺳﻂ، ε A = ∪{Bε(x).x ∈ A} = {y : d(y, A) < ε} t − در ﻃﻮل ﻣﻘﺎﻟﻪ زﯾﺮ دﻧﺒﺎﻟﻪ n ε enlargmeat از ﮔﻮی ﺑﺎز (Bε(x ﺑﻪ ﺻﻮرت زﯾﺮ ﺗﻌﺮﯾﻒ ﻣﯽ ﺷﻮد: ﺑﺮای ﻫﺮ ٢ ≤ n ١ Bε = Bε(x) ε ١−n n Bε := (Bε (x)) . در ﭘﺎﯾﺎن ﺗﻌﺮﯾﻒﻣﯽ ﮐﻨﯿﻢ ﻣﺆﻟﻔﻪ ε-زﻧﺠﯿﺮﭘﺬﯾﺮ (ε − cheinable conponeat) را در x ∈ X ﺑﻪ ﺻﻮرت، ∪ ∞ n Bε = Bε . n∈N ﺑﺨﺎﻃﺮ ﺑﯿﺎورﯾﺪ ﮐﻪ ﻓﻀﺎی ﻣﺘﺮﯾﮏ ﯾﮏ زﯾﺮ ﻣﺠﻤﻮﻋﻪ ﮐﺮاﻧﺪار (ﻣﺘﺮﯾﮏ) ﮔﻔﺘﻪﻣﯽ ﺷﻮد. اﮔﺮ ﻗﻄﺮ ﻣﺘﻨﺎﻫﯽ ﺑﺎﺷﺪ. ﺑﻪ اﯾﻦ ﻓﻀﺎ ﮐﻪ ﻣﺸﻤﻮل در ﯾﮏ ﮔﻮی ﺑﺎز ﺑﺎﺷﺪ. اﯾﻦ ﻣﻔﻬﻮم از ﮐﺮاﻧﺪاری ﮐﺎﻣﻼ ﻃﺒﯿﻌﯽ اﺳﺖ اﻣﺎ ﺑﺮﺧﯽ از آﻧﻬﺎ ﻧﺎﺳﺎزﮔﺎرﻧﺪ. ﺑﻪ ﻋﻨﻮان ﻣﺜﺎل ﻣﺘﺮ ﮐﺮاﻧﺪار، ﭘﺎﯾﺎﻧﯽ ﯾﮑﻨﻮاﺧﺖ ﻧﯿﺴﺖ. ﺑﻪ اﯾﻦ ﻣﻌﻨﺎ ﮐﻪ ﺗﺤﺖ ﯾﮑﻨﻮاﺧﺘﯽ ﻫﻤﺌﻮﻣﻮرﻓﯿﺴﻢ ﻣﺤﻔﻮظ ﻧﯿﺴﺖ. ﺑﻨﺎﺑﺮاﯾﻦ در زﻣﯿﻨﻪ ﻓﻀﺎﻫﺎی ﻣﺘﺮﯾﮏﻧﺮم دار، ﮐﺮاﻧﺪاری ﯾﮏ وﯾﮋﮔﯽ ﯾﮑﻨﻮاﺧﺘﯽ و در ﺣﻘﯿﻘﺖ زﯾﺮﻣﺠﻤﻮﻋﻪ ﻫﺎی ﮐﺮاﻧﺪاری اﺳﺖ ﮐﻪ ﺑﺎ اﺳﺘﻔﺎده از ﺗﻮاﺑﻊ ﭘﯿﻮﺳﺘﻪ ﯾﮑﻨﻮاﺧﺖ ﻣﺸﺨﺺﻣﯽ ﺷﻮد. در واﻗﻊ ﺑﻪ آﺳﺎﻧﯽﻣﯽ ﺑﯿﻨﯿﻢ ﮐﻪ زﯾﺮﻣﺠﻤﻮﻋﻪ B از ﻏﻀﺎیﻧﺮم دار X ﺗﻮﺳﻂ ﻧﺮم ﮐﺮاﻧﺪار اﺳﺖ اﮔﺮ و ﻓﻘﻂ اﮔﺮ ﺑﺮای ﻫﺮ ﺗﺎﺑﻊ ﺣﻘﯿﻘﯽ ﭘﯿﻮﺳﺘﻪ ﯾﮑﻨﻮاﺧﺖ f از f(B) ،X ﺑﺎ ﻣﺘﺮ ﻣﻌﻤﻮﻟﯽ در R ﮐﺮاﻧﺪار ﺑﺎﺷﺪ. ﻫﻤﺎﻧﻄﻮر ﮐﻪ در ﻣﻘﺪﻣﻪ ﮔﻔﺘﻪ ﺷﺪ، آﺗﺴﻮﺟﯽ در [١] ﻣﻔﻬﻮم ﺑﻮرﺑﺎﮐﯽ ﮐﺮاﻧﺪاری را در ﻓﻀﺎی ﻣﺘﺮی ﻣﻄﺮح ﮐﺮد (ﺗﺤﺖ ﻧﺎم ﺑﻪ ﻃﻮر ﻣﺘﻨﺎﻫﯽ زﻧﺠﯿﺮﭘﺬﯾﺮ ﻓﻀﺎی ﻣﺘﺮﯾﮏ) و ﻧﺸﺎن داد ﮐﻪ آﻧﻬﺎ ﻓﻘﻂ ﻓﻀﺎﻫﺎی ﻣﺘﺮﯾﮏ ﻫﺴﺘﻨﺪ ﮐﻪ ﺑﺮ روی آن ﺗﺼﻮﯾﺮ ﮐﺮاﻧﺪار ﺗﺤﺖ ﻫﺮ ﺗﺎﺑﻊ ﺣﻘﯿﻘﯽ ﭘﯿﻮﺳﺘﻪ ﯾﮑﻨﻮاﺧﺖ ﺗﻌﺮﯾﻒﻣﯽ ﺷﻮد. اﯾﻦ ﻓﻀﺎﻫﺎ در ﻗﺎﻟﺐ ﻓﻀﺎﻫﺎی ﯾﮑﻨﻮاﺧﺖ ﺗﻮﺳﻂ ﺣﺠﻤﻦ در [۵] ﻣﻄﺮح ﺷﺪه اﺳﺖ ﮐﻪ آﻧﻬﺎ را ﮐﺮاﻧﺪار ﻧﺎﻣﯿﺪه.

ﺗﻌﺮﯾﻒ ٢ . ١. زﯾﺮ ﻣﺠﻤﻮﻋﻪ B از ﻓﻀﺎی ﻣﺘﺮﯾﮏ (X, d)، زﯾﺮﻣﺠﻤﻮﻋﻪ ﺑﻮرﺑﺎﮐﯽ-ﮐﺮاﻧﺪار X ﻧﺎﻣﯿﺪه ﻣﯽ ﺷﻮد اﮔﺮ p , ··· , p ∈ X m ∈ N ε > ﺑﺮای ﻫﺮ ٠ ∪ وﺟﻮد داﺷﺘﻪ ﺑﺎﺷﺪ و ﯾﮏ ﻣﺠﻤﻮﻋﻪ ﻣﺘﻨﺎﻫﯽ از ﻧﻘﺎط k ١ ﺑﻪ ﻃﻮرﯾﮑﻪ B ⊂ k Bm(p ) ε i ١=i . ﺗﻮﺟﻪ داﺷﺘﻪ ﺑﺎﺷﯿﺪ ﮐﻪ ﺧﺎﻧﻮاده B از ﻫﻤﻪ زﯾﺮﻣﺠﻤﻮﻋﻪ ﻫﺎی ﺑﻮرﺑﺎﮐﯽ-ﮐﺮاﻧﺪار از X ﺑﻪ ﺷﮑﻞ ﺑﺮﻧﻮﻟﻮژی در X اﺳﺖ ﮐﻪ B در ﺷﺮاﯾﻂ زﯾﺮ اﺳﺖ (i) ﺑﺮای ﻫﺮ x ∈ X ﻣﺠﻤﻮﻋﻪ x} ∈ B} (ii) اﮔﺮ B ∈ B و A ⊆ B ﺑﻨﺎﺑﺮاﯾﻦ A ∈ B (iii) اﮔﺮ A, B ∈ B ﭘﺲ A ∪ B ∈ B ∈ ≤ از ﺳﻮﯾﯽ دﯾﮕﺮ اﮔﺮ در ﺗﻌﺮﯾﻒ ٢ . ١ ﺑﺎﻻ داﺷﺘﻪ ﺑﺎﺷﯿﻢ ١ = m (ﯾﺎ m m٠ ﺑﺮای ﻫﺮ N m٠) ﺑﺮای ﻫﺮ ٠ < ε ﺑﻨﺎﺑﺮاﯾﻦ ﻣﺎ ﻣﻔﻬﻮم ﮐﻼﺳﯿﮑﯽ اززﯾﺮﻣﺠﻤﻮﻋﻪ ﻫﺎی ﮐﻼ ﮐﺮاﻧﺪار درﯾﺎﻓﺖﻣﯽ ﮐﯿﻨﻢ. از اﯾﻨﺮو ﻫﺮ زﯾﺮﻣﺠﻤﻮﻋﻪ ﮐﻼ ﮐﺮاﻧﺪار در ﺑﻮرﺑﺎﮐﯽ-ﮐﺮاﻧﺪار ﺧﺎص اﺳﺖ و ﺑﻪ روﺷﻨﯽﻣﯽ ﺑﯿﻨﯿﻢ ﮐﻪ ﻫﺮ زﯾﺮﻣﺠﻤﻮﻋﻪ ﺑﻮرﺑﺎﮐﯽ ﮐﺮاﻧﺪار در ﻣﻔﻬﻮم ﻣﻌﻤﻮﻟﯽ ﮐﺮاﻧﺪار ﺧﺎص اﺳﺖ.

٨ ﻧﻮع ﺟﺪﯾﺪ ﮐﺎﻣﻠﺴﺎزی در ﻓﻀﺎی ﻣﺘﺮﯾﮏ

٣. ﻧﺘﺎﯾﺞ اﺻﻠﯽ ﺑﺮای اﻃﻼﻋﺎت ﺑﯿﺸﺘﺮ در ﻣﻮرد ارﺗﺒﺎط ﻣﯿﺎن اﯾﻦ ﺳﻪ ﻣﻔﻬﻮم ﻣﺘﻔﺎوت ﮐﺮاﻧﺪاری ﺑﻪ[۴] ﻣﺮاﺟﻌﻪﻣﯽ ﮐﻨﯿﻢ ﺑﻪ ﻃﻮری ﮐﻪ ﻫﻤﻪ آﻧﻬﺎ را ﺑﻪ ﻃﻮر ﮐﻠﯽ ﺑﺮﻧﻮﻟﻮژی ﺟﻬﺎﻧﯽﻣﯽ ﻧﺎﻣﯿﻢ. ﺗﻮﺟﻪ داﺷﺘﻪ ﺑﺎﺷﯿﺪ ﮐﻪ ﯾﮏ زﯾﺮﻣﺠﻤﻮﻋﻪ ﺑﻮرﺑﺎﮐﯽ-ﮐﺮاﻧﺪار وﯾﮋﮔﯽ ذاﺗﯽ ﻧﯿﺴﺖ ﯾﻌﻨﯽ ﺑﺴﺘﮕﯽ ﺑﻪ ﻓﻀﺎی ﻣﺤﯿﻄﯽ دارد ﮐﻪ در آن ﻫﺴﺘﯿﻢ. در ﺣﻘﯿﻘﺖ ﺧﻮاﻫﯿﻢ دﯾﺪ ﮐﻪ ﯾﮏ زﯾﺮﻣﺠﻤﻮﻋﻪ در X ﻣﯽ ﺗﻮاﻧﺪ ﺑﻮرﺑﺎﮐﯽ-ﮐﻮﺷﯽ ﺑﺎﺷﺪ اﻣﺎ ﻧﻪ ﺑﻪ ﺧﻮدی ﺧﻮد.

ﻗﻀﯿﻪ٣ . ١. ﻓﺮض ﮐﻨﯿﺪ (X, d) ﻓﻀﺎی ﻣﺘﺮﯾﮏ ﺑﺎﺷﺪ. ﻋﺒﺎرات زﯾﺮ ﻣﻌﺎدﻟﻨﺪ: (١) X ﯾﮏ ﻓﻀﺎی ﻣﺘﺮﯾﮏ ﺑﻮرﺑﺎﮐﯽ-ﮐﺮاﻧﺪار اﺳﺖ. (٢) ﺑﺮای ﻫﺮ ﺗﺎﺑﻊ ﺣﻘﯿﻘﯽ ﭘﯿﻮﺳﺘﻪ f از f(X) ،X در R ﮐﺮاﻧﺪار اﺳﺖ. ′ ′ (٣) ﺑﺮای ﻫﺮ ﻣﺘﺮ d ﺑﻪ ﻃﻮر ﯾﮑﻨﻮاﺧﺖ ﻣﻌﺎدل ﺑﻪ d ،d-ﮐﺮاﻧﺪار در X اﺳﺖ. (۴) ﻫﺮ ﭘﻮﺷﺶ ﯾﮑﻨﻮاﺧﺖ ﺳﺘﺎره-ﻣﺘﻨﺎﻫﯽ از X ﻣﺘﻨﺎﻫﯽ اﺳﺖ. ﻣﺎ در ﺣﺎل ﺗﻮﺻﯿﻒدﻧﺒﺎﻟﻪ ای اززﯾﺮﻣﺠﻤﻮﻋﻪ ﻫﺎی ﺑﻮرﺑﺎﮐﯽ-ﮐﺮاﻧﺪار از ﻓﻀﺎی ﻣﺘﺮﯾﮏ ﻫﺴﺘﯿﻢ. اوﻻ ﺑﻪ ﻣﻌﺮﻓﯽ ﻣﻔﺎﻫﯿﻢ ﺑﻮرﺑﺎﮐﯽ-ﮐﻮﺷﯽ دﻧﺒﺎﻟﻪ ﺑﻮرﺑﺎﮐﯽ-ﮐﻮﺷﯽ ﻫﻤﭙﺎﯾﺎن ﻧﯿﺎز دارﯾﻢ.

ﺗﻌﺮﯾﻒ٣ . ٢. ﻓﺮض ﮐﻨﯿﺪ (X, d) ﻓﻀﺎی ﻣﺘﺮﯾﮏ ﺑﺎﺷﺪ. دﻧﺒﺎﻟﻪ xn)n∈N) در X ﺑﻮرﺑﺎﮐﯽ-ﮐﻮﺷﯽ اﺳﺖ اﮔﺮ ﺑﺮای ∈ m ∈ ∈ ∈ ﻫﺮ ٠ < ε وﺟﻮد داﺷﺘﻪ ﺑﺎﺷﺪ m N و N n٠ ﺑﻪ ﻃﻮرﯾﮑﻪ ﺑﺮای ﻫﺮ p X داﺷﺘﻪ ﺑﺎﺷﯿﻢ (xn Bε (p ≥ ﺑﺮای ﻫﺮ n n٠. از ﺳﻮﯾﯽ دﯾﮕﺮ، xn)n∈N) ﺑﻮرﺑﺎﮐﯽ-ﮐﻮﺷﯽﻫﻢ ﭘﺎﯾﺎن در X اﺳﺖ اﮔﺮ ﺑﺮای ﻫﺮ ٠ < ε وﺟﻮد ∈ m ∈ ⊂ ∈ داﺷﺘﻪ ﺑﺎﺷﺪ m N و زﯾﺮ ﻣﺠﻤﻮﻋﻪ Nε N ﺑﻪ ﻃﻮرﯾﮑﻪ ﺑﺮای ﻫﺮ p X داﺷﺘﻪ ﺑﺎﺷﯿﻢ (xn Bε (p ﺑﺮای ﻫﺮ ∈ .n Nε ﻗﻀﯿﻪ٣ . ٣. ﺑﺮای ﻓﻀﺎی ﻣﺘﺮﯾﮏ (X, d) و B ⊂ X ﻋﺒﺎرات زﯾﺮ ﻣﻌﺎدﻟﻨﺪ: (١) B زﯾﺮﻣﺠﻤﻮﻋﻪ ﺑﻮرﺑﺎﮐﯽ-ﮐﺮاﻧﺪار در X اﺳﺖ. (٢) ﻫﺮ زﯾﺮﻣﺠﻤﻮﻋﻪ ﺷﻤﺎرا از B زﯾﺮﻣﺠﻤﻮﻋﻪ ﺑﻮرﺑﺎﮐﯽ-ﮐﺮاﻧﺪار در X اﺳﺖ. (٣) ﻫﺮ دﻧﺒﺎﻟﻪ در B ﯾﮏ زﯾﺮدﻧﺒﺎﻟﻪ ﺑﻮرﺑﺎﮐﯽ-ﮐﻮﺷﯽ در X اﺳﺖ. (۴) ﻫﺮ دﻧﺒﺎﻟﻪ در B، ﺑﻮرﺑﺎﮐﯽ-ﮐﻮﺷﯽﻫﻢ ﭘﺎﯾﺎن در X اﺳﺖ. ··· ··· ﻣﺜﺎل٣ . ۴. در اﻋﺪاد ﺣﻘﯿﻘﯽ R ﺑﺎ ﻣﺘﺮ ﻣﻌﻤﻮﻟﯽ، ﺷﺎﻣﻞ دﻧﺒﺎﻟﻪ ,qn, n , ,٣ ,q٣ ,٢ ,q٢ ,١ ,q١ ﺑﻪ ﻃﻮری ﮐﻪ ∩ { ··· ···} ,qn , ,q٢ ,q١ ﺷﻤﺎرش ﭘﺬﯾﺮ از ﻣﺠﻤﻮﻋﻪ (١ ,٠) Q اﺳﺖ ﺑﻨﺎﺑﺮاﯾﻦ اﯾﻦ دﻧﺒﺎﻟﻪ ﻣﺠﻤﻮﻋﻪ ﮐﺮاﻧﺪار ﻧﯿﺴﺖ و از اﯾﻨﺮو ﯾﮏ زﯾﺮﻣﺠﻤﻮﻋﮥ ﺑﻮرﺑﺎﮐﯽ -ﮐﺮاﻧﺪار ﻧﯿﺴﺖ و ﻧﻪ ﯾﮏ زﯾﺮﻣﺠﻤﻮﻋﻪ ﺑﻮرﺑﺎﮐﯽ-ﮐﻮﺷﯽ در R. از ﺳﻮﯾﯽ دﯾﮕﺮ ﺑﻪ آﺳﺎﻧﯽ ﭼﮏﻣﯽ ﮐﻨﯿﻢ ﮐﻪ دﻧﺒﺎﻟﻪﻫﻢ ﭘﺎﯾﺎن ﮐﻮﺷﯽ (ﻧﻪ ﮐﻮﺷﯽ) اﺳﺖ و از اﯾﻨﺮو دﻧﺒﺎﻟﻪ ﺑﻮرﺑﺎﮐﯽ-ﮐﻮﺷﯽ در R اﺳﺖ. ∥ · ∥ ﻣﺜﺎل٣ . ۵. ﻓﺮض ﮐﻨﯿﺪ ( ,l٢) ﻓﻀﺎی ﻫﯿﻠﺒﺮت ﮐﻼﺳﯿﮏ ازﻫﻤﻪ یدﻧﺒﺎﻟﻪ ﻫﺎیﺟﻤﻊ ﭘﺬﯾﺮ ﻣﺮﺑﻊ اﻋﺪاد ﺣﻘﯿﻘﯽ ﺑﺎﺷﺪو { ∈ } ﻓﺮض ﮐﻨﯿﺪ B = en : n N ﭘﺎﯾﻪ اﺳﺘﺎﻧﺪارد ﺑﺎﺷﺪ، B ﺑﻪ ﻋﻨﻮان زﯾﺮﻣﺠﻤﻮﻋﻪ l٢ در ﮐﻞ ﻓﻀﺎ ﺑﻮرﺑﺎﮐﯽ-ﮐﺮاﻧﺪار اﺳﺖ. ﯾﮏ زﯾﺮﻣﺠﻤﻮﻋﻪ ﺑﻮرﺑﺎﮐﯽ-ﮐﺮاﻧﺪار از ﺧﻮدش ﻧﯿﺴﺖ. زﯾﺮا ﻓﻀﺎی ﻣﺘﺮﯾﮏ ﮔﺴﺴﺘﻪ ﯾﮑﻨﻮاﺧﺖ ﻧﺎﻣﺘﻨﺎﻫﯽ اﺳﺖ. در ﺣﻘﯿﻘﺖ ∈ m { } ∥ · ∥ در ﻓﻀﺎی (B, B) و ﺑﺮای ١ = ε دارﯾﻢ Bε (en) = en، ﺑﺮای ﻫﺮ m, n N. از اﯾﻨﺮو دﻧﺒﺎﻟﻪ en)n∈N) در l٢، ﺑﻮرﺑﺎﮐﯽ -ﮐﻮﺷﯽ (ﮐﻮﺷﯽ ﻧﯿﺴﺖ) اﺳﺖ. اﻣﺎ در ﺧﻮدش ﺑﻮرﺑﺎﮐﯽ-ﮐﻮﺷﯽ ﻧﯿﺴﺖ. ﻋﻼوه ﺑﺮ اﯾﻦ اﯾﻦ دﻧﺒﺎﻟﻪ در l٢، ﺑﻮرﺑﺎﮐﯽ-ﮐﻮﺷﯽﻫﻢ ﭘﺎﯾﺎﯾﻦ اﺳﺖ اﻣﺎ ﮐﻮﺷﯽﻫﻢ ﭘﺎﯾﺎن ﻧﯿﺴﺖ.

ﻣﺮاﺟﻊ 1. Atsuji, M. Uniform continuity of continuous functions of metric spaces. Pancific J. Math. 8, 1958, 11-16.

2. Beer, G. Between compactness and completeness. Topology Appl. 155, 2008, 503, 514.

3. Beer, G. Between the cofinally complete spaces and the UC spaces. Houston J. Math. 38,2012, 999-1015.

٩ س. دﻫﻤﺮده و ج. ﺟﻤﺎﻟﺰاده

4. Garrido, I. and A. S. Merono. Some classes of bounded sets in metric spaces. In: Con-tribucionces matematicas en homenaje a Juan Tarres, eds. M. Castrillon et al. Universidad Complutense de Madrid, 2012, 179-186.

5. Hejcman, J. Boundedness in uniform spaces and topological groups. Czechoslovak Math. J. 9, 1959, 544-563.

١٠ اﺑﺮﮔﺮوﻫﻮاره ﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ

∗ ﺷﺮﯾﻔﻪ دﯾﺎﻟﯽ١ و ﻏﻼﻣﺮﺿﺎ رﺿﺎﯾﯽ٢

ﭼﮑﯿﺪه. اﺑﺮﮔﺮوه ﻫﺎ ﺗﻌﻤﯿﻤﯽ ازﮔﺮوه ﻫﺎ ﻫﺴﺘﻨﺪ. اﮔﺮ ﯾﮏ اﺑﺮﻋﻤﻞ دوﺗﺎﯾﯽ ﻗﺎﺑﻞ ﺗﻌﺮﯾﻒ ﻫﻤﺮاه ﺑﺎ ﭼﻨﺪارزﺷﯽ ﺑﻮدن ﺑﺎﺷﺪ، دراﯾﻦ ﺻﻮرت ﺑﻪ ﯾﮏ اﺑﺮﮔﺮوه ﻣﻨﺠﺮﻣﯽ ﺷﻮد. اﻧﮕﯿﺰه ﺑﺮای ﺗﻌﻤﯿﻢ ﻧﻈﺮﯾﻪ ﮔﺮوه ﺑﻪ ﻃﻮر ﻃﺒﯿﻌﯽ ازﻣﺴﺎﺋﻞ ﻣﺨﺘﻠﻒ در ﺟﺒﺮ ﻧﺎﺟﺎﺑﺠﺎﯾﯽ ﻧﺘﯿﺠﻪﻣﯽ ﺷﻮد اﻧﮕﯿﺰۀ دﯾﮕﺮی ﺑﺮﮔﺮﻓﺘﻪ از ﻫﻨﺪﺳﻪ اﺳﺖ. در ﭼﻨﺪﯾﻦ ﺷﺎﺧﻪ از رﯾﺎﺿﯿﺎت ﺑﺎﻣﺜﺎل ﻫﺎی ﻣﻬﻤﯽ از ﺳﺎﺧﺘﺎرﻫﺎی ﺗﻮﭘﻮﻟﻮژی- ﺟﺒﺮی ﻣﻮاﺟﻪ ﺷﺪﯾﻢ. ﻣﺎﻧﻨﺪ: ﮔﺮوه ﻫﺎ،ﺣﻠﻘﻪ ﻫﺎ،ﻣﯿﺪان ﻫﺎ،ﮔﺮوﻫﻮاره ﻫﺎی ﺗﻮﭘﻮﻟﻮژی و ... .در اﯾﻦ ﻧﻮﺷﺘﻪ، اﻧﻮاع ﻣﺨﺘﻠﻒ از ﭘﯿﻮﺳﺘﮕﯽاﺑﺮﻋﻤﻞ ﻫﺎ ﻣﻌﺮﻓﯽ ﺧﻮاﻫﯿﻢ ﮐﺮد. ﺑﺮای ﻣﺜﺎل: راﺑﻄﻪ ﺑﯿﻦ ﺷﺒﻪ ﭘﯿﻮﺳﺘﮕﯽ، ﺷﺒﻪ ﭘﯿﻮﺳﺘﮕﯽ ﻗﻮی واﺑﺮﻋﻤﻞ ﻫﺎی ﭘﯿﻮﺳﺘﻪ ﻣﻄﺎﻟﻌﻪ ﺧﻮاﻫﯿﻢ ﮐﺮد. ﻫﺪف ﮐﻠﯽ ﺗﻌﻤﯿﻢ ﻣﻔﻬﻮم ﮔﺮوﻫﻮاره ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺑﻪ اﺑﺮﮔﺮوﻫﻮاره ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ اﺳﺖ.

١. ﭘﯿﺶ ﮔﻔﺘﺎر اﺑﺮﮔﺮوه ﻫﺎ ﺗﻌﻤﯿﻤﯽ ازﮔﺮوه ﻫﺎ( ﻣﺠﻤﻮﻋﻪ ﺑﺎ ﯾﮏ ﻋﻤﻞ دوﺗﺎﯾﯽ روی آن و ﺗﻌﺪادی از ﺷﺮاﯾﻂ ﺑﺮﻗﺮار اﺳﺖ) ﻫﺴﺘﻨﺪ. اﮔﺮ اﯾﻦ ﻋﻤﻞ دوﺗﺎﯾﯽ ﻗﺎﺑﻞ ﺗﻌﺮﯾﻒ ﻫﻤﺮاه ﺑﺎ ﭼﻨﺪ ارزﺷﯽ ﮐﺮدن ﺑﺎﺷﺪ دراﯾﻦ ﺻﻮرت ﺑﻪ ﯾﮏ اﺑﺮﮔﺮوه ﻣﻨﺠﺮﻣﯽ ﺷﻮد. اﯾﺪه ﺑﺮای ﺗﻌﻤﯿﻢ دادن ﻧﻈﺮﯾﮥ ﮔﺮوهﺑﻪ ﻃﻮر ﻃﺒﯿﻌﯽ از ﭼﻨﺪﯾﻦ ﻣﺴﺎﺋﻞ در ﺟﺒﺮ ﺗﻌﻮﯾﺾ ﻧﺎﭘﺬﯾﺮ ﻧﺘﯿﺠﻪ ﮔﺮﻓﺖ. اﯾﺪۀ دﯾﮕﺮی ﺑﺮای اﯾﻦ ﺗﺤﻘﯿﻖ از ﻫﻨﺪﺳﻪ ﺑﺪﺳﺖ آﻣﺪ. ﻧﻈﺮﯾﻪاﺑﺮﮔﺮوه ﻫﺎ در ﺳﺎل ١٩٣۴ ﺑﺎ ﻣﻘﺎﻟﻪ ﻣﺎرﺗﯽ در ﮐﻨﮕﺮه ﻫﺸﺘﻢ رﯾﺎﺿﯿﺪاﻧﺎن اﺳﮑﺎﻧﺪﯾﻨﺎوی در اﺳﺘﮑﻬﻠﻢ[٢] ﭘﺪﯾﺪار ﺷﺪ. ﭘﺲ از آن در ﺳﺮاﺳﺮ دﻫﮥ ۴٠ ﺑﺎ ﻫﻤﮑﺎری ﻧﻮﯾﺴﻨﺪﮔﺎن ﻣﺨﺘﻠﻒ ﻣﺨﺼﻮﺻﺎً در ﻓﺮاﻧﺴﻪ، اﯾﺘﺎﻟﯿﺎ، ﯾﻮﻧﺎن واﯾﺎﻻت ﻣﺘﺤﺪه ﺗﻮﺳﻌﻪ ﯾﺎﻓﺖ. ﺑﺮای ﻣﺜﺎل، ﻃﯽدﻫﻪ ﻫﺎی زﯾﺮ ﻣﻄﺎﻟﺐ وﻧﺘﺎﯾﺞ ﺟﺪﯾﺪ ﻇﺎﻫﺮ ﺷﺪ. اﻣﺎ ﺑﯿﺸﺘﺮ از ﻫﻤﻪ ﺷﮑﻮﻓﺎﯾﯽاﻧﺒﻮه ﺗﺮ از اﺑﺮﺳﺎﺧﺘﺎرﻫﺎ در ٣٠ ﺳﺎل ﮔﺬﺷﺘﻪ دﯾﺪه ﺷﺪه اﺳﺖ. در ﻃﯽ اﯾﻦﺳﺎل ﻫﺎ،اﺑﺮﮔﺮوه ﻫﺎ در ﺟﺒﺮ، ﻫﻨﺪﺳﻪ، ﻣﺤﺪب( ﮐﻮژی) و ﻧﻈﺮﯾﮥﻣﺎﺷﯿﻦ ﻫﺎ اﺳﺘﻔﺎده ﺷﺪه اﺳﺖ. ﻣﺴﺌﻠﻪ ﻫﺎی ﺗﺮﮐﯿﺒﯽ از رﻧﮓ آﻣﯿﺰی، ﺗﺌﻮری ﺷﺒﮑﻪ، ﺟﺒﺮ ﺑﻮﻟﯽ، ﻣﻨﻄﻖ و ... ﻣﻮرد اﺳﺘﻔﺎده ﺑﻮده اﺳﺖ ( [۴]).

2010 Mathematics Subject Classification. Primary 99X99, Secondary 99Y99. واژﮔﺎن ﮐﻠﯿﺪی. اﺑﺮﮔﺮوﻫﻮاره؛ اﺑﺮﻋﻤﻞ؛ ﭘﯿﻮﺳﺘﮕﯽ؛ ﺗﻮﭘﻮﻟﻮژی اﺑﺮﻋﻤﻞﭘﯿﻮﺳﺘﻪ ﻧﻤﺎ؛ اﺑﺮﮔﺮوﻫﻮارۀ ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ. . ∗ ﺳﺨﻨﺮان

١١ ش. دﯾﺎﻟﯽ و غ.رﺿﺎﯾﯽ

٢. ﻣﻔﺎﻫﯿﻢ ﻣﻘﺪﻣﺎﺗﯽ

اﺑﺘﺪا ﺷﺮاﯾﻂ ﮐﻠﯽ و ﺗﻌﺎرﯾﻒ ﻧﻈﺮﯾﮥ اﺑﺮﺳﺎﺧﺘﺎرﻫﺎ ﯾﺎدآوریﻣﯽ ﮐﻨﯿﻢ ([۴] ﺑﺒﯿﻨﯿﺪ). ∗ ﺗﻌﺮﯾﻒ٢ . ١. ﻓﺮض ﮐﻨﯿﺪ H ﯾﮏ ﻣﺠﻤﻮﻋﮥ ﻧﺎﺗﻬﯽ و (P (H ﻣﺠﻤﻮﻋﮥ ﺗﻤﺎم زﯾﺮﻣﺠﻤﻮﻋﻪ ﻫﺎی ﻧﺎﺗﻬﯽ ازH ﺑﺎﺷﺪ. در ∗ اﯾﻦ ﺻﻮرت زوج ( · ,H) ﯾﮏ اﺑﺮﮔﺮوﻫﻮاره اﺳﺖ، ﺑﻪ ﻃﻮری ﮐﻪ (H −→ P (H : · ﯾﮏ اﺑﺮﻋﻤﻞ دوﺗﺎﯾﯽ روی ﻣﺠﻤﻮﻋﮥ H اﺳﺖ. اﮔﺮﺑﻪ ازای ﻫﺮ a · (b · c) = (a · b) · c ،a, b, c ∈ H ( ﻗﺎﻋﺪۀﺷﺮﮐﺖ ﭘﺬﯾﺮی) ﺑﺮﻗﺮار ﺑﺎﺷﺪ آﻧﮕﺎه (· ,H) ﻧﯿﻢ اﺑﺮﮔﺮوه اﺳﺖ. ﻧﯿﻢ اﺑﺮﮔﺮوه (· ,H) ﯾﮏ اﺑﺮﮔﺮوه ﻧﺎﻣﯿﺪهﻣﯽ ﺷﻮد اﮔﺮ ﭘﯿﺮو ﻗﺎﻋﺪۀ ﻫﻤﺎﻧﯽ ﺑﺎﺷﺪﺑﻪ ازای ̸ ∅ ̸ ⊆ · · ∈ ﻫﺮ a H = H∪ = H a ،a H. در اداﻣﻪ، ﺑﺮای A = = B ،A, B H ﺗﻌﺮﯾﻒ ﻣﯽ ﮐﻨﯿﻢ {A · B = {a · b : a ∈ A, b ∈ B. ﻋﺒﺎرت ﻫﺎی aA و Aa ﺑﺮای a}· A} و {A ·{a اﺳﺘﻔﺎده ﺷﺪه اﺳﺖ. ﺑﻪ ﻃﻮر ﮐﻠﯽ ﻣﺠﻤﻮﻋﮥ ﺗﮏ ﻋﻀﻮی {a}∩ﺑﺎ ﻋﻨﺼﺮ a ﺷﻨﺎﺳﺎﯾﯽ ﺷﺪه اﺳﺖ. ﻋﺒﺎرت A ≈ B ﯾﻌﻨﯽ A ﻣﺠﻤﻮﻋﮥ B را ﻗﻄﻊﻣﯽ ﮐﻨﺪ. ﺑﻪ ﻋﺒﺎرﺗﯽ دﯾﮕﺮ ∅ ̸= A B.

ﺑﻪ وﺳﯿﻠﻪ ﺗﻌﺮﯾﻒ ﮔﺮوﻫﻮاره ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ [١]ﻧﻤﻮﻧﻪ ﻫﺎی زﯾﺮ را ﻣﻌﺮﻓﯽﻣﯽ ﮐﻨﯿﻢ.

ﺗﻌﺮﯾﻒ٢ . ٢. ﻓﺮض ﮐﻨﯿﺪ (· ,H) ﯾﮏ اﺑﺮﮔﺮوﻫﻮاره و (H, τ) ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺑﺎﺷﺪ. اﺑﺮﻋﻤﻞ ” · ” ﭘﯿﻮﺳﺘﻪ ﻧﻤﺎ ﯾﺎ ﺑﻪ اﺧﺘﺼﺎر P - ﭘﯿﻮﺳﺘﻪﻣﯽ ﻧﺎﻣﯿﻢ اﮔﺮﺑﻪ ازای ﻫﺮ O ∈ τ ﻣﺠﻤﻮﻋﮥ {O∗ = {(x, y) ∈ H × H : x · y ⊆ O در H × H ﺑﺎز ﺑﺎﺷﺪ. ” · ” (H.τ) (H, ·) ﺗﻌﺮﯾﻒ٢ . ٣. ﻓﺮض ﮐﻨﯿﺪ اﺑﺮﮔﺮوﻫﻮاره و ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺑﺎﺷﺪ. اﺑﺮﻋﻤﻞ ﭘﯿﻮﺳﺘﻪ ﻧﻤﺎی ∗ﻗﻮی ﯾﺎ ﺑﻪ اﺧﺘﺼﺎر SP -ﭘﯿﻮﺳﺘﻪ ﻧﺎﻣﯿﺪهﻣﯽ ﺷﻮدﺑﻪ ازای ﻫﺮ O ∈ τ ﻣﺠﻤﻮﻋﮥ {O = {(x, y) ∈ H × H : x · y ≈ O در H × H ﺑﺎز اﺳﺖ. اﺑﺮﻋﻤﻞ ” · ” ﭘﯿﻮﺳﺘﻪ ﻧﻤﺎ اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮﺑﻪ ازای ﻫﺮ O ∈ τ و ﻫﺮ زوج x, y) ∈ H × H) ﮐﻪ x · y ⊆ o، ﺑﺮای ﻫﺮ x ∈ U و y ∈ V وﺟﻮد دارد U, V ∈ τ ﺑﻪ ﻃﻮرﯾﮑﻪ u · v ⊆ O . ﻣﺸﺎﺑﻬﺎً، اﺑﺮﻋﻤﻞ ” · ”، SP -ﭘﯿﻮﺳﺘﻪ اﺳﺖ اﮔﺮ وﻓﻘﻂ اﮔﺮﺑﻪ ازای ﻫﺮ O ∈ τ و ﻫﺮ زوج x, y) ∈ H × H) وﺟﻮد دارد U, V ∈ τ و x ∈ U و y ∈ V دراﯾﻦ ﺻﻮرت x · y ≈ O ﺑﻪ ﻃﻮری ﮐﻪﺑﻪ ازای ﻫﺮ u ∈ U و v ∈ V دارﯾﻢ u · v ≈ O.

τ∗ (H, τ) (H, ·) ∗ﺗﻌﺮﯾﻒ ٢ . ۴. ﻓﺮض ﮐﻨﯿﺪ ﯾﮏ اﺑﺮﮔﺮوﻫﻮاره و ﯾﮏ ﻓﻀﺎی ∗ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ و ﯾﮏ ﺗﻮﭘﻮﻟﻮژی روی (P (H ﺑﺎﺷﺪ. اﺑﺮﻋﻤﻞ ” · ”، ∗τ - ﭘﯿﻮﺳﺘﻪ ﻣﯽ ﻧﺎﻣﯿﻢ اﮔﺮ ﻧﮕﺎﺷﺖ (H × H −→ P (H : · ﻧﺴﺒﺖ ﺑﻪ ﺗﻮﭘﻮﻟﻮژی ﻫﺎی τ × τ و∗τ ﭘﯿﻮﺳﺘﻪ ﺑﺎﺷﺪ. ∗ ﺗﻌﺮﯾﻒ٢ . ۵. ﻓﺮض ﮐﻨﯿﺪ (· ,H) ﯾﮏ اﺑﺮﮔﺮوﻫﻮاره، (H, τ) ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژی و ∗τ ﯾﮏ ﺗﻮﭘﻮﻟﻮژی روی (P (H ﺑﺎﺷﺪ. ﺳﻪ ﺗﺎﯾﯽ (H, ·, τ) ﯾﮏ ﺗﻮﭘﻮﻟﻮژی ﻧﻤﺎ ﯾﺎ اﺑﺮﮔﺮوﻫﻮاره ﺗﻮﭘﻮﻟﻮژی ﻧﻤﺎی ﻗﻮیﻣﯽ ﻧﺎﻣﯿﻢ اﮔﺮ اﺑﺮﻋﻤﻞ ” · ” ﺑﻪ ﺗﺮﺗﯿﺐ ﭘﯿﻮﺳﺘﻪ ﻧﻤﺎ ﯾﺎ ﭘﯿﻮﺳﺘﻪ ﻗﻮی ﺑﺎﺷﺪ. ﭼﻬﺎرﺗﺎﯾﯽ ∗H, ·, τ, τ)، اﺑﺮ ﮔﺮوﻫﻮاره ∗τ - ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﻧﺎﻣﯿﻢ اﮔﺮ اﺑﺮﻋﻤﻞ ” · ”، ∗τ -ﭘﯿﻮﺳﺘﻪ ﺑﺎﺷﺪ.

ﻟﻢ٢ . ۶. ﻓﺮض ﮐﻨﯿﺪ (d (X, τ ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژی ﺑﺎﺷﺪ، دراﯾﻦ ﺻﻮرت ﺧﺎﻧﻮاده U ، ﺑﺮای ﻫﺮ V ∈ τ ﻣﺘﺸﮑﻞ از ∗ ∗ P (H) τυ Sv = {U ∈ P (H): U ⊆ V ﻫﻤﻪﻣﺠﻤﻮﻋﻪ ﻫﺎی ∗ ﭘﺎﯾﻪ ای از ﺗﻮﭘﻮﻟﻮژی روی اﺳﺖ. اﯾﻦ ﺗﻮﭘﻮﻟﻮژی را ﺗﻮﭘﻮﻟﻮژی ﭘﺎﺋﯿﻨﯽ روی (P (H ﻣﯽ ﻧﺎﻣﯿﻢ.

ﻗﻀﯿﻪ ٢ . ٧. ﻓﺮض ﮐﻨﯿﻢ (· ,H) ﯾﮏ اﺑﺮﮔﺮوﻫﻮاره و (H, τ) ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺑﺎﺷﺪ، آﻧﮕﺎه ﺳﻪ ﺗﺎﯾﯽ (H, ·, τ) اﺑﺮﮔﺮوﻫﻮارۀ ﺗﻮﭘﻮﻟﻮژی ﻧﻤﺎ اﺳﺖ اﮔﺮ و ﻓﻘﻂ اﮔﺮ ﭼﻬﺎرﺗﺎﯾﯽ ( H, ·, τ, τU) ﯾﮏ اﺑﺮﮔﺮوﻫﻮارۀ τU -ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺑﺎﺷﺪ. { ∈ P∗ ≈ } L ﻟﻢ٢ . ٨. اﮔﺮ (H, τ) ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺑﺎﺷﺪ، آﻧﮕﺎه ﺧﺎﻧﻮادۀ ﺷﺎﻣﻞ ﻫﻤﮥ ﻣﺠﻤﻮﻋﮥ IV = U (H): U V ∗ و V ∈ τ زﯾﺮﭘﺎﯾﻪ ای ﺑﺮای τL روی (P (H اﺳﺖ. ﮐﻪ اﯾﻦ ﻧﻮع ﺗﻮﭘﻮﻟﻮژی ﭘﺎﺋﯿﻨﯽﻣﯽ ﻧﺎﻣﯿﻢ.

ﻗﻀﯿﻪ ٢ . ٩. ﻓﺮض ﮐﻨﯿﻢ (· ,H) اﺑﺮﮔﺮوﻫﻮاره و (H, τ) ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺑﺎﺷﺪ، آﻧﮕﺎه ﺳﻪ ﺗﺎﯾﯽ (H, ·, τ) اﺑﺮﮔﺮوﻫﻮارۀ ﺗﻮﭘﻮﻟﻮژی ﻧﻤﺎی ﻗﻮی اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ ﭼﻬﺎرﺗﺎﯾﯽ (H, ·, τ, τL) ﯾﮏ اﺑﺮﮔﺮوﻫﻮارۀ τL -ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺑﺎﺷﺪ.

١٢ اﺑﺮﮔﺮوﻫﻮاره ﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ

٣. ﻧﺘﺎﯾﺞ اﺻﻠﯽ در ﺳﺎل ١٩٢٢ ﯾﮏ رﯾﺎﺿﯿﺪان اﺗﺮﯾﺸﯽ ﺑﻪ ﻧﺎم وﯾﺘﻮرﯾﺲ [۵] ﯾﮏ ﺗﻮﭘﻮﻟﻮژی روی ﻣﺠﻤﻮﻋﮥ ﺗﻤﺎم زﯾﺮﻣﺠﻤﻮﻋﻪ ﻫﺎی ﺑﺴﺘﻪ ﻧﺎﺗﻬﯽ از ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ (H, τ) ﺗﻌﺮﯾﻒ ﮐﺮد. در ﭘﻮﺷﺸﯽ از ﯾﮏ ﻓﻀﺎی ﻣﺘﺮﯾﮏ ﺑﺴﺘﻪ- ﻓﺸﺮده X اﯾﻦ ﺗﻮﭘﻮﻟﻮژی ﺑﺮاﺑﺮ ﺑﺎ ﺗﻮﭘﻮﻟﻮژی اﻟﻘﺎﯾﯽﺑﻪ وﺳﯿﻠﮥ ”ﻣﺘﺮ ﻫﺎﺳﺪورف” ﺧﻮاﻧﺪهﻣﯽ ﺷﻮد و روﯾﮑﺮدشﻣﯽ ﺗﻮاﻧﺪ ﺑﺎﻣﺠﻤﻮﻋﻪ ای از ﺗﻤﺎمزﯾﺮﻣﺠﻤﻮﻋﻪ ﻫﺎی ﻧﺎﺗﻬﯽ (H, τ) ﮐﺎرﺑﺮدی ﺑﺎﺷﺪ. ﺑﻪ روش زﯾﺮ آن را ﺗﺸﺮﯾﺢﻣﯽ ﮐﻨﯿﻢ: ∈ N ··· ﻟﻢ٣ . ١. ﻓﺮض ﮐﻨﯿﻢ (H, τ) ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺑﺎﺷﺪ. ﺑﻪ ازای ﻫﺮ Uk, ,U٢,U١ و k وﺑﻪ ازای ∈ ··· k , ,٢ ,١ = Ui τ ،iﺗﻌﺮﯾﻒﻣﯽ ﮐﻨﯿﻢ: { } ∪k V ··· ∈ P∗ ⊂ ≈ Uk) = B (H): B Ui,B Ui, ,U٢,U١) ١=i P∗ V ··· B ﺧﺎﻧﻮادۀ از ﻫﻤﮥﻣﺠﻤﻮﻋﻪ ﻫﺎی (Uk, ,U٢,U١) ﺗﺸﮑﯿﻞ ﯾﮏ ﭘﺎﯾﻪ ﺑﺮای ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ( H), τV) ) ﻣﯽ دﻫﺪ. ∗ ﺗﻌﺮﯾﻒ٣ . ٢. ﺗﻮﭘﻮﻟﻮژی τV در ﻟﻢ ﻗﺒﻞ ﺗﻮﭘﻮﻟﻮژی وﯾﺘﻮرﯾﺲ روی (P (H ﻧﺎﻣﯿﺪهﻣﯽ ﺷﻮد. ﻟﻢ٣ . ٣. ﺗﻮﭘﻮﻟﻮژی وﯾﺘﻮرﯾﺲ τV ﺗﻈﺮﯾﻒ ﻣﺸﺘﺮک ﺗﻮﭘﻮﻟﻮژی ﻫﺎی ﺑﺎﻻﯾﯽ τU و ﭘﺎﺋﯿﻨﯽ τL اﺳﺖ. ﻗﻀﯿﻪ٣ . ۴. اﮔﺮ (H, τ) ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ و (· ,H) ﯾﮏ اﺑﺮﮔﺮوﻫﻮاره ﺑﺎﺷﺪ. ﺳﻪ ﺗﺎﯾﯽ (H, ·, τ) ﻫﻢ اﺑﺮﮔﺮوﻫﻮارۀ ﺗﻮﭘﻮﻟﻮژی ﻧﻤﺎ و ﻫﻢ اﺑﺮﮔﺮوﻫﻮارۀ ﺗﻮﭘﻮﻟﻮژی ﻧﻤﺎی ﻗﻮی اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ ﭼﻬﺎرﺗﺎﯾﯽ ( H, ·, τ, τV) ﯾﮏ τV -اﺑﺮﮔﺮوﻫﻮاره ﺑﺎﺷﺪ.

ﻣﺮاﺟﻊ 1. R. Ameri, Topological (transposition) hypergroups, Italian Journal Pure Appllied Mathematics (13) (2003) 171 –176.

2. F. Marty, Sur une généralisation de la notion de groupe. in: Huitième Congr. Math. Scan., 1934, Stockholm, pp. 45–49.

3. B. Davvaz, J. Zhan, K.H. Kim, Fuzzy-hypernear-rings, Computers Mathematics with Applications 59 (8) (2010) 2846–2853.

4. P. corsini,V. Leoreanu, Applications of Hyperstructur theory, Kluwer Academic Publishers, Dordercht, Hardbound, 2003.

5. L. Vietoris, Bereiche zweiter Ordnung, Monatshefte für Mathematik und Physik. Akademische Verlags-Gesellschaft 32 (1922) 258–280.

١٣ ﻣﺘﺮی ﭘﺬﯾﺮی ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ

∗ ﺷﺮﯾﻔﻪ ﻣﻨﺼﻮری١

١ آدرس ١ [email protected] ٢ ﮔﺮوه رﯾﺎﺿﯽ، داﻧﺸﮑﺪه ﻋﻠﻮم رﯾﺎﺿﯽ، داﻧﺸﮕﺎه ﺳﯿﺴﺘﺎن و ﺑﻠﻮﭼﺴﺘﺎن

ﭼﮑﯿﺪه. در اﯾﻦ ﻣﻘﺎﻟﻪ، ﻣﻬﻤﺘﺮﯾﻦ ﺳﺎﺧﺖ و ﺳﺎز ﻣﺮﺑﻮط ﺑﻪ ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی را ﺗﻮﺳﻂ ﭘﯿﺶ ﻧﺮم ﻫﺎ ﻣﻌﺮﻓﯽ ﻣﯽ ﮐﻨﯿﻢ و ﺷﺮط ﻻزم و ﮐﺎﻓﯽ ﺑﺮایﻣﺘﺮی ﭘﺬﯾﺮی (ﺗﻮﺳﻂ ﻣﺘﺮﯾﮏ ﻫﺎی راﺳﺖ- ﭘﺎﯾﺎ )ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی اراﺋﻪﻣﯽ دﻫﯿﻢ. ﺛﺎﺑﺖ ﻣﯽ ﮐﻨﯿﻢ ﮐﻪ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی G ﻣﺘﺮی ﭘﺬﯾﺮ (ﺗﻮﺳﻂ ﻣﺘﺮﯾﮏ راﺳﺖ -ﭘﺎﯾﺎ ) اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ G ﺷﻤﺎرای ﻧﻮع اول ﺑﺎﺷﺪ.

١. ﭘﯿﺶ ﮔﻔﺘﺎر در اﯾﻦ ﻣﻘﺎﻟﻪ ﻫﻤﮥ ﻓﻀﺎﻫﺎ ﻫﺎﺳﺪورف اﺳﺖ. d را ﯾﮏ ﻣﺘﺮ روی ﻣﺠﻤﻮﻋﮥ ﻧﺎﺗﻬﯽ X ﮔﻮﺋﯿﻢ ﻫﺮﮔﺎه ﺗﺎﺑﻊ → d : X ×X R واﺟﺪ ﺳﻪ ﺧﺎﺻﯿﺖ زﯾﺮ ﺑﺎﺷﺪ: (M١)ﺑﻪ ازای ﻫﺮ x و y از X، ٠ ⩽ (d(x, y و ٠ = (d(x, yاﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ x = y؛ (M٢)ﺑﻪ ازای ﻫﺮ x و y از d(x, y) = d(y, x) ،X؛ (M٣)ﺑﻪ ازای ﻫﺮ x و y و z از d(x, y) + d(y, z) ⩾ d(x, z) ،X . ∈ ﺣﺎل، اﮔﺮ X ﻣﺠﻤﻮﻋﻪ ای ﻧﺎﺗﻬﯽ، d ﻣﺘﺮﯾﮑﯽ روی X و r ﻋﺪدﺣﻘﯿﻘﯽ ﻣﺜﺒﺘﯽ ﺑﺎﺷﺪ. ﺑﻪ ﻋﻼوه، X x٠ در { ∈ | } اﯾﻦ ﺻﻮرت ﻣﺠﻤﻮﻋﮥ r > (x X d(x, x٠ را ﯾﮏ ﮔﻮی ﺑﻪ ﻣﺮﮐﺰ x٠ و ﺷﻌﺎع r در X ﻣﯽ ﻧﺎﻣﯿﻢ و آن را ﺑﺎ { ∈ ∈ R+} (r ,Bd(x٠ ﻧﺸﺎنﻣﯽ دﻫﯿﻢ و ﮔﺮداﯾﮥ X, r r): x٠ ,β = Bd(x٠ در ﺧﻮاص ﭘﺎﯾﻪ ﺻﺪقﻣﯽ ﮐﻨﺪ T ﺗﻮﭘﻮﻟﻮژﯾﯽ ﮐﻪ ﺗﻮﺳﻂ اﯾﻦ ﭘﺎﯾﻪ ﺗﻮﻟﯿﺪﻣﯽ ﺷﻮد ﺗﻮﭘﻮﻟﻮژی ﻣﺘﺮی ﺑﺮروی X ﮐﻪ ﺑﻮﺳﯿﻠﮥ ﻣﺘﺮﯾﮏ d اﻟﻘﺎ ﺷﺪه اﺳﺖ و آن را ﺑﺎ d ﻧﺸﺎنﻣﯽ دﻫﻨﺪ. و زوج (X, d) را ﻓﻀﺎی ﻣﺘﺮی ﮔﻮﯾﻨﺪ. در ﺣﺎﻟﺖ ﮐﻠﯽ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ( X, T) راﻣﺘﺮی ﭘﺬﯾﺮ ﮔﻮﯾﻨﺪ T T ﻫﺮﮔﺎه ﻣﺘﺮﯾﮑﯽ ﻣﺎﻧﻨﺪ d روی X ﻣﻮﺟﻮد ﺑﺎﺷﺪﺑﻪ ﻃﻮری ﮐﻪ ﺗﻮﭘﻮﻟﻮژی ﻣﺘﺮی d ﺑﺎ ﺗﻮﭘﻮﻟﻮژی ﯾﮑﺴﺎن ﺑﺎﺷﺪ (ﻣﺮاﺟﻌﻪ ﺑﻪ [۴] ﺷﻮد ). اﯾﻨﮏ ﻓﺮضﻣﯽ ﮐﻨﯿﻢ G ﯾﮏ ﻣﺠﻤﻮﻋﮥ ﻧﺎﺗﻬﯽ ﺑﺎﺷﺪ. ﺳﻪ ﺗﺎﯾﯽ ( G, ∗, T) را ﯾﮏ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی (ﮔﺮوه ﭘﺎراﺗﻮﭘﻮﻟﻮژی) ﮔﻮﺋﯿﻢ ﻫﺮﮔﺎه (∗ ,G) ﯾﮏ ﮔﺮوه، ( G, T) ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ و دو ﺷﺮط زﯾﺮ (ﻓﻘﻂ ﺷﺮط ١) ﺑﺮﻗﺮار ﺑﺎﺷﺪ: ١. ﻋﻤﻞ ﮔﺮوه ﯾﻌﻨﯽ ﻧﮕﺎﺷﺖ G × G → G : ∗ ﭘﯿﻮﺳﺘﻪ ﺑﺎﺷﺪ؛ − ٢. ﻧﮕﺎﺷﺖوارون ﮔﯿﺮی ﮔﺮوه inv : G → G ﺑﻪ ازای ﻫﺮ g ∈ G ﺑﺎ ﺿﺎﺑﻄﮥ ١ inv(g) = g ﭘﯿﻮﺳﺘﻪ ﺑﺎﺷﺪ.

2010 Mathematics Subject Classification. Primary 99X99; Secondary 99Y99. واژﮔﺎن ﮐﻠﯿﺪی. ﻣﺘﺮی ﭘﺬﯾﺮی؛ﭘﯿﺶ ﻧﺮم؛ ﻣﺘﺮﯾﮏ راﺳﺖ- ﭘﺎﯾﺎ؛ ﮔﺮوه ﭘﺎراﺗﻮﭘﻮﻟﻮژی . ∗ ﺳﺨﻨﺮان

١۴ ش. ﻣﻨﺼﻮری

ﻣﯽ ﺧﻮاﻫﯿﻢ ﻧﺸﺎن دﻫﯿﻢ ﮐﻪﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی ﺷﻤﺎرای ﻧﻮع اولﻣﺘﺮی ﭘﺬﯾﺮ (ﺗﻮﺳﻂ ﻣﺘﺮﯾﮏ ﻫﺎی راﺳﺖ ﭘﺎﯾﺎ ) ﻫﺴﺘﻨﺪ. در ﺑﺨﺶ ( ٢ ) ﯾﮏﭘﯿﺶ ﻧﺮم را رویﮔﺮوه ﻫﺎ ﺗﻌﺮﯾﻒﻣﯽ ﮐﻨﯿﻢ وﺑﺮﺧﯽ از ﮐﺎﺑﺮدﻫﺎی آن را اراﺋﻪﻣﯽ دﻫﯿﻢ . ﻧﺸﺎنﻣﯽ دﻫﯿﻢ ﮐﻪ ﭘﯿﺶ ﻧﺮم ﭘﯿﻮﺳﺘﻪ ای روی ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی وﺟﻮد دارد ﯾﮑﯽ از ﻣﻬﻤﺘﺮﯾﻦ ﮐﺎرﺑﺮدﻫﺎی ﭘﯿﺶ ﻧﺮم ﻫﺎ روی ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژیﻣﯽ ﺗﻮان ﺑﻪ ﺳﺎﺧﺖ ﭘﺎﯾﻪ ﺑﺮای آﻧﻬﺎ اﺷﺎره ﮐﺮد (ﻟﻢ ٢ . ٧). در ﺑﺨﺶ (٣) ﺛﺎﺑﺖ ﺧﻮاﻫﯿﻢ ﮐﺮد ﮐﻪ ﺑﺎ اﺳﺘﻔﺎده از اﯾﻦ ﭘﺎﯾﻪﻣﯽ ﺗﻮان ﯾﮏ ﺗﻮﭘﻮﻟﻮژی ﻣﺘﺮی ﺳﺎﺧﺖ ﮐﻪ ﺑﺎ ﺗﻮﭘﻮﻟﻮژی اﺻﻠﯽﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺷﻤﺎرای ﻧﻮع اول ﯾﮑﺴﺎن اﺳﺖ (ﻗﻀﯿﮥ ٢ . ٨)).

٢. ﻧﺘﺎﯾﺞ ﻣﻘﺪﻣﺎﺗﯽ در اﯾﻦ ﺑﺨﺶ ﺑﺮﺧﯽ از ﻣﻘﺪﻣﺎت و ﻧﺘﺎﯾﺞ ﮐﺎرﺑﺮدی ﺑﺮای ﻣﺘﺮی ﭘﺬﯾﺮی ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی اراﺋﻪ ﻣﯽ دﻫﯿﻢ ﮐﻪ اﺑﺘﺪا ﻣﯽ ﺧﻮاﻫﯿﻢ ﺑﺎ ﻣﻌﺮﻓﯽ و ﮐﺎرﺑﺮدﭘﯿﺶ ﻧﺮم ﻫﺎ اﯾﻦ ﻣﻄﻠﺐ را ﺷﺮوع ﮐﻨﯿﻢ:

ﺗﻌﺮﯾﻒ٢ . ١. ﻓﺮض ﮐﻨﯿﻢ G ﯾﮏ ﮔﺮوه ﺑﺎ ﻋﻨﺼﺮ ﻫﻤﺎﻧﯽ e ﺑﺎﺷﺪ دراﯾﻦ ﺻﻮرت، ﻫﺮ ﺗﺎﺑﻊ ﻣﺎﻧﻨﺪ N : G × G → R را ﮐﻪ واﺟﺪ ﺳﻪ ﺷﺮط زﯾﺮ ﺑﺎﺷﺪ، ﯾﮏﭘﯿﺶ ﻧﺮم روی ﮔﺮوه G ﮔﻮﺋﯿﻢ. (PN١) ٠ = (N(e؛ (PN٢)ﺑﻪ ازای ﻫﺮ x و y از N(xy) ≤ N(x) + N(y) ،G؛ − (PN٣)ﺑﻪ ازای ﻫﺮ x از N(x) ،G = (١ N(x. ﺑﻮﺿﻮح، ﻫﺮﭘﯿﺶ ﻧﺮم روی ﮔﺮوﻫﯽ ﻣﺎﻧﻨﺪ G ﻧﺎﻣﻨﻔﯽ اﺳﺖ.

ﮔﺰاره ٢ . ٢. اﮔﺮ N ﯾﮏﭘﯿﺶ ﻧﺮم در ﮔﺮوﻫﯽ ﻣﺎﻧﻨﺪ G ﺑﺎﺷﺪ. آﻧﮕﺎهﺑﻪ ازای ﻫﺮ x و y از G، .(١−N(x) − N(y)| ≤ N(xy| { ∈ } ﮔﺰاره ٢ . ٣. ﺑﺮای ﻫﺮﭘﯿﺶ ﻧﺮم N در G، ﻣﺠﻤﻮﻋﮥ ٠ = (ZN = x G : N(x زﯾﺮﮔﺮوﻫﯽ از G اﺳﺖ.

ﯾﮏ روش ﺑﺴﯿﺎر ﺳﺎده ﺑﺮای ﺳﺎﺧﺖﭘﯿﺶ ﻧﺮم درﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی وﺟﻮد دارد ﮐﻪ ﻟﻢ زﯾﺮ ﺣﺎﮐﯽ از اﯾﻦ ﻣﺴﺄﻟﻪ اﺳﺖ:

ﻟﻢ٢ . ۴. ﻓﺮض ﮐﻨﯿﺪ f ﯾﮏ ﺗﺎﺑﻊ ﮐﺮاﻧﺪار ﺣﻘﯿﻘﯽ روی ﮔﺮوه G ﺑﺎﺷﺪ. آﻧﮕﺎه ﺗﺎﺑﻊ Nf ﺑﺮ G ﺑﻪ ازای ﻫﺮ x از G، ﮐﻪ {| − | ∈ } ﺗﻮﺳﻂ ﺿﺎﺑﻄﮥ Nf (x) = sup f(yx) f(y) : y G ﺗﻌﺮﯾﻒ ﺷﺪه اﺳﺖ ﯾﮏﭘﯿﺶ ﻧﺮم در G اﺳﺖ.

ﺑﻪ ﻃﻮر ﮐﻠﯽ،ﭘﯿﺶ ﻧﺮم ﻫﺎ رویﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی ﻟﺰوﻣﺎً ﭘﯿﻮﺳﺘﻪ ﻧﯿﺴﺘﻨﺪ. ﮔﺰاره ﺑﻌﺪی ﻫﺮ ﭼﻨﺪ ﺳﺎده، وﻟﯽ ﻣﻔﯿﺪ ﻣﻮاﻗﻊ ﻣﯽ ﺷﻮد. ﮔﺰاره ٢ . ۵. ﭘﯿﺶ ﻧﺮم N در ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی G ﭘﯿﻮﺳﺘﻪ اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮﺑﻪ ازای ﻫﺮ ﻋﺪد ﻣﺜﺒﺘﯽ ﻣﺎﻧﻨﺪ ϵ، ﻫﻤﺴﺎﯾﮕﯽ U از ﻋﻨﺼﺮ ﻫﻤﺎﻧﯽ e ﻣﻮﺟﻮد اﺳﺖﺑﻪ ﻃﻮری ﮐﻪﺑﻪ ازای ﻫﺮ x از N(x) < ϵ ،U. در ﺣﺎل ﺣﺎﺿﺮ، ﭼﮕﻮﻧﮕﯽ ﺳﺎﺧﺖﭘﯿﺶ ﻧﺮم ﭘﯿﻮﺳﺘﻪ در ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی G را ﻧﺸﺎنﻣﯽ دﻫﯿﻢ. در اﺑﺘﺪا ﺗﻌﺮﯾﻒ زﯾﺮ را ﻣﯽ آورﯾﻢ:

ﺗﻌﺮﯾﻒ٢ . ۶. ﻓﺮض ﮐﻨﯿﻢ N ﯾﮏﭘﯿﺶ ﻧﺮم در ﮔﺮوه G ﺑﺎﺷﺪ. دراﯾﻦ ﺻﻮرت ﻣﺠﻤﻮﻋﮥ { ∈ } BN (ϵ) = x G : N(x) < ϵ ﮐﻪ ϵ ﻋﺪدی ﻣﺜﺒﺖ ﺑﺎﺷﺪ را ﯾﮏ N -ﮔﻮی ﺑﻪ ﺷﻌﺎع ϵ ﻣﯽ ﻧﺎﻣﻨﺪ. (BN (ϵ ﻣﺠﻤﻮﻋﻪ ﻫﺎی ﺑﺎزی ﻫﺴﺘﻨﺪ ﻫﺮﮔﺎه N ﯾﮏﭘﯿﺶ ﻧﺮم ﭘﯿﻮﺳﺘﻪ ﺑﺎﺷﺪ. { ∈ } ﻟﻢ٢ . ٧. ﻓﺮض ﮐﻨﯿﻢ Un : n ω دﻧﺒﺎﻟﻪ ای ازﻫﻤﺴﺎﯾﮕﯽ ﻫﺎی ﻣﺘﻘﺎرن ﺑﺎز ﻋﻨﺼﺮ ﻫﻤﺎﻧﯽ e در ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی G U n ∈ ω ⊃ ٢ G N U ﺑﺎﺷﺪﺑﻪ ﻃﻮری ﮐﻪﺑﻪ ازای ﻫﺮ ، n ١+n . آﻧﮕﺎهﭘﯿﺶ ﻧﺮﻣﯽ ﻣﺎﻧﻨﺪ ﺑﺮروی ﻣﻮﺟﻮد اﺳﺖﺑﻪ ﻃﻮری ﮐﻪ ﺷﺮط زﯾﺮ {ﺑﺮﻗﺮار اﺳﺖ: } { } ٢ ∋ ⊃ ⊃ ١ ∋ x G : N(x) < n Un x G : N(x) < (٢n (PN۴ ٢ . ﺑﻨﺎﺑﺮاﯾﻦ،ﭘﯿﺶ ﻧﺮم N ﭘﯿﻮﺳﺘﻪ اﺳﺖ. ﻋﻼوه ﺑﺮاﯾﻦ، اﮔﺮﻣﺠﻤﻮﻋﻪ ﻫﺎی Un ﭘﺎﯾﺎ ﺑﺎﺷﻨﺪ، دراﯾﻦ ﺻﻮرتﭘﯿﺶ ﻧﺮم N − در G راﻣﯽ ﺗﻮانﺑﻪ ﮔﻮﻧﻪ ای اﻧﺘﺨﺎب ﮐﺮد ﮐﻪﺑﻪ ازای ﻫﺮ x و y از G، ﺷﺮط (N(y = (١ N(xyx ﺑﺮﻗﺮار ﺑﺎﺷﺪ.

١۵ ﻣﺘﺮی ﭘﺬﯾﺮی ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی

ﻗﻀﯿﻪ٢ . ٨ (ﻣﺎرﮐﻒ ). ﻓﺮضﻣﯽ ﮐﻨﯿﻢ G ﯾﮏ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی و e ﯾﮏ ﻋﻨﺼﺮ ﻫﻤﺎﻧﯽ از G ﺑﺎﺷﺪ، آﻧﮕﺎهﺑﻪ ازای ﻫﺮ ﻫﻤﺴﺎﯾﮕﯽ ﺑﺎز U از e،ﭘﯿﺶ ﻧﺮمﭘﯿﻮﺳﺘﻪ ای ﻣﺎﻧﻨﺪ N در G ﻣﻮﺟﻮد اﺳﺖﺑﻪ ﻃﻮری ﮐﻪ ﮔﻮی واﺣﺪ BN ﻣﺸﻤﻮل در U اﺳﺖ( ﻣﺮاﺟﻌﻪ ﺑﻪ [٢] ﺷﻮد).

٣. ﻧﺘﺎﯾﺞ اﺻﻠﯽ در اﯾﻦ ﺑﺨﺶ ﮐﺎرﺑﺮد ﺑﺴﯿﺎر ﻣﻬﻤﯽ از ﻟﻢ ٢ . ٧ را ﺑﯿﺎنﻣﯽ ﮐﻨﯿﻢ . ﺑﺎ اﺳﺘﻔﺎده از اﯾﻦ ﻟﻢ (و ﯾﺎ ﺑﺎ اﺳﺘﻔﺎده از ﻗﻀﯿﮥ ﻣﺎرﮐﻒ ٢ . ٨ ﮐﻪ در اﺛﺒﺎت ﻗﻀﯿﮥ زﯾﺮ ﺑﻪ اﻧﺪازۀ ﻟﻢ ٢ . ٧ ﮐﺎراﯾﯽ ﻧﺪارد ) ﻧﺸﺎنﻣﯽ دﻫﯿﻢ ﮐﻪﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژیﻣﺘﺮی ﭘﺬﯾﺮﻧﺪ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ آﻧﻬﺎ ﺷﻤﺎرای ﻧﻮع اول ﺑﺎﺷﻨﺪ ﮐﻪ در اﺑﺘﺪا ﺗﻌﺮﯾﻒ زﯾﺮ را ﺑﯿﺎنﻣﯽ ﮐﻨﯿﻢ:

ﺗﻌﺮﯾﻒ٣ . ١. ﻣﺘﺮﯾﮏ d را روی ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی G راﺳﺖ-ﭘﺎﯾﺎ (ﭼﭗ-ﭘﺎﯾﺎ ) ﮔﻮﺋﯿﻢ ﻫﺮﮔﺎهﺑﻪ ازای ﻫﺮ x, y, z ∈ G ، .( d(zx, zy) = d(x, y) ) d(xz, yz) = d(x, y) ﻗﻀﯿﻪ٣ . ٢ (ﺑﯿﺮﺧﻮف - ﮐﺎﮐﻮﺗﺎﻧﯽ ). ﯾﮏ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی G ﻣﺘﺮی ﭘﺬﯾﺮ (ﺗﻮﺳﻂ ﻣﺘﺮﯾﮏ ﻫﺎی راﺳﺖ-ﭘﺎﯾﺎ) اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ آن ﺷﻤﺎرای ﻧﻮع اول ﺑﺎﺷﺪ.

ﺑﺮﻫﺎن. ﻓﺮضﻣﯽ ﮐﻨﯿﻢ G ﯾﮏ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺷﻤﺎرای ﻧﻮع اول ﺑﺎﺷﺪ. ﻣﯽ ﺧﻮاﻫﯿﻢ ﺛﺎﺑﺖ ﮐﻨﯿﻢ ﮐﻪ G ﻣﺘﺮی ﭘﺬﯾﺮ { ∈ } اﺳﺖ. ﺑﺮای اﯾﻦ ﻣﻨﻈﻮر، ﭘﺎﯾﮥ ﺷﻤﺎرای Wn : n ω ﺑﺮای ﻓﻀﺎی G در ﻧﻘﻄﮥ e را ﺛﺎﺑﺖ در ﻧﻈﺮﻣﯽ ﮔﯿﺮﯾﻢ. ﺑﺎ { ∈ } اﺳﺘﻔﺎده از اﺳﺘﻘﺮاء، ﯾﮏ دﻧﺒﺎﻟﻪ ﻣﺎﻧﻨﺪ Un : n ω ازﻫﻤﺴﺎﯾﮕﯽ ﻫﺎی ﺑﺎز ﻣﺘﻘﺎرن e ﺑﺪﺳﺖﻣﯽ آورﯾﻢﺑﻪﻃﻮری ﮐﻪ ⊂ ⊂ ∈ ﺑﻪ ازای ﻫﺮ Un Wn ،n ω و Un ١+Un. اﯾﻦ دﻧﺒﺎﻟﻪ ﻧﯿﺰ ﯾﮏ ﭘﺎﯾﻪ ﺑﺮای G در ﻧﻘﻄﮥ e اﺳﺖ. دراﯾﻦ ﺻﻮرت n ∈ ω G N ﺑﻨﺎﺑﺮ ﻟﻢ ٢ {. ٧ ﭘﯿﺶ ﻧﺮمﭘﯿﻮﺳﺘﻪ ای ﻣﺎﻧﻨﺪ} در ﻣﻮﺟﻮد اﺳﺖﺑﻪ ﻃﻮری ﮐﻪﺑﻪ ازای ﻫﺮ ، ⊃ ١ ∋ ١ ١ BN ( n ) BN ( n ) = x G : N(x) < n Un ٢ ٢ . از اﯾﻨﺠﺎ ﻧﺘﯿﺠﻪﻣﯽ ﮔﯿﺮﯾﻢ ﮐﻪﻣﺠﻤﻮﻋﻪ ﻫﺎی ﺑﺎز ٢ ﻧﯿﺰ ﺗﺸﮑﯿﻞ ﯾﮏ ﭘﺎﯾﻪ ﺑﺮای G در ﻧﻘﻄﮥ e ﻣﯽ دﻫﻨﺪ. − × → R اﯾﻨﮏ،ﺑﻪ ازای x و y دﻟﺨﻮاه از G، ﺗﺎﺑﻊ ϱN : G G ﺑﺎ ﺿﺎﺑﻄﮥ (١ ϱN (x, y) = N(xy ﺗﻌﺮﯾﻒ ﻣﯽ ﮐﻨﯿﻢ و ﻧﺸﺎنﻣﯽ دﻫﯿﻢ ϱN ﯾﮏ ﻣﺘﺮ روی G اﺳﺖ ﮐﻪ ﺗﻮﭘﻮﻟﻮژی اﺻﻠﯽ G را ﺗﻮﻟﯿﺪﻣﯽ ﮐﻨﺪ. ﭼﻮن N ﭘﯿﺶ ﻧﺮم و ﻫﺮ − ≥ ﭘﯿﺶ ﻧﺮم ﻧﺎﻣﻨﻔﯽ اﺳﺖ،ﺑﻪ ازای ﻫﺮ x و y از G، ٠ (١ ϱN (x, y) = N(xy. از اﺻﻞ ﻣﻮﺿﻮع (PN٣) ﺗﻌﺮﯾﻒ − ∈ ٢ . ١ﻧﺘﯿﺠﻪﻣﯽ ﮔﯿﺮﯾﻢ ﮐﻪﺑﻪ ازای ﻫﺮ x G، ٠ = (N(e = (١ ϱN (x, x) = N(xx. اﯾﻨﮏ ﻓﺮضﻣﯽ ﮐﻨﯿﻢ ∋ ١ ١− N(xy ) = < n n ω ϱN (x, y) = ٠ ﭘﺲﺑﻪ ازای ﻫﺮ ، ٢ ٠ ∩. از اﯾﻨﺠﺎ ﻧﺘﯿﺠﻪﻣﯽ ﮔﯿﺮﯾﻢ ﮐﻪﺑﻪ ازای ﻫﺮ ∋ ⊃ ١ ∋ ١− { } ١− xy = e e = ∈ Un xy BN ( n ) Un n ω ، ٢ . از ﻃﺮﻓﯽ دﯾﮕﺮ ﭼﻮن n ω ، . ﺑﻪ اﯾﻦ ﻣﻌﻨﯽ ﮐﻪ x = y. دراﯾﻦ ﺻﻮرت اﺻﻞ ﻣﻮﺿﻮع (M١) ﺗﻌﺮﯾﻒ ﻣﺘﺮ ﺑﺮﻗﺮار اﺳﺖ. واﺿﺢ اﺳﺖ ﮐﻪ، (ϱN (x, y) = ϱN (y, x. دراﯾﻦ ﺻﻮرت ϱN در اﺻﻞ ﻣﻮﺿﻮع (M٢) ﺻﺎدق اﺳﺖ. ﺑﺮای ﺗﺤﻘﯿﻖ در اﺻﻞ ﻣﻮﺿﻮع (M٣) ﻓﺮضﻣﯽ ﮐﻨﯿﻢ x و y و z اﻋﻀﺎی دﻟﺨﻮاﻫﯽ از G ﺑﺎﺷﻨﺪ. دراﯾﻦ ﺻﻮرت دارﯾﻢ: − (١ ϱN (x, z) = N(xz (١−١yz−N(xy = − − (ﺑﻨﺎﺑﺮ (PN٢) ﺗﻌﺮﯾﻒ ٢ . ١) (١ N(yz + (١ N(xy ≥ = ϱN (x, y) + ϱN (y, z) ∈ ﺑﻨﺎﺑﺮاﯾﻦ، ϱN ﯾﮏ ﻣﺘﺮ روی G اﺳﺖ. ﻣﺘﺮﯾﮏ ϱN ﯾﮏ ﻣﺘﺮﯾﮏ راﺳﺖ- ﭘﺎﯾﺎﺳﺖ، زﯾﺮاﺑﻪ ازای ﻫﺮ x, y, z G، − − − .(ϱN (x, y = (١ N(xy = (١ ١y ϱN (xz, yz) = N(xzz

واﺿﺢ اﺳﺖ ﮐﻪ (ϱN ،BN (ϵ -ﻫﻤﺴﺎﯾﮕﯽ ﮔﻮی ﻣﺎﻧﻨﺪ از e ﺑﻪ ﺷﻌﺎع ϵ ﻣﯽ ﺑﺎﺷﺪ. از اﯾﻨﺠﺎ ﻧﺘﯿﺠﻪﻣﯽ ﮔﯿﺮﯾﻢ ﮐﻪ ϱN -ﻫﻤﺴﺎﯾﮕﯽ ﮔﻮی ﻣﺎﻧﻨﺪ از ﯾﮏ ﻧﻘﻄﮥ x از G ﺑﻪ ﺷﻌﺎع ϵ دﻗﯿﻘﺎً ﻣﺠﻤﻮﻋﮥ BN (ϵ)x اﺳﺖ. اﯾﻨﮏ ﻧﻘﻄﮥ x از G را ١ G e G BN ( n ) دﻟﺨﻮاه در ﻧﻈﺮﻣﯽ ﮔﯿﺮﯾﻢ. ﭼﻮنﻣﺠﻤﻮﻋﻪ ﻫﺎی ٢ ﺗﺸﮑﯿﻞ ﯾﮏ ﭘﺎﯾﻪ ﺑﺮای در ﻣﯽ دﻫﻨﺪ و ﻫﻤﭽﻨﯿﻦ ﯾﮏ ١ ϱN x G BN ( n )x ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ اﺳﺖ،ﻣﺠﻤﻮﻋﻪ ﻫﺎی ٢ ﻧﯿﺰ ﺗﺸﮑﯿﻞ ﯾﮏ ﭘﺎﯾﻪ ﺑﺮای در ﻣﯽ دﻫﻨﺪ. ﺑﻪ ﻋﺒﺎرﺗﯽ دﯾﮕﺮ،

١۶ ش. ﻣﻨﺼﻮری

١ -ﻫﻤﺴﺎﯾﮕﯽ ﮔﻮی ﻣﺎﻧﻨﺪ از e ﺑﻪ ﺷﻌﺎع n ﺗﺸﮑﯿﻞ ﯾﮏ ﭘﺎﯾﻪ ﺑﺮای ﻓﻀﺎی G در ﻧﻘﻄﮥ x ﻣﯽ دﻫﻨﺪ. ﺑﻨﺎﺑﺮاﯾﻦ، ﻣﺘﺮﯾﮏ ٢ □ ϱN ﺗﻮﭘﻮﻟﻮژی اﺻﻠﯽ ﻓﻀﺎی G را ﺗﻮﻟﯿﺪﻣﯽ ﮐﻨﺪ. ﺑﻪ اﯾﻦ ﻣﻌﻨﯽ ﮐﻪ، G ﻣﺘﺮی ﭘﺬﯾﺮ اﺳﺖ. در واﻗﻊﻣﯽ ﺗﻮاﻧﯿﻢ ﻗﻀﯿﮥ ٣ . ٢ راﺑﻪ ﺻﻮرت زﯾﺮ ﺗﮑﻤﯿﻞ ﮐﻨﯿﻢ: ﻧﺘﯿﺠﻪ٣ . ٣. ﻫﺮ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺷﻤﺎرای ﻧﻮع اول، ﻣﺘﺮﯾﮏ ﻫﺎی راﺳﺖ- ﭘﺎﯾﺎ و ﭼﭗ -ﭘﺎﯾﺎ را اذﻋﺎنﻣﯽ ﮐﻨﺪﺑﻪ ﻃﻮری ﮐﻪ ﺗﻮﭘﻮﻟﻮژی ﻫﺮ دوی آﻧﻬﺎ ﺑﺎ ﺗﻮﭘﻮﻟﻮژی اﺻﻠﯽ ﯾﮑﺴﺎن اﺳﺖ. ﺗﻮﺟﻪ٣ . ۴. ﮔﺮوه ﻫﺎی ﭘﺎرا ﺗﻮﭘﻮﻟﻮژی ﻧﯿﺎزی ﻧﯿﺴﺖ ﮐﻪﻣﺘﺮی ﭘﺬﯾﺮ ﺑﺎﺷﻨﺪ. ﻣﺜﺎل٣ . ۵ (ﺧﻂ ﺳﻮر ﺟﻨﻔﺮی). ﺧﻂ ﺳﻮر ﺟﻨﻔﺮی ﯾﮏ ﮔﺮوه ﭘﺎرا ﺗﻮﭘﻮﻟﻮژﯾﮏ اﺳﺖ ﮐﻪﻣﺘﺮی ﭘﺬﯾﺮ ﻧﯿﺴﺖ وﻟﯽ ﺑﻪ آﺳﺎﻧﯽ ﺛﺎﺑﺖﻣﯽ ﺷﻮد ﮐﻪ ﻫﺮ زﯾﺮ ﻓﻀﺎی ﻓﺸﺮده ﺧﻂ ﺳﻮرﺟﻨﻔﺮی ﺷﻤﺎراﺳﺖ و درﻧﺘﯿﺠﻪﻣﺘﺮی ﭘﺬﯾﺮ اﺳﺖ.

ﻣﺮاﺟﻊ 1. A. Arhangel’skii, M. Tkachenko, Topological groups and related structures, Atlantis Press, Paris; World Sci. Publ., NJ, (2008).

2. A. A. Markov, On the existence of periodic connected topological groups, Izv. Akad. Nauk SSSR 8, pp. 225-232, (1944).

3. G. Birkhoff, A note on topological groups, Comput. Math. 3, pp. 427-130, (1936).

4. R. Engelking, General Topology, Heldermann Verlag, Berlin,( 1989).

١٧ ﻣﺘﺮی ﭘﺬﯾﺮی ﻧﯿﻢ ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی ﮐﻠﯿﻔﻮرد

∗ ﺷﺮﯾﻔﻪ ﻣﻨﺼﻮری ١

١ آدرس ١ [email protected] ٢ ﮔﺮوه رﯾﺎﺿﯽ، داﻧﺸﮑﺪه ﻋﻠﻮم رﯾﺎﺿﯽ، داﻧﺸﮕﺎه ﺳﯿﺴﺘﺎن و ﺑﻠﻮﭼﺴﺘﺎن

ﭼﮑﯿﺪه. در اﯾﻦ ﻣﻘﺎﻟﻪ ﺛﺎﺑﺖﻣﯽ ﮐﻨﯿﻢ ﮐﻪﻧﯿﻢ ﮔﺮوه ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژی S ﻣﺘﺮی ﭘﺬﯾﺮ اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ S ﯾﮏ M -ﻓﻀﺎ و ﻣﺠﻤﻮﻋﮥ ﺧﻮدﺗﻮان ﻫﺎی آن در S ﻣﺘﺮی ﭘﺬﯾﺮ ﺑﺎﺷﺪ و ﻫﻤﭽﻨﯿﻦ، ﻧﺸﺎن ﻣﯽ دﻫﯿﻢ ﮐﻪ ﻣﻌﯿﺎر ﻣﺘﺮی ﭘﺬﯾﺮی ﺑﺮای ﻧﯿﻢ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی ﮐﻠﯿﻔﻮرد ﻓﺸﺮده ﺷﻤﺎرا ﺑﺮﻗﺮار اﺳﺖ.

١. ﻣﻘﺪﻣﻪ ﻣﯽ داﻧﯿﻢ ﮐﻪ ﺷﺮط ﻻزم و ﮐﺎﻓﯽ ﺑﺮایﻣﺘﺮی ﭘﺬﯾﺮیﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی در ﻣﻘﺎﻻت ﻣﺘﻌﺪدی اراﺋﻪ ﺷﺪه اﺳﺖ از آن ﺟﻤﻠﻪ ﻣﯽ ﺗﻮان ﺑﻪ ﻗﻀﯿﮥﻣﺘﺮی ﭘﺬﯾﺮی ﺑﯿﺮﺧﻮف -ﮐﺎﮐﺎﺗﻮﻧﯽ اﺷﺎره ﮐﺮد [١]: ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی G ﻣﺘﺮی ﭘﺬﯾﺮ اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ G ﺷﻤﺎرای ﻧﻮع اول ﺑﺎﺷﺪ. در اﯾﻦ ﻣﻘﺎﻟﻪ ﺷﺮط ﻻزم و ﮐﺎﻓﯽ ﺑﺮایﻣﺘﺮی ﭘﺬﯾﺮی ﻧﯿﻢﮔﺮوه ﻫﺎی ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژی وﻧﯿﻢ ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی ﮐﻠﯿﻔﻮرد را اراﺋﻪﻣﯽ دﻫﯿﻢ در ﻗﻀﯿﮥ (٢ . ۴) ﺛﺎﺑﺖﻣﯽ ﮐﻨﯿﻢ ﮐﻪﻧﯿﻢ ﮔﺮوه ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژی S ﻣﺘﺮی ﭘﺬﯾﺮ اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ S ﯾﮏ M -ﻓﻀﺎ و زﯾﺮﻣﺠﻤﻮﻋﻪ ای ازﺧﻮدﺗﻮان ﻫﺎﯾﺶ در S ﻣﺘﺮی ﭘﺬﯾﺮ ﺑﺎﺷﺪ. ﺳﺮاﻧﺠﺎم، در ﻗﻀﯿﮥ (٣ . ١) ﺛﺎﺑﺖ ﺧﻮاﻫﯿﻢ ﮐﺮد ﮐﻪﻧﯿﻢ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی ﮐﻠﯿﻔﻮرد S ﻣﺘﺮی ﭘﺬﯾﺮ اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ زﯾﺮﻣﺠﻤﻮﻋﻪ ای ازﺧﻮدﺗﻮان ﻫﺎی آن در S ﻣﺘﺮی ﭘﺬﯾﺮ ﺑﺎﺷﺪ. ﻫﻤﮥ ﻓﻀﺎﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ در اﯾﻦ ﻣﻘﺎﻟﻪ ﻣﻨﻈﻢ در ﻧﻈﺮﻣﯽ ﮔﯿﺮﯾﻢ.

2010 Mathematics Subject Classification. Primary 22A15, Secondary 54E35; 54E18; 54D30. واژﮔﺎن ﮐﻠﯿﺪی. ﻣﺘﺮی ﭘﺬﯾﺮی؛ﻧﯿﻢ ﮔﺮوه ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژی؛ ﺧﻮدﺗﻮان؛ M -ﻓﻀﺎ . ∗ ﺳﺨﻨﺮان

١٨ ش. ﻣﻨﺼﻮری

٢. ﻣﻔﺎﻫﯿﻢ و ﻧﺘﺎﯾﺞ ﻣﻘﺪﻣﺎﺗﯽ در اﯾﻦ ﺑﺨﺶ ﺑﺮﺧﯽ از ﺗﻌﺎرﯾﻒ و ﻧﺘﺎﯾﺞ ﮐﻤﮑﯽ را اراﺋﻪﻣﯽ دﻫﯿﻢ. ﻧﯿﻢ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی S، ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژی ﺑﺎ ﯾﮏ ﻋﻤﻞ ﺷﺮﮐﺖ ﭘﺬﯾﺮ ﭘﯿﻮﺳﺘﻪ S × S → S : · اﺳﺖ. ﻋﻨﺼﺮ e از S را ﺧﻮدﺗﻮان ﻧﺎﻣﯿﻢ ﻫﺮﮔﺎه ee = e وزﯾﺮﻣﺠﻤﻮﻋﻪ ای از ﻫﻤﻪﺧﻮدﺗﻮان ﻫﺎی S را ﺗﻮﺳﻂ E ﻧﻤﺎﯾﺶﻣﯽ دﻫﯿﻢ. ﻧﯿﻢ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی S، ﯾﮏﻧﯿﻢ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی ﮐﻠﯿﻔﻮرد اﺳﺖ ﻫﺮﮔﺎه ﮐﻠﯿﻔﻮرد ﺟﺒﺮی ﺑﺎﺷﺪ؛ ﺑﻪ اﯾﻦ ﻣﻌﻨﯽ ﮐﻪﺑﻪ ﺻﻮرت − اﺟﺘﻤﺎﻋﯽ از ﮔﺮوه ﻫﺎ ﺑﺎﺷﺪ. ﺑﻪ ازای ﻫﺮ ﻋﻀﻮ ﻣﺎﻧﻨﺪ x از ﻧﯿﻢ ﮔﺮوه ﮐﻠﯿﻔﻮرد S ﯾﮏ ﻋﻨﺼﺮ ﻣﻨﺤﺼﺮﺑﻔﺮد S ∋ ١ x − − − − − − − ﻣﻮﺟﻮد ﺑﺎﺷﺪﺑﻪ ﻃﻮری ﮐﻪ ١x = x xx و ١ x = ١ ١xx x و ١x x = ١ xx. ﻋﻨﺼﺮ ١ x را ﻣﻌﮑﻮس − − x ﻧﺎﻣﯿﻢ. ﻧﮕﺎﺷﺖ S → S :(·) ﺑﺎ ﺿﺎﺑﻄﮥ ١ x → x : ١ (·) را ﯾﮏ ﻣﻌﮑﻮس روی S ﮔﻮﺋﯿﻢ. ﻧﮕﺎﺷﺖ ﻣﻌﮑﻮس درﻧﯿﻢ ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی ﮐﻠﯿﻔﻮرد ﻟﺰوﻣﺎً ﭘﯿﻮﺳﺘﻪ ﻧﯿﺴﺖ (ﻣﺮاﺟﻌﻪ ﺑﻪ [٣] ﺷﻮد). ﯾﮏﻧﯿﻢ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی ﮐﻠﯿﻔﻮرد ﺑﺎ ﻋﻤﻞ ﻣﻌﮑﻮس ﭘﯿﻮﺳﺘﻪ را ﯾﮏ ﻧﯿﻢ ﮔﺮوه ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژی ﻣﯽ ﻧﺎﻣﯿﻢ. ﺑﺮای ﻧﯿﻢ ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی ﮐﻠﯿﻔﻮرد ﻓﺸﺮدۀ ﺷﻤﺎرا ﻋﻤﻞ ﻣﻌﮑﻮس ﭘﯿﻮﺳﺘﻪدﻧﺒﺎﻟﻪ ای اﺳﺖ. ﯾﺎدآوریﻣﯽ ﮐﻨﯿﻢ ﮐﻪ ﺗﺎﺑﻊ f : X → Y ﺑﯿﻦ ﻓﻀﺎﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ، Y (f(x ))∞ X (x )∞ ﭘﯿﻮﺳﺘﮥدﻧﺒﺎﻟﻪ ای اﺳﺖ ﻫﺮﮔﺎهﺑﻪ ازای ﻫﺮ دﻧﺒﺎﻟﻪ ﻣﺎﻧﻨﺪ ١=n n در ، ﯾﮏ دﻧﺒﺎﻟﮥ ١=n n در ﺑﺎﺷﺪ ﮐﻪ (limn→∞ f(xn) = f(limn→∞ xn. ﺑﺮﻫﺎن ﻗﻀﯿﮥ زﯾﺮ ﺗﻮﺳﻂ Gutik و P agon و Repovs در [٣] ﺛﺎﺑﺖ ﺷﺪه اﺳﺖ. ﻗﻀﯿﻪ٢ . ١. ﺑﺮای ﻫﺮﻧﯿﻢ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی ﮐﻠﯿﻔﻮرد ﻓﺸﺮدۀ ﺷﻤﺎرای S ﻋﻤﻞ ﻣﻌﮑﻮس S → S :(·)، ﭘﯿﻮﺳﺘﮥدﻧﺒﺎﻟﻪ ای اﺳﺖ. ﺑﺮﺧﯽ از ﺷﺮاﯾﻂ دﯾﮕﺮ ﻧﯿﺰ ﺗﻀﻤﯿﻦﻣﯽ ﮐﻨﻨﺪ ﮐﻪ ﯾﮏﻧﯿﻢ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی ﮐﻠﯿﻔﻮرد ﻓﺸﺮدۀ ﺷﻤﺎرا، ﯾﮏﻧﯿﻢ ﮔﺮوه ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژی اﺳﺖ. ﮐﻪ از آن ﺟﻤﻠﻪ ﺷﺮاﯾﻂ، ﺧﺎﺻﯿﺖﺑﻪ ﻃﻮر ﻣﺘﻨﺎوب ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ اﺳﺖ. ﻧﯿﻢ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی S راﺑﻪ ﻃﻮر ﻣﺘﻨﺎوب ﺗﻮﭘﻮﻟﻮژﯾﮏ ﻧﺎﻣﯿﻢ ﻫﺮﮔﺎه ﻫﺮ ﻋﻨﺼﺮ x ∈ S ﺑﻪ ﻃﻮر ﻣﺘﻨﺎوب ﺗﻮﭘﻮﻟﻮژﯾﮏ ﺑﺎﺷﺪ ﺑﻪ اﯾﻦ ﻣﻌﻨﺎ ﮐﻪ؛ ﺑﻪ ازای ﻫﺮ n ∈ ≥ ⊂ ﻫﻤﺴﺎﯾﮕﯽ Ox S از x ﯾﮏ ﻋﺪد ﺻﺤﯿﺢ ﻣﺎﻧﻨﺪ ٢ n ﻣﻮﺟﻮد ﺑﺎﺷﺪﺑﻪ ﻃﻮری ﮐﻪ x Ox. اﺛﺒﺎت ﻗﻀﯿﮥ زﯾﺮ را ﻣﯽ ﺗﻮان در [٣] ﯾﺎﻓﺖ. ﻗﻀﯿﻪ٢ . ٢. ﻓﺮض ﮐﻨﯿﻢ S ﯾﮏﻧﯿﻢ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﮐﻠﯿﻔﻮرد ﻓﺸﺮدۀ ﺷﻤﺎرا ﺑﺎﺷﺪ. اﮔﺮ S ﺑﻪ ﻃﻮر ﻣﺘﻨﺎوب ﺗﻮﭘﻮﻟﻮژﯾﮏ و در ﻫﺮ ﺧﻮدﺗﻮان e ∈ E ﺷﻤﺎرای ﻧﻮع اول ﺑﺎﺷﺪ آﻧﮕﺎه S ﯾﮏﻧﯿﻢ ﮔﺮوه ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژی اﺳﺖ. ﺑﺎ اﺳﺘﻔﺎده از اﺻﻠﯿﺖﻧﯿﻢ ﮔﺮو ه ﻫﺎی ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژی ﻧﺸﺎنﻣﯽ دﻫﯿﻢ ﮐﻪﻧﯿﻢ ﮔﺮوه ﻫﺎی ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژی دارای ﮐﺮان ﺑﺎﻻی (S) ≤ ∆(E) · ψ(E,S)∆ ﻫﺴﺘﻨﺪ. ﯾﺎدآوریﻣﯽ ﮐﻨﯿﻢ ﮐﻪ ﺑﺮای ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژی X و زﯾﺮﻣﺠﻤﻮﻋﮥ A ⊂ X، • ﻣﺸﺨﺼﻪ ﻧﻤﺎی (ψ(A, X از A در X، ﺑﺮاﺑﺮ ﺑﺎ ﮐﻮﭼﮑﺘﺮﯾﻦ ﮐﺎردﯾﻨﺎل |U| ( ﮐﻪ در آن U ﺧﺎﻧﻮاده ای از زﯾﺮﻣﺠﻤﻮﻋﻪ ﻫﺎی ﺑﺎز X ﺑﺎﺷﺪ ) اﺳﺖﺑﻪ ﻃﻮری ﮐﻪ U = A∩. { ∈ } × • ﻋﺪد ﻗﻄﺮی (X) = ψ(∆X ,X X)∆، ﺑﺮاﺑﺮ ﺑﺎ ﻣﺸﺨﺼﻪ ﻧﻤﺎی ﻗﻄﺮ X = (x, x): x X∆ در ﻣﺮﺑﻊ X × X اﺳﺖ. ≤ N ﮔﻮﺋﯿﻢ ﮐﻪ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ X دارای Gδ -ﻗﻄﺮ اﺳﺖ ﻫﺮﮔﺎه ٠ (X)∆. ﺑﺮای ﻧﯿﻢ ﮔﺮوه ﻫﺎی ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژی اﺛﺒﺎت ﻗﻀﯿﮥ زﯾﺮ راﻣﯽ ﺗﻮان در ﻗﻀﯿﮥ (١٠)٣ · ٢ از [٢] ﯾﺎﻓﺖ . ﻗﻀﯿﻪ٢ . ٣. اﮔﺮ S ﯾﮏﻧﯿﻢ ﮔﺮوه ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژی و {E = {e ∈ S : ee = e زﯾﺮﻣﺠﻤﻮﻋﻪ ای ازﺧﻮدﺗﻮان ﻫﺎی S ﺑﺎﺷﺪ آﻧﮕﺎه (S) ≤ ∆(E) · ψ(E,S)∆. در ﭘﺎﯾﺎن اﯾﻦ ﺑﺨﺶ ﻧﺸﺎنﻣﯽ دﻫﯿﻢ ﮐﻪ ﻣﺤﮏﻣﺘﺮی ﭘﺬﯾﺮیﺑﻪ ﻃﻮر ﮐﻠﯽ ﺑﺮایﻧﯿﻢ ﮔﺮوﻫﻬﺎی ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژی ﮐﻪ M -ﻓﻀﺎ ﻫﺴﺘﻨﺪ ﺑﺮﻗﺮار اﺳﺖ. ﯾﺎدآوریﻣﯽ ﮐﻨﯿﻢ، ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژی X ﯾﮏ M -ﻓﻀﺎ اﺳﺖ ﻫﺮﮔﺎه ﯾﮏ دﻧﺒﺎﻟﻪ ازﭘﻮﺷﺶ ﻫﺎی U U U ﺑﺎز X ﻣﺎﻧﻨﺪ n)n∈ω ) ﻣﻮﺟﻮد ﺑﺎﺷﺪﺑﻪ ﻃﻮری ﮐﻪ ﻫﺮ ﭘﻮﺷﺶ ١+n ﺗﻈﺮﯾﻒﺳﺘﺎره ای از ﭘﻮﺷﺶ n ﺑﺎﺷﺪ وﺑﻪ ازای ∈ U ∈ ∈ ﻫﺮ x X و n ω دﻧﺒﺎﻟﮥ (xn (x, n دارای ﯾﮏ ﻧﻘﻄﮥ ﺣﺪی در X ﺑﺎﺷﺪ. ﻓﻀﺎﻫﺎی ﻓﺸﺮدۀ ﺷﻤﺎرا، M · · -ﻓﻀﺎ ﻫﺴﺘﻨﺪ( ﺑﺒﯿﻨﯿﺪ ﺑﺨﺶ ۵ ٣ از [۵] ). ﺑﻨﺎﺑﺮ ﻗﻀﯿﮥ ٨ ٣ از [۵]، ﻫﺮ M -ﻓﻀﺎ ﺑﺎ ﯾﮏ Gδ -ﻗﻄﺮ،ﻣﺘﺮی ﭘﺬﯾﺮ اﺳﺖ. اﯾﻨﮏ ﺑﺎ ﺗﺮﮐﯿﺐ اﯾﻦ واﻗﻌﯿﺖ و ﻗﻀﯿﮥ ٢ . ٣، ﻗﻀﯿﮥﻣﺘﺮی ﭘﺬﯾﺮی ﺑﺮایﻧﯿﻢ ﮔﺮوه ﻫﺎی ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژیﺑﻪ ﺻﻮرت زﯾﺮ دارﯾﻢ:

١٩ ﻣﺘﺮی ﭘﺬﯾﺮی ﻧﯿﻢ ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی ﮐﻠﯿﻔﻮرد

ﻗﻀﯿﻪ٢ . ۴. ﻧﯿﻢ ﮔﺮوه ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژی S ﻣﺘﺮی ﭘﺬﯾﺮ اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ S ﯾﮏ M -ﻓﻀﺎ و E ﯾﮏ Gδ -ﻣﺠﻤﻮﻋﮥ ﻣﺘﺮی ﭘﺬﯾﺮ در S ﺑﺎﺷﺪ.

٣. ﻣﺘﺮی ﭘﺬﯾﺮیﻧﯿﻢ ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی ﮐﻠﯿﻔﻮرد ﻓﺸﺮده ﺷﻤﺎرا ﻗﻀﯿﮥ ٢ . ۴ ﻧﺸﺎنﻣﯽ دﻫﺪ ﮐﻪﻧﯿﻢ ﮔﺮوه ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژی ﻓﺸﺮده ﺷﻤﺎرا S ﻣﺘﺮی ﭘﺬﯾﺮ اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ ﻣﺠﻤﻮﻋﮥ E از ﺧﻮدﺗﻮان ﻫﺎی S ﯾﮏ Gδ -ﻣﺠﻤﻮﻋﮥ ﻣﺘﺮی ﭘﺬﯾﺮ در S ﺑﺎﺷﺪ. در اﯾﻦ ﺑﺨﺶ، اﯾﻦ ﻣﻌﯿﺎر ﻣﺘﺮی ﭘﺬﯾﺮی را ﺑﺮای ﻧﯿﻢ ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﮐﻠﯿﻔﻮرد ﻓﺸﺮدۀ ﺷﻤﺎرا ﺗﻌﻤﯿﻢﻣﯽ دﻫﯿﻢ. ﻗﻀﯿﻪ٣ . ١. ﻧﯿﻢ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی ﮐﻠﯿﻔﻮرد ﻓﺸﺮده ﺷﻤﺎرای S ﻣﺘﺮی ﭘﺬﯾﺮ اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ ﻣﺠﻤﻮﻋﮥ E ازﺧﻮدﺗﻮان ﻫﺎی S ﯾﮏ Gδ -ﻣﺠﻤﻮﻋﮥﻣﺘﺮی ﭘﺬﯾﺮ در S ﺑﺎﺷﺪ. ﺑﺮﻫﺎن. ﻓﺮضﻣﯽ ﮐﻨﯿﻢ E ﯾﮏ Gδ -ﻣﺠﻤﻮﻋﮥﻣﺘﺮی ﭘﺬﯾﺮ در S ﺑﺎﺷﺪ. ﭘﯿﻮﺳﺘﮕﯽ ﻋﻤﻞﻧﯿﻢ ﮔﺮوه ﻫﺎ روی S ﻧﺸﺎن ﻣﯽ دﻫﺪ ﮐﻪ ﻣﺠﻤﻮﻋﮥﺧﻮدﺗﻮان ﻫﺎی {E = {e ∈ S : ee = e در S ﺑﺴﺘﻪ اﺳﺖ و در ﻧﺘﯿﺠﻪ E ﻓﺸﺮدۀ ﺷﻤﺎرا اﺳﺖ. دراﯾﻦ ﺻﻮرت ﺑﻨﺎﺑﺮﻣﺘﺮی ﭘﺬﯾﺮی، ﻓﻀﺎی ﻓﺸﺮدۀ ﺷﻤﺎرای E ﻓﺸﺮده اﺳﺖ. ﻓﻀﺎی S در ﻫﺮ ﻧﻘﻄﮥ e ∈ E ﺷﻤﺎرای ﻧﻮع اول اﺳﺖ. ﺑﺮﻫﺎن. ﭼﻮن E ﯾﮏ Gδ -ﻣﺠﻤﻮﻋﮥﻣﺘﺮی ﭘﺬﯾﺮ در S اﺳﺖ. ﻣﯽ ﺗﻮان ﻧﺸﺎن داد ﮐﻪ ﺗﮏ ﻋﻀﻮی S S G {e} ⊂ E ﯾﮏ δ -ﻣﺠﻤﻮﻋﻪ در اﺳﺖ. از ﻃﺮﻓﯽ دﯾﮕﺮ ﻣﻨﻈﻢ اﺳﺖ دراﯾﻦ ﺻﻮرتﻣﯽ ∩ﺗﻮان دﻧﺒﺎﻟﮥ ﮐﺎﻫﺸﯽ { } ﺷﻤﺎرای Un)n∈ω) از زﯾﺮﻣﺠﻤﻮﻋﻪ ﻫﺎی ﺑﺎز S را ﺑﻪ ﮔﻮﻧﻪ ای اﻧﺘﺨﺎب ﮐﺮد ﮐﻪ n∈ω Un = e . اﯾﻨﮏ ﻓﺸﺮده ⊂ ∈ ⊂ ﺷﻤﺎراﯾﯽ S ﻧﺘﯿﺠﻪﻣﯽ دﻫﺪ ﮐﻪﺑﻪ ازای ﻫﺮ ﻫﻤﺴﺎﯾﮕﯽ U S از n ω ،eای وﺟﻮد داردﺑﻪ ﻃﻮری ﮐﻪ Un U. ﯾﺎ { } ﺑﻪ ﻋﺒﺎرﺗﯽ دﯾﮕﺮ، Un n∈ω ﯾﮏ ﻫﻤﺴﺎﯾﮕﯽﭘﺎﯾﻪ ای در e اﺳﺖ. ﺑﻨﺎﺑﺮاﯾﻦ S در e ﺷﻤﺎرای ﻧﻮع اول اﺳﺖ. ﻣﻌﮑﻮس S → S :(·) در ﻫﺮ ﺧﻮدﺗﻮان e ∈ E ﭘﯿﻮﺳﺘﻪ اﺳﺖ. ﺑﺮﻫﺎن. ﺑﻨﺎﺑﺮ ﻗﻀﯿﮥ ٢ . ١، ﻋﻤﻞ ﻣﻌﮑﻮس ﭘﯿﻮﺳﺘﻪدﻧﺒﺎﻟﻪ ای اﺳﺖ. دراﯾﻦ ﺻﻮرت ﻋﻤﻞ ﻣﻌﮑﻮس در ﻫﺮ ﻧﻘﻄﮥ x ∈ S ﺑﺎ داﺷﺘﻦ ﻫﻤﺴﺎﯾﮕﯽﭘﺎﯾﻪ ای ﺷﻤﺎرا، ﭘﯿﻮﺳﺘﻪ اﺳﺖ. ﺑﻪ وﯾﮋه، {ﻋﻤﻞ ﻣﻌﮑﻮس در ﻫﺮ }ﺧﻮدﺗﻮان e ∈ E ﭘﯿﻮﺳﺘﻪ اﺳﺖ. ادﻋﺎ ٣ : ﺑﺮای ﻫﺮ ﺧﻮدﺗﻮان e ∈ E، ﮔﺮوه ﻣﺎﮐﺴﯿﻤﺎل ∈ − e = ١ He = x S : xx ﯾﮏ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮑﯽﻣﺘﺮی ﭘﺬﯾﺮ اﺳﺖ. ﺑﺮﻫﺎن. ﭘﯿﻮﺳﺘﮕﯽ ﻋﻤﻞ ﻧﯿﻢ ﮔﺮوه روی S ﻧﺸﺎنﻣﯽ دﻫﺪ ﮐﻪ He ﯾﮏ ﮔﺮوه ﭘﺎراﺗﻮﭘﻮﻟﻮژﯾﮑﯽ اﺳﺖ. ﺑﻨﺎﺑﺮ ادﻋﺎی ٣، ﻣﻌﮑﻮس S در ﻫﺮ ﺧﻮدﺗﻮان e ﭘﯿﻮﺳﺘﻪ اﺳﺖ. · · در ﻧﺘﯿﺠﻪ ﻋﻤﻞ ﻣﻌﮑﻮس ﮔﺮوه ﭘﺎراﺗﻮﭘﻮﻟﻮژی He در e ﭘﯿﻮﺳﺘﻪ اﺳﺖ. ﭘﺲ ﺑﻨﺎﺑﺮ ﻗﻀﯿﮥ ١٢ ٣ ١ از [١]، He ﯾﮏ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ اﺳﺖ. ﭼﻮن S در e ﺷﻤﺎرای ﻧﻮع اول اﺳﺖ، ﮔﺮوه ﭘﺎراﺗﻮﭘﻮﻟﻮژﯾﮑﯽ He ﻧﯿﺰ ﺷﻤﺎرای ﻧﻮع اول اﺳﺖ. · · اﯾﻨﮏ ﺑﻨﺎﺑﺮ ﻗﻀﯿﮥ ﺑﯿﺮﺧﻮف -ﮐﺎﮐﺎﺗﻮﻧﯽ (ﻗﻀﯿﮥ ١٢ ٣ ٣ از [١]) ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی He ﻣﺘﺮی ﭘﺬﯾﺮ اﺳﺖ. ﻧﯿﻢ ﮔﺮوه S ﺑﻪ ﻃﻮر ﻣﺘﻨﺎوب ﺗﻮﭘﻮﻟﻮژﯾﮏ اﺳﺖ. ﺑﺮﻫﺎن. ﯾﮏ ﻧﻘﻄﻪ ﻣﺎﻧﻨﺪ a ∈ S در ﻧﻈﺮﻣﯽ ﮔﯿﺮﯾﻢ و ﻣﺠﻤﻮﻋﮥ (ﺑﺴﺘﻪ) ﻫﻤﮥ ﻧﻘﺎط n ﺣﺪی از دﻧﺒﺎﻟﮥ {a : n ∈ N} در S را ﺗﻮﺳﻂ A ﻧﻤﺎﯾﺶﻣﯽ دﻫﯿﻢ. ﭘﯿﻮﺳﺘﮕﯽ ﻋﻤﻞﻧﯿﻢ ﮔﺮوه روی S ﻧﺸﺎنﻣﯽ دﻫﺪ ﮐﻪ A ﯾﮏزﯾﺮﻧﯿﻢ ﮔﺮوهﺟﺎﺑﺠﺎﯾﯽ ﭘﺬﯾﺮ ﺑﺴﺘﻪ در S اﺳﺖ. اﮐﻨﻮن ﻧﮕﺎﺷﺖ ﺗﺼﻮﯾﺮ از S ﺑﻪ ﺗﻮی E را ﺗﻮﺳﻂ π : S → E − − ﻧﻤﺎﯾﺶ داده و ﺑﺎ ﺿﺎﺑﻄﮥ ١x x = ١ π(x) = xx ﺗﻌﺮﯾﻒﻣﯽ ﮐﻨﯿﻢ واﺿﺢ اﺳﺖ ﮐﻪ، ﻧﮕﺎﺷﺖ π ﻟﺰوﻣﺎً ﭘﯿﻮﺳﺘﻪ ﻧﯿﺴﺖ. ادﻋﺎﻣﯽ ﮐﻨﯿﻢ ﮐﻪ π(A) ⊂ E دارای ﻋﻨﺼﺮ ﻣﯿﻨﯿﻤﺎل اﺳﺖ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ زوج ﻣﺮﺗﺐ x, y روی E ﮐﻪﺑﻪ ﺻﻮرت x ≤ y اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ xy = x = yx ﺗﻌﺮﯾﻒﻣﯽ ﮐﻨﯿﻢ. ﺑﻨﺎﺑﺮ ﻟﻢ زرن، ﮐﺎﻓﯽ اﺳﺖ ﺛﺎﺑﺖ ﮐﻨﯿﻢ ﮐﻪ ﻫﺮ زﯾﺮﻣﺠﻤﻮﻋﮥ ﻣﺮﺗﺐ ﺟﺰﺋﯽ L از (π(A دارای ﮐﺮان ﭘﺎﺋﯿﻦ در (π(A اﺳﺖ. ﺑﻪ ازای ﻫﺮ ﻋﻨﺼﺮ λ ∈ L ﻣﺨﺮوط ﭘﺎﺋﯿﻨﯽ ↓ λ = {e ∈ E : e ≤ λ} = {e ∈ E : λeλ = e} ↓ را در ∩ﻧﻈﺮﻣﯽ ﮔﯿﺮﯾﻢ و ﻣﺸﺎﻫﺪهﻣﯽ ﮐﻨﯿﻢ ﮐﻪ آن زﯾﺮﻣﺠﻤﻮﻋﮥﺑﺴﺘﻪ ای از E اﺳﺖ. اﯾﻨﮏ زﯾﺮ ﻣﺠﻤﻮﻋﮥ ﺑﺴﺘﮥ = L ↓ ↓ ↓ λ∈L λ را در ﻧﻈﺮﻣﯽ ﮔﯿﺮﯾﻢ. ﺑﻨﺎﺑﺮ ﻓﺸﺮدﮔﯽ E، ﻫﺮ ﻫﻤﺴﺎﯾﮕﯽ ﺑﺎز از L ﺷﺎﻣﻞ ﯾﮏ ﻣﺨﺮوط ﭘﺎﺋﯿﻨﯽ λ ( { } ⊂ ↓ ∈ ﮐﻪ λ L) اﺳﺖ. ﭼﻮن L∩ ﯾﮏ Gδ -ﻣﺠﻤﻮﻋﮥ ﺑﺴﺘﻪ در E اﺳﺖ ﯾﮏ دﻧﺒﺎﻟﮥ ﮐﺎﻫﺸﯽ λn n∈ω L وﺟﻮد ∈ ∈ ↓ ↓ داردﺑﻪ ﻃﻮری ﮐﻪ L = n∈ω λn . ﺑﻪ ازای ﻫﺮ n ω ﯾﮏ ﻧﻘﻄﮥ an A راﺑﻪ ﮔﻮﻧﻪ ای اﻧﺘﺨﺎبﻣﯽ ﮐﻨﯿﻢ ﮐﻪ ١− ١− π(an) = λn و ﻣﺸﺎﻫﺪهﻣﯽ ﮐﻨﯿﻢ ﮐﻪ λnanλn = (anan )an(an an) = an. ﭘﺲ an در زﯾﺮﻣﺠﻤﻮﻋﮥ ≤ ≥ { ∈ } ﺑﺴﺘﮥ λnSλn = x S : λnxλn = x از S ﻗﺮار دارد. دراﯾﻦ ﺻﻮرتﺑﻪ ازای ﻫﺮ λm λn ،m n و ∈ ⊂ درﻧﺘﯿﺠﻪ am λmSλm λnSλn. اﯾﻨﮏ ∩ﺑﻨﺎﺑﺮ ﻓﺸﺮدۀ ﺷﻤﺎراﯾﯽ A، دﻧﺒﺎﻟﮥ an)n∈ω) دارای ﻧﻘﻄﮥ ﺣﺪی ﻣﺎﻧﻨﺪ ∈ ∞a ﮐﻪ ﻣﺘﻌﻠﻖ ﺑﻪ زﯾﺮﻣﺠﻤﻮﻋﮥ ﺑﺴﺘﮥ∩n∈ω λnSλn از S اﺳﺖ. ﻓﺮضﻣﯽ ﮐﻨﯿﻢ (λ∞ = π(x∞) π(A. ادﻋﺎ ∈ ↓ ↓ ﻣﯽ ﮐﻨﯿﻢ ﮐﻪ λ∞ n∈ω λn = L. ﺑﻪ اﯾﻦ ﻣﻌﻨﯽ ﮐﻪ ∞λ ﯾﮏ ﮐﺮان ﭘﺎﺋﯿﻦ ﺑﺮای L در (π(A اﺳﺖ. در − − ∈ ∈ واﻗﻊ،ﺑﻪ ازای ﻫﺮ a∞ λnSλ∞ ،n ω ﻧﺸﺎنﻣﯽ دﻫﺪ ﮐﻪ ∞١a∞ = λ∞١a∞λn = a∞λ∞λn = a.

٢٠ ش. ﻣﻨﺼﻮری

≤ ﺑﻪ ﻃﻮر ﻣﺸﺎﺑﻪ، ∞λnλ∞ = λ. ﺑﻪ ﻋﺒﺎرﺗﯽ دﯾﮕﺮ، λ∞ λn. ﭘﺲ ∞λ ﯾﮏ ﮐﺮان ﭘﺎﺋﯿﻦ از زﻧﺠﯿﺮ L اﺳﺖ. ﺑﻨﺎﺑﺮ ﻟﻢ زرن، ﻣﺠﻤﻮﻋﮥ (π(A دارای ﯾﮏ ﻋﻨﺼﺮ ﻣﯿﻨﯿﻤﺎل ﻣﺎﻧﻨﺪ e ∈ E اﺳﺖ. ﻓﺮضﻣﯽ ﮐﻨﯿﻢ b ∈ A ﯾﮏ ﻋﻨﺼﺮ ﺑﺎ ﺧﺎﺻﯿﺖ π(b) = e ﺑﺎﺷﺪ. ادﻋﺎ ﻣﯽ ﮐﻨﯿﻢ ﮐﻪ b ﺑﻪ ﻃﻮر ﻣﺘﻨﺎوب ﺗﻮﭘﻮﻟﻮژﯾﮏ اﺳﺖ. ﺑﻨﺎﺑﺮ ﻓﺸﺮده ﺷﻤﺎراﯾﯽ S، n دﻧﺒﺎﻟﮥ {b : n ∈ N} دارای ﯾﮏ ﻧﻘﻄﮥ ﺣﺪی ﻣﺎﻧﻨﺪ c ﮐﻪ ﻣﺘﻌﻠﻖ ﺑﻪزﯾﺮﻧﯿﻢ ﮔﺮوه ﺑﺴﺘﮥ A ∩ eSe از S اﺳﺖ. ﻧﺘﯿﺠﻪ − − − − ﻣﯽ ﮔﯿﺮﯾﻢ ﮐﻪ (π(c = ١ cc = ١ eπ(c) = ecc و (١c = π(c ١c = c π(c)e = c. ﺑﻪ اﯾﻦ ﻣﻌﻨﯽ ﮐﻪ π(c) ≤ e. اﮐﻨﻮن ﺑﻨﺎﺑﺮ ﻣﯿﻨﯿﻤﺎل ﺑﻮدن e در ﻣﺠﻤﻮﻋﮥ (π(c) = e ،π(A. ﺑﻨﺎﺑﺮاﯾﻦ، ﻧﻘﻄﮥ c ﯾﮏ ﻧﻘﻄﮥ ﺣﺪی از n ∞ (n ) ∈ H (b ) دﻧﺒﺎﻟﮥ ١=n در ﮔﺮوه ﺗﻮﭘﻮﻟﻮژیﻣﺘﺮی ﭘﺬﯾﺮ e اﺳﺖ. ﭘﺲﻣﯽ ﺗﻮان ﯾﮏ دﻧﺒﺎﻟﮥ ﻋﺪدی اﻓﺰاﯾﺸﯽ ﻣﺎﻧﻨﺪ k k ω nk − ∞ راﺑﻪ ﮔﻮﻧﻪ ای اﻧﺘﺨﺎب ﮐﺮد ﮐﻪ = (limk→∞(nk+ nk و دﻧﺒﺎﻟﮥ b )k∈ω) ﻫﻤﮕﺮا ﺑﻪ c ﺑﺎﺷﺪ. در ﻧﺘﯿﺠﻪ − − ١ nk+ nk −→ دﻧﺒﺎﻟﮥ ١ k∈ω e = cc( ١ b) و درزﯾﺮﻧﯿﻢ ﮔﺮوه A ﻗﺮار دارد. n در ﺣﺎل ﺣﺎﺿﺮ،ﻣﯽ ﺑﯿﻨﯿﻢ ﮐﻪ ﺧﻮدﺗﻮان e ﻣﺘﻌﻠﻖ ﺑﻪ ﻣﺠﻤﻮﻋﮥ ﺑﺴﺘﮥ A از ﻧﻘﺎط ﺣﺪی دﻧﺒﺎﻟﮥ a )n∈ω) اﺳﺖ. ﭼﻮن S در mk −→ e ﺷﻤﺎرای ﻧﻮع اول اﺳﺖ، ﯾﮏ دﻧﺒﺎﻟﻪ ﻋﺪدی اﻓﺰاﯾﺸﯽ mk)k∈ω) وﺟﻮد داردﺑﻪ ﻃﻮری ﮐﻪ a )k∈ω e). ﺑﻨﺎﺑﺮ − − − − mk mk mk −→ ﭘﯿﻮﺳﺘﮕﯽ ﻣﻌﮑﻮس در e، دﻧﺒﺎﻟﮥ a )k∈ω e). ﭘﺲ ١ limk→∞ a a = aa = ١ e = ee. − m m − ﭼﻮن ١ limk→∞ a k = e = aa، دﻧﺒﺎﻟﮥ k∈ω( ١+a k) ﺑﻪ ١a = a aa ﻣﯿﻞﻣﯽ ﮐﻨﺪ. ﻣﺸﺎﻫﺪهﻣﯽ ﮐﻨﯿﻢ ﻋﻨﺼﺮ a ﺑﻪ ﻃﻮر ﻣﺘﻨﺎوب ﺗﻮﭘﻮﻟﻮژﯾﮏ اﺳﺖ. از ادﻋﺎی ٣ و ٣ و ﻗﻀﯿﮥ ٢ . ٢ ﻧﺘﯿﺠﻪﻣﯽ ﮔﯿﺮﯾﻢ ﮐﻪ S ﯾﮏﻧﯿﻢ ﮔﺮوه ﮐﻠﯿﻔﻮرد ﺗﻮﭘﻮﻟﻮژی اﺳﺖ. اﯾﻨﮏ ﺑﺎ اﺳﺘﻔﺎده از ﻓﺸﺮده ﺷﻤﺎراﯾﯽ، ﻓﻀﺎی S ﯾﮏ M - ﻓﻀﺎ اﺳﺖ دراﯾﻦ ﺻﻮرت ﺑﻨﺎﺑﺮ ﻗﻀﯿﮥ ، S ﻣﺘﺮی ﭘﺬﯾﺮ اﺳﺖ.

ﻣﺮاﺟﻊ 1. A. Arhangel’skii, M. Tkachenko, Topological groups and related structures, Atlantis Press, Paris; World Sci. Publ., NJ, 2008.

2. T. Banakh, On cardinal invariants and metrizability of topological inverse Clifford semigroups, Topology Appl. 128:1 (2003) 13–48.

3. T. Banakh, O. Gutik, On the continuity of the inversion in countably compact inverse topological semigroups, SemigroupForum,68 (2004), 411–418.

4. R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.

5. G. Gruenhage, Generalized metric spaces, in: Handbook of set-theoretic topology, 423–501, North-Holland, Amsterdam,1984.

٢١ ﮔﺮوه ﻫﺎی ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی R-ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ

∗ زﻫﺮا ﻣﻌﯿﺮی ١ و ﺟﻮاد ﺟﻤﺎﻟﺰاده٢

ﭼﮑﯿﺪه. در اﯾﻦ ﻣﻘﺎﻟﻪ R-ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮی را ﺑﺮ روی ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﺑﺮرﺳﯽﻣﯽ ﮐﻨﯿﻢ. ﺑﺮای ﻫﺮ ۴, ۵ ,٣ ,٢ ,١ = i ﺧﺎﻧﻮاده ای ازﮔﺮوه ﻫﺎی ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی Ri-ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ G را ﺑﺎ اﯾﻦ ﮐﻪ ﺑﺮای ﻫﺮ ﺗﺎﺑﻊ ﺣﻘﯿﻘﯽ ﻣﻘﺪار روی G، ﻫﻤﻮﻣﻮرﻓﯿﺴﻢ → ﭘﯿﻮﺳﺘﻪ p : G H ﺑﻪ روی ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی H راﺑﻪ ﻃﻮرﯾﮑﻪ H ﮔﺮوه دارای ﺧﺎﺻﯿﺖ ﺟﺪاﺳﺎزی Ti را داﺷﺘﻪ ﺑﺎﺷﺪ و ﺷﺮط ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮی ﻧﯿﺰ ﺑﺮﻗﺮار ﺑﺎﺷﺪ را ﺗﻌﺮﯾﻒﻣﯽ ﮐﻨﯿﻢ. ﻧﺸﺎنﻣﯽ دﻫﯿﻢ ﮐﻪ ﺧﻂ ﺳﻮرﺟﻨﻔﺮی ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی R ﻟﯿﻨﺪﻟﻮﻓﯽ اﺳﺖ ﮐﻪ ١ -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ ﻧﯿﺴﺖ. ﺑﺎ اﯾﻦ ﺣﺎل ﻫﺮ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﻣﻨﻈﻢ (ﻫﺎﺳﺪورف) ﻟﯿﻨﺪﻟﻮف ﻣﻮﺿﻌﺎ R R w-ﻣﺤﺪود، ٣ -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ اﺳﺖ (٢ -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ) اﺳﺖ. ﻫﻤﭽﻨﯿﻦ ﻧﺸﺎنﻣﯽ دﻫﯿﻢ ﮐﻪ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﻟﯿﻨﺪﻟﻮف ﻣﻨﻈﻢ ﻣﻮﺿﻌﺎ w-ﻣﺤﺪود ﺑﺎ زﯾﺮﮔﺮوﻫﯽ از ﺣﺎﺻﻞ ﺿﺮب ﮔﺮوه ﻫﺎی ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﻣﺠﺰای ﻣﺘﺮ ﭘﺬﯾﺮ ﺑﻪ ﻃﻮر ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ اﯾﺰوﻣﻮرﻓﯿﺴﻢ اﺳﺖ.

١. ﻣﻘﺪﻣﻪ ﺑﻪﻧﯿﻢ ﮔﺮوه H ﻣﺠﻬﺰ ﺑﻪ ﯾﮏ ﺗﻮﭘﻮﻟﻮژی،ﻧﯿﻢ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﮔﻔﺘﻪﻣﯽ ﺷﻮد ﻫﺮﮔﺎه ﺗﺎﺑﻊ ﻋﻤﻞ در H ﭘﯿﻮﺳﺘﻪ ﺑﺎﺷﺪ. ﯾﮏ ﻧﯿﻢ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﮐﻪ ﺑﻪ ﻟﺤﺎظ ﺟﺒﺮی ﮔﺮوه ﺑﺎﺷﺪ را ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﻧﺎﻣﻨﺪ و ﻫﺮ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺑﺎ ﻣﻌﮑﻮس ﭘﯿﻮﺳﺘﻪ را ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی ﮔﻮﯾﻨﺪ. اﻫﻤﯿﺖ ﻣﻔﺎﻫﯿﻢ اﺧﯿﺮ در ﻣﻘﺎﻻﺗﯽ ﮐﻪ ﺗﻮﺳﻂ واﻟﯿﺲ [١]، اﻟﯿﺲ [۴] ﮔﺮدآوریﺷﺪه اﻧﺪ ﻣﺸﻬﻮد اﺳﺖ. اﺧﯿﺮاﮔﺮوه ﻫﺎی ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺑﻪ زﻣﯿﻨﻪ ای از ﺗﺤﻘﯿﻘﺎت ﻗﻮی ﻣﺒﺪل ﺷﺪه اﺳﺖ. در اﯾﻦ ﻣﻘﺎﻟﻪﮔﺮوه ﻫﺎی ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژﯾﮑﯽ R-ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ را ﻣﻄﺎﻟﻌﻪﻣﯽ ﮐﻨﯿﻢ. ﭘﻨﺘﺮاﮔﯿﻦ [٢] ﻧﺘﯿﺠﻪ ﺟﺎﻟﺒﯽ در اﯾﻦ راﺑﻄﻪ ﺛﺎﺑﺖ ﻧﻤﻮده اﺳﺖ: ﺑﺮای ﻫﺮ ﺗﺎﺑﻊ ﭘﯿﻮﺳﺘﻪ f :→ R روی ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ G، ﻫﻤﻮﻣﻮرﻓﯿﺴﻢ ﭘﯿﻮﺳﺘﻪ ϕ : G → H ﺑﺮ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺷﻤﺎرای دوم H و ﺗﺎﺑﻊ ﭘﯿﻮﺳﺘﻪ h : H → R ﻣﻮﺟﻮد اﺳﺖ ﺑﻄﻮرﯾﮑﻪ f = hoϕ، اﯾﻦ ﻧﺘﯿﺠﻪ ﺑﺮایﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﻓﺸﺮده ﻧﻤﺎ ﻗﺎﺑﻞ اﺛﺒﺎت اﺳﺖ. ﻧﺘﯿﺠﻪ ﻣﺮﺑﻮﻃﻪ ﺗﻮﺳﻂ ﮐﺎﻣﻔﻮرت رز[۵] ﮔﺮدآوری ﺷﺪه اﺳﺖ اﯾﻦ ﻧﺘﺎﯾﺞ ﻧﻮﯾﺴﻨﺪه را ﺑﺮای ﻣﻌﺮﻓﯽ ﮐﻼسﮔﺮوه ﻫﺎی R-ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ درﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺗﺮﻏﯿﺐﻣﯽ ﮐﻨﺪ. ﮐﻼسﮔﺮوه ﻫﺎی R-ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ ﮔﺴﺘﺮده اﺳﺖ.

2010 Mathematics Subject Classification. Primary 47A55; Secondary 39B52, 34K20, 39B82. واژﮔﺎن ﮐﻠﯿﺪی. ﮐﻼ w-ﻣﺤﺪود؛ R-ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ؛ ﻟﯿﻨﺪﻟﻮف؛ Z-ﻧﺸﺎﻧﻨﺪه . ∗ ﺳﺨﻨﺮان

٢٢ ز. ﻣﻌﯿﺮی و ج. ﺟﻤﺎﻟﺰاده

ﺑﻌﻨﻮان ﻣﺜﺎل، ﺷﺎﻣﻞ ﻫﻤﻪﮔﺮوه ﻫﺎی ﻟﯿﻨﺪﻟﻮف وزﯾﺮﮔﺮوه ﻫﺎی دﻟﺨﻮاه ازﮔﺮوه ﻫﺎی Σ-ﻟﯿﻨﺪﻟﻮف اﺳﺖ. ﻫﻤﻪﮔﺮوه ﻫﺎی ﭘﯿﺶ ﻓﺸﺮده وزﯾﺮﮔﺮوه ﻫﺎی دﻟﺨﻮاه ازﮔﺮوه ﻫﺎی σ-ﻓﺸﺮده، R-ﻓﺎﮐﺘﻮرﭘﺬﯾﺮﻧﺪ ( [٣]see for instance ) ﻫﻤﺎﻧﻨﺪ ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺑﻪ ﯾﮏ ﮔﺮوه ﭘﺎراﺗﻮﭘﻮﻟﻮژﯾﮑﯽ R ،H-ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ ﮔﻮﯾﻨﺪ ﻫﺮ ﮔﺎه ﻫﺮ ﺗﺎﺑﻊ ﺣﻘﯿﻘﯽ ﻣﻘﺪار ﭘﯿﻮﺳﺘﻪ در H ﯾﮏ ﻫﻤﻮﻣﻮرﻓﯿﺴﻢ ﭘﯿﻮﺳﺘﻪ از H ﺑﺮ ﮔﺮوه ﭘﺎراﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﺷﻤﺎرای ﺳﻮم ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ ﺑﺎﺷﺪ. ﭼﻮنﮐﻼس ﻫﺎی R R R T١، ﻫﺎﺳﺪورف وﮔﺮوه ﻫﺎی ﭘﺎراﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﻣﻨﻈﻢ دوﺑﺪو ﻣﺠﺰا ﻫﺴﺘﻨﺪ ﻣﻔﺎﻫﯿﻢ -ﻓﺎﮐﺘﻮرﭘﺬﯾﺮی در ١ و ٢ و را ﺗﻔﮑﯿﮏﻣﯽ ﮐﻨﯿﻢ. R در ﻧﻈﺮ ﮔﺮﻓﺘﻦ ﯾﮏ ﮐﻼس ازﮔﺮوه ﻫﺎی ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ۵/٣ ﻃﺒﯿﻌﯽ ﺑﻪ ﻧﻈﺮﻣﯽ رﺳﺪ اﻣﺎ ﺑﺮایﮔﺮوه ﻫﺎی ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژﯾﮏ R R ﺗﯿﺨﻮﻧﻒ، ٣ -ﻓﺎﮐﺘﻮرﭘﺬﯾﺮی و ۵/٣ -ﻓﺎﮐﺘﻮرﭘﺬﯾﺮیﻣﻨﻄﺒﻖ اﻧﺪ در اﯾﻦ ﻣﻘﺎﻟﻪ ﻧﺸﺎن ﺧﻮاﻫﯿﻢ داد ﻫﺮ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژﯾﮑﯽ R R ١ -ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ ﮐﻼ w-ﻣﺤﺪود اﺳﺖ و ﻫﺮ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﻣﻨﻈﻢ ﻟﯿﻨﺪﻟﻮف ﮐﻼ w-ﻣﺤﺪود ٣ -ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ اﺳﺖ. ﻣﯽ داﻧﯿﻢ ﮐﻪ ﻫﻤﻪ ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﻟﯿﻨﺪﻟﻮف R-ﻓﺎﮐﺘﻮرﭘﺬﯾﺮﻧﺪ ﺑﻄﻮر ﺧﺎص ﻫﺮ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﻣﻨﻈﻢ ﮐﺎﻣﻼ R ﻟﯿﻨﺪﻟﻮف ٣ -ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ اﺳﺖ.

٢. ﻣﻔﺎﻫﯿﻢ ﻣﻘﺪﻣﺎﺗﯽ − ﺑﺮای ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی H ﺑﺎ ﺗﻮﭘﻮﻟﻮژی τ، ﺗﻮﭘﻮﻟﻮژی ﺗﻮام آﻧﺮا ﺑﺎ ١ τ ﻧﺸﺎنﻣﯽ دﻫﯿﻢ و ﺑﺼﻮرت ′ − − − {u ∈ τ : ١ u} = ١ τ ﺗﻌﺮﯾﻒﻣﯽ ﮐﻨﯿﻢ ﺑﻨﺎﺑﺮاﯾﻦ (١ H = (H, τ ﯾﮏ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی اﺳﺖ و ﻣﻌﮑﻮس ١−H′ H x → x ١− H τ ∗ = τ ∨ τ ∗ ∗ﻫﻤﻮﻣﻮرﻓﯿﺴﻤﯽ از ﺑﻪ اﺳﺖ. ﮐﺮان ﺑﺎﻻﯾﯽ ﮔﺮوه ∗ﺗﻮﭘﻮﻟﻮژی روی اﺳﺖ و ( H = (H, τ را ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮏ واﺑﺴﺘﻪ ﺑﻪ H ﻣﯽ ﻧﺎﻣﯿﻢ. ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮏ واﺑﺴﺘﻪ H ﻫﺎﺳﺪورف اﺳﺖ. ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی H را w-ﻣﺤﺪود (ﭘﯿﺶ ﻓﺸﺮده) ﮔﻮﯾﯿﻢ اﮔﺮ ﺑﺮای ﻫﺮ ﻫﻤﺴﺎﯾﮕﯽ U از ﻋﻨﺼﺮ ﻫﻤﺎﻧﯽ در H ﻣﺠﻤﻮﻋﻪ ﺷﻤﺎرای (ﻣﺘﻨﺎﻫﯽ) C ⊆ H ﭼﻨﺎن ﻣﻮﺟﻮد ﺑﺎﺷﺪ ﮐﻪ CU = H = UC واﺿﺢ اﺳﺖ ﮐﻪ اﮔﺮ ﻫﺮ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ σ-ﻓﺸﺮده، w-ﻣﺤﺪود اﺳﺖ (ﯾﮏ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ﮐﻪ اﺟﺘﻤﺎع ﺷﻤﺎرا از ﻣﺠﻤﻮﻋﻪ ﻫﺎی ﻓﺸﺮده اﺳﺖ را σ-ﻓﺸﺮده ﮔﻮﯾﻨﺪ) ﻓﻀﺎی X را Σ-ﻓﻀﺎی ﻟﯿﻨﺪﻟﻮف ﮔﻮﯾﻨﺪ ﻫﺮ ﮔﺎه ﻓﻀﺎﻫﺎی Y و M و ﻧﮕﺎﺷﺖ ﭘﯿﻮﺳﺘﻪ f : Y → X و g : Y → M ﺑﻪﮔﻮﻧﻪ ای ﺑﺎﺷﻨﺪ ﮐﻪ M ﯾﮏ ﻓﻀﺎی ﺷﻤﺎرای ﻧﻮع دوم و ﻧﮕﺎﺷﺖ g ﮐﺎﻣﻞ ﺑﺎﺷﺪ ﺑﻌﺒﺎرت دﯾﮕﺮ X ﺗﺼﻮﯾﺮﭘﯿﻮﺳﺘﻪ ای از p-ﻓﻀﺎی ﻟﯿﻨﺪﻟﻮف Y ﺑﺎﺷﺪ. ﺑﻮﺿﻮح ﻫﺮ ﻓﻀﺎی σ-ﻓﺸﺮده ﻫﻤﺎﻧﻨﺪ ﻫﺮ ﻓﻀﺎ ﺑﺎ ﺷﺒﮑﻪ ﺷﻤﺎرا ﯾﮏ Σ-ﻓﻀﺎی ﻟﯿﻨﺪﻟﻮف اﺳﺖ. ﻓﺮض ﮐﻨﯿﺪ X و Y دو ﻓﻀﺎی دﻟﺨﻮاه و f : X → Y ﯾﮏ ﺗﺎﺑﻊ ﺑﺎﺷﺪ، f را ﯾﮏ ﻧﺸﺎﻧﻨﺪه X در Y ﮔﻮﯾﯿﻢ ﻫﺮﮔﺎه X ﺑﺎ زﯾﺮﻓﻀﺎﯾﯽ از Y ﻫﻤﺴﺎﻧﺮﯾﺨﺖ ﺑﺎﺷﺪ در اﯾﻦ ﺣﺎﻟﺖ X در Y ﻧﺸﺎﻧﻨﺪه ﺷﺪه اﺳﺖ. زﯾﺮﻓﻀﺎی Y از X را C-ﻧﺸﺎﻧﻨﺪه ﮔﻮﯾﯿﻢ ﻫﺮﮔﺎه ﺗﺎﺑﻊ ﺣﻘﯿﻘﯽ ﻣﻘﺪار ﭘﯿﻮﺳﺘﻪ ﺑﺮ X را ﺑﺘﻮان ﺑﻪ ﺗﺎﺑﻊ ﺣﻘﯿﻘﯽ ﻣﻘﺪار ﭘﯿﻮﺳﺘﻪ ﺑﺮ X ﺗﻮﺳﯿﻊ دﻫﯿﻢ. (χ(H را ﮐﻮﭼﮑﺘﺮﯾﻦ ﻋﺪد اﺻﻠﯽ از ﭘﺎﯾﻪ ﻣﻮﺿﻌﯽ از ﻋﻨﺼﺮ ﻫﻤﺎﻧﯽ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی H ﻗﺮارﻣﯽ دﻫﯿﻢ ﮐﻪ ﻣﺸﺨﺼﻪ H ﻧﺎﻣﯿﺪه ﻣﯽ ﺷﻮد و (w(H ﮐﻮﭼﮑﺘﺮﯾﻦ ﻋﺪد اﺻﻠﯽ ﭘﺎﯾﻪ H اﺳﺖ ﮐﻪ وزن H ﻧﺎﻣﯿﺪهﻣﯽ ﺷﻮد. ﮔﺰاره ٢ . ١. ﻓﺮض ﮐﻨﯿﺪ H ﯾﮏ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﻫﺎﺳﺪورف ﺑﺎﺷﺪ: ∗ (a) اﮔﺮ σ ،H-ﻓﺸﺮده ﺑﺎﺷﺪ آﻧﮕﺎه H ﻧﯿﺰ σ-ﻓﺸﺮده اﺳﺖ. ∗ (b) اﮔﺮ H ﯾﮏ Σ-ﻓﻀﺎی ﻟﯿﻨﺪﻟﻮف ﺑﺎﺷﺪ آﻧﮕﺎه H ﻧﯿﺰ Σ-ﻓﻀﺎی ﻟﯿﻨﺪﻟﻮف اﺳﺖ. ∗ (c) اﮔﺮ H دارای ﺷﺒﮑﻪ ﺷﻤﺎرا ﺑﺎﺷﺪ آﻧﮕﺎه H ﻧﯿﺰ دارای ﺷﺒﮑﻪ ﺷﻤﺎراﻣﯽ ﺑﺎﺷﺪ.

٣. R-ﻓﺎﮐﺘﻮرﭘﺬﯾﺮی درﮔﺮوه ﻫﺎی ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی R-ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ ﺧﺎﻧﻮاده ﺑﺰرﮔﯽ ازﮔﺮوه ﻫﺎ را ﺑﺎ ﺧﻮاص ﺟﺎﻟﺐ و وﯾﮋه ﺗﺸﮑﯿﻞﻣﯽ دﻫﻨﺪ. ﺑﺮای ﮔﺮوه ﻫﺎی ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﺗﻌﺮﯾﻔﯽ ﺷﺒﯿﻪ R-ﻓﺎﮐﺘﻮرﭘﺬﯾﺮی درﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی را اراﺋﻪﻣﯽ دﻫﯿﻢ. از اﯾﻨﺮو اﺻﻮل ﺟﺪاﺳﺎزی ﺑﺮ رویﮔﺮوه ﻫﺎی ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی را ﻣﺸﺨﺺﻣﯽ ﮐﻨﯿﻢ ﭼﻮن ﺑﺮﺧﻼفﮔﺮوه ﻫﺎی ﺗﻮﭘﻮﻟﻮژی وﻗﺘﯽ ﮐﻪ i ≤ j ≤ ε ≥ ٠ ⇒ ﻧﻤﯽ ﺗﻮاﻧﯿﻢ Ti = Tj را داﺷﺘﻪ ﺑﺎﺷﯿﻢ. R ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی H را ﺑﺮای ۴, ۵ ,٣ ,٢ ,١ = i ،i -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ ﮔﻮﯾﻨﺪ اﮔﺮ H ﯾﮏ ﻓﻀﺎی Ti ﺑﺎﺷﺪ و ﺑﺮای ﻫﺮ ﺗﺎﺑﻊ ﺣﻘﯿﻘﯽ ﻣﻘﺪار f روی H ﻫﻤﻮﻣﻮرﻓﯿﺴﻢ ﭘﯿﻮﺳﺘﻪ p : H → k ﺑﺮ روی ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی k ﺑﺎ وزن ﺷﻤﺎرا ﮐﻪ در ﺷﺮط ﺟﺪاﺳﺎزی Ti ﺻﺪق ﮐﻨﺪ و ﺗﺎﺑﻊ ﺣﻘﯿﻘﯽ ﻣﻘﺪار g روی k را ﺑﻪﮔﻮﻧﻪ ای ﭘﯿﺪا ﮐﻨﯿﻢ ﮐﻪ f = gop R ≤ ≤ R واﺿﺢ اﺳﺖ ﮐﻪ ﻫﺮ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی j -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ ﮐﻪ ۵/٣ i < j ١، i -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ اﺳﺖ. ﭼﻮن ﻫﺮ

٢٣ ﮔﺮوه ﻫﺎی ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی R-ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ

ﻓﻀﺎی ﻣﻨﻈﻢ ﺷﻤﺎرای ﻧﻮع دوم ﻧﺮﻣﺎل اﺳﺖ از اﯾﻨﺮو ﺗﯿﺨﻮﻧﻮف اﺳﺖ، اﯾﻦ ﻫﻤﻮاره واﺿﺢ اﺳﺖ ﮐﻪ ﻫﺮ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی R R ﺗﯿﺨﻮﻧﻮف ٣ -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ، ۵/٣ -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ اﺳﺖ. R اﯾﻦ دﻟﯿﻞ ﺑﺮای اﯾﻨﮑﻪ ﺑﺎ ٣ -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮی در ﻣﻮردﮔﺮوه ﻫﺎی ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﺳﺮوﮐﺎر دارﯾﻢ ﺑﯿﺎن ﺷﺪه اﺳﺖ. در زﯾﺮ اﯾﻦ ′ − ﻣﻮﺿﻮع ﮐﻪ ﻋﻤﻠﮕﺮ ﻣﻌﮑﻮس ١ x −→ x در ﮔﺮوه ﻫﺎی ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﻫﻤﻮﻣﻮرﻓﯿﺴﻤﯽ از H ﺑﻪ H اﺳﺖ اﺳﺘﻔﺎده ﻣﯽ ﮐﻨﯿﻢ. ′ ∈ { } R ﮔﺰاره ٣ . ١. اﮔﺮ H ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی i -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ ﺑﺎﺷﺪ ﮐﻪ ۴, ۵ ,٣ ,٢ ,١ i آﻧﮕﺎه H ﻧﯿﺰ ﭼﻨﯿﻦ اﺳﺖ. ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ اﯾﻨﮑﻪ ﻫﺮ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی ﻟﯿﻨﺪﻟﻮف، R-ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ اﺳﺖ، ﺧﻂ ﺳﻮرﺟﻨﻔﺮی ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی اﺳﺖ ﮐﻪ R ١ -ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ ﻧﯿﺴﺖ. R در ﻣﺜﺎل زﯾﺮ ﻧﺸﺎنﻣﯽ دﻫﯿﻢ ﮐﻪ درﮔﺮوه ﻫﺎی ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﺣﺘﯽ ﺗﺮﮐﯿﺒﯽ از ﺧﺎﺻﯿﺖ ﻟﯿﻨﺪﻟﻮف و ﭘﯿﺶ ﻓﺸﺮدﮔﯽ ، ١ - ﻓﺎﮐﺘﻮرﭘﺬﯾﺮی را ﻧﺘﯿﺠﻪﻧﻤﯽ دﻫﺪ. ﻣﺜﺎل٣ . ٢. ﻓﺮض ﮐﻨﯿﻢ S ﯾﮏ ﺧﻂ ﺳﻮرﺟﻨﻔﺮی ﺑﺎﺷﺪ و Z ﮔﺮوﻫﯽ از اﻋﺪاد ﺻﺤﯿﺢ ﺑﺎﺷﺪ ﮐﻪ زﯾﺮﮔﺮوه ﺑﺴﺘﻪ ای از S S اﺳﺖ، ﺑﻨﺎﺑﺮاﯾﻦ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﺧﺎرج ﻗﺴﻤﺘﯽ T = Z ﺑﻪ ﻃﻮر ﺟﺒﺮی ﮔﺮوه ﺟﻤﻌﯽ دوری اﺳﺖ ﮐﻪ دارایﭘﺎﯾﻪ ای ١ ∋ ﻣﻮﺿﻌﯽ در ﻋﻨﺼﺮ ﻫﻤﺎﻧﯽ ٠ ﺗﻮﻟﯿﺪ ﺷﺪه ﺑﻪ وﺳﯿﻠﻪﺑﺎزه ﻫﺎیﻧﯿﻢ ﺑﺎز ( n ,٠] اﺳﺖ ﮐﻪ n N. ﮔﺮوه ﭘﯿﺮا ﺗﻮﭘﻮﻟﻮژی T R ﻣﻨﻈﻢ،ﺑﻪ ﻃﻮر ارﺛﯽ ﻟﯿﻨﺪﻟﻮف،ﺑﻪ ﻃﻮر ارﺛﯽ ﻣﺠﺰا ﺷﺪه و ﭘﯿﺶ ﻓﺸﺮده اﺳﺖ. ادﻋﺎﻣﯽ ﮐﻨﯿﻢ ﮐﻪ T ﻧﻤﯽ ﺗﻮاﻧﺪ ١ -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ ﺑﺎﺷﻨﺪ. S ﺧﻂ ﺳﻮرﺟﻨﻔﺮی و ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﺧﺎرج ﻗﺴﻤﺘﯽ Z ﯾﮏ ﻓﻀﺎی ﺷﻤﺎرای ﻧﻮع اولﻧﺮﻣﺎل ﻣﺘﺮ ﻧﺎﭘﺬﯾﺮ اﺳﺖ. ﻟﺬا R S از اﯾﻦ ﻣﻮﺿﻮع ﮐﻪ ﻫﯿﭻ ﯾﮏ از S و T = Z، ١ -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ ﻧﯿﺴﺖﻣﯽ ﺗﻮان ﮔﺰاره زﯾﺮ را ﺑﯿﺎن ﮐﺮد. R ﮔﺰاره ٣ . ٣. ﻫﺮ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﺑﻄﻮر ﮐﺎﻣﻞ ﻣﻨﻈﻢ ١ -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ دارای ﺧﺎﺻﯿﺖ (w(H) = χ(H ﻣﯽ ﺑﺎﺷﺪ. R درﮔﺰاره ی ذﯾﻞ ﺷﺮط ﻻزم ﻧﻪ ﮐﺎﻓﯽ ﺑﺮای ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ ١ -ﻓﺎﮐﺘﻮرﭘﺬﯾﺮی اﺳﺖ. R ﮔﺰاره ٣ . ۴. ﻫﺮ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ١ -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ ﮐﻼ w-ﻣﺤﺪود اﺳﺖ. R ﻗﻀﯿﻪ٣ . ۵. ﻫﺮ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﻣﻨﻈﻢ ﻟﯿﻨﺪﻟﻮف و ﮐﻼ w-ﻣﺤﺪود H، ٣ -ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ اﺳﺖ. R ﻗﻀﯿﻪ٣ . ۶. ﻓﺮض ﮐﻨﯿﺪ H ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﻫﺎﺳﺪورف ﮐﻼ w-ﻣﺤﺪود اﺳﺖ، اﮔﺮ H ﻟﯿﻨﺪﻟﻮف ﺑﺎﺷﺪ آﻧﮕﺎه ٢ - ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ اﺳﺖ. R ﻗﻀﯿﻪ٣ . ٧. ﻓﺮض ﮐﻨﯿﺪ H ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﻫﺎﺳﺪورف و ﻟﯿﻨﺪﻟﻮف ﯾﺎﺷﺪ آﻧﮕﺎه ٢ -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ ﮐﻼ w-ﻣﺤﺪود ﺑﺎﺷﺪ. ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ اﯾﻨﮑﻪ ﻫﺮ ﮔﺮوه ﺗﻮﭘﻮﻟﻮژی ﭘﯿﺶ ﻓﺸﺮده R-ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ اﺳﺖ اﯾﻦ ﺳﻮال ﻣﻄﺮحﻣﯽ ﺷﻮد ﮐﻪ آﯾﺎ ﻫﺮ ﮔﺮوه R R ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﮐﺎﻣﻼ ﭘﯿﺶ ﻓﺸﺮده، ٣ -ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ ﯾﺎ ٢ -ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ اﺳﺖ؟ ﭘﺎﺳﺦ ﻣﺜﺒﺖ اﺳﺖ زﯾﺮا ﻫﺮ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﮐﻼ ﭘﯿﺶ ﻓﺸﺮده ﮔﺮوه ﺗﻮﭘﻮﻟﻮژﯾﮑﯽ اﺳﺖ. R ﻧﺘﯿﺠﻪ٣ . ٨. ﻫﺮ ﮔﺮوه ﭘﯿﺮا ﺗﻮﭘﻮﻟﻮژی ﻣﻨﻈﻢ ﺑﺎ ﺷﺒﮑﻪ ﺷﻤﺎرا، ٣ -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ اﺳﺖ. R ﻧﺘﯿﺠﻪ٣ . ٩. ﻫﺮ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﻫﺎﺳﺪورف ﺑﺎ ﺷﺒﮑﻪ ﺷﻤﺎرا، ٢ -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ اﺳﺖ. R R زﯾﺮ ﮔﺮوه Z-ﻧﺸﺎﻧﻨﺪه از ﯾﮏ ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی i -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ ﮐﻪ ٣ ,٢ ,١ = i i -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ ﺧﻮاﻫﺪ ﺑﻮد. ﻗﻀﯿﻪ٣ . ١٠. ﻓﺮضﻣﯽ ﮐﻨﯿﻢ H ﮔﺮوﻫﯽ ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﺑﺎﺷﺪ ﺑﻄﻮرﯾﮑﻪ ﯾﮏ Σ-ﻓﻀﺎی ﻟﯿﻨﺪﻟﻮف اﺳﺖ آﻧﮕﺎه ﻫﺮ زﯾﺮ ﮔﺮوه R از H، ٣ -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ ﺧﻮاﻫﺪ ﺑﻮد. R ﻧﺘﯿﺠﻪ٣ . ١١. ﻫﺮ زﯾﺮ ﮔﺮوه از ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﻣﻨﻈﻢ σ-ﻓﺸﺮده، ٣ -ﺗﺠﺰﯾﻪ ﭘﺬﯾﺮ اﺳﺖ. ﮔﺰاره ذﯾﻞ ﺑﺮایﮔﺮوه ﻫﺎی ﭘﺎراﺗﻮﭘﻮﻟﻮژی ﻫﺎﺳﺪورف در ﺗﻀﺎد ﺑﺎﻧﺘﯿﺠﻪ ی ﻓﻮق اﺳﺖ.

R ﮔﺰاره ٣ . ١٢. ﻫﺮ زﯾﺮ ﮔﺮوه از ﮔﺮوه ﭘﯿﺮاﺗﻮﭘﻮﻟﻮژی ﻫﺎﺳﺪورف σ-ﻓﺸﺮده، ٢ -ﻓﺎﮐﺘﻮرﭘﺬﯾﺮ اﺳﺖ.

٢۴ ز. ﻣﻌﯿﺮی و ج. ﺟﻤﺎﻟﺰاده

ﻣﺮاﺟﻊ 1. A. D. Wallace, The structure of topological semigroups, Bull. Amer. Math. Soc. 61 (1955) 95-112.

2. L. S. Pontryagin, Nepreryvnye gruppy, Moscow, 1938; English translation: Topological Groups, Princeton University Press. Princeton. 1939.

3. M. G. Tkachenko, Subgroups, quotient groups and products of R-factorizable groups, Topology Proc. 16 (1991) 201-231.

4. R. Ellis, Locally compact transformation groups, Duke Math. Soc. 8 (1957) 372-373.

5. W. W. Comfort, K. A. Ross, Pseudocompactness and uniform continuity in topological gruop, Mat. Stud. 6 (1996) 39-40.

٢۵ Extended Abstracts

The 5th Seminar on Harmonic Analysis and Applications

Ferdowsi University of Mashhad, Iran 18-19th January 2017 Contents

Author Index 1

Plenary Presentations 3

Short Presentations 12

Posters 205 Author Index

A. A. Arefijamaal, 162 J. Saadatmandan, 47 A. A. Arefijamaal , 181 A. A. Khadem Maboudi, 90 L. Najarpisheh , 178 A. A. Khosravi , 96 M. A. Bahmani, 52 A. Alinejad, 23 M. Akbari Tootkaboni , 12 A. Askari Hemmat, 27, 86 M. Amin khah , 27 A. Bagheri Salec, 47 M. Choubin, 222 A. H. Mokhtari, 52 M. Choubin , 230 A. K. Mirmostafaee, 108 M. Essmaili, 71 A. N. Motlagh, 123, 127 M. Fashandi, 78 A. Niknam, 154 M. H. Ahmadi Gandomani, 19 A. Niknam , 189 M. H. Heydari, 189 A. R. Khoddami, 93 M. J. Mehdipour, 19 A. Rabeie , 145 M. Janfada, 119 A. Rahimi, 226 M. Mehmandust, 218 A. Razghandi, 162 M. Nasehi , 134 A. Safapour, 174 M. Nazarianpoor , 138 A. Sahleh , 178 M. Niazi , 141 B. Sadathoseyni, 197 M. R. Miri, 141 M. Ramezanpour, 158 F. Afkhami, 18 M. Rostami, 23 F. Arabyani Neyshaburi, 39 M. Sadeghi, 205 F. Esmaeelzadeh, 67, 166 M. Shamsabadi , 181, 185 F. Ghomanjani, 82 M. Zare, 201 F. Khosravi , 100 M. Zarebnia, 31 F. Mohammadi, 205 F. Roohi Afrapoli, 166 N. Aalipour, 205 F. Soltani Sarvestani , 189 N. Gholami, 214 N. Mohammadian, 112 G. Sadeghi, 138, 170 N. Salahzehi, 234 N. Salahzehi , 230 H. Javanshiri , 3 N. Tavallaei , 9 H. Lakzian , 104 H. Najafi, 131 O. Baghani, 43 H. R.Ebrahimi Vishki, 75 O. Baghani , 210

J. Keshavarzian, 12 P. Kasprzak, 100

1 2 AUTHOR INDEX

P. Rahimkhani , 150

R. A. Kamyabi-Gol, 7, 35, 201 R. Amiri, 31 R. Ebrahimi Vishki, 141 R. Faal, 75 R. Raisi Tousi, 27, 31

S. Barootkoob, 55, 59 S. Jokar , 193 S. M. Tabatabaie , 193, 197 S. Mohammadzadeh , 116 S. Movahed, 222 S. Zebarjad, 238

T. Derikvand, 63

V. R. Morshedi, 119

Y. Ordokhani, 150

Z. Amiri, 35 Z. Ghazvini, 210 Z. Kalateh Bojdi, 86 Z. Rahmati Nasrabad, 154 Z. Yazdani Fard, 174 Plenary Presentations The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

Plenary Presentation

STATE OF THE ART IN THE STUDY OF THE SECOND DUALS OF MEASURE ALGEBRAS

Hossein Javanshiri

Abstract I will present some recent investigation on the second duals of measures algebras of locally compact groups. In details, in the year 2013, Rasoul Nasr-Isfahani and I published a paper in which we studied the nature of the strong dual of measure algebras with certain locally convex topologies. This talk is about subsequent advances made in the study of the duals structures of measure algebras, and about the theory of generalised function. I shall put particular emphasis on more recent results.

2010 Mathematics subject classiﬁcation: Primary: 43A10, 46H05; Secondary: 43A22. Keywords and phrases: Locally compact groups, measure algebra, generalised functions, second dual.

1. Introduction

All over this note G is a locally compact group and the notation C0(G) refers to the space of all functions vanishing at infinity. The letter M(G) means that the measure algebra of G consisting of all complex regular Borel measures on G with the total variation norm and the convolution product “∗" defined by the formula ∫ ∫ ⟨µ ∗ ν, g⟩ = g(xy) dµ(x) dν(y) G G for all µ, ν ∈ M(G) and g ∈ C0(G). It is folklore that M(G) is the first dual space of C0(G) for the pairing ∫ ( ) ⟨µ, g⟩ := g(x) dµ(x) µ ∈ M(G), g ∈ C0(G) . G

In the last thirty years, researches on the second dual of Banach algebras have mostly focused around the Banach algebras related to locally compact groups, and have been dealt with by Lau and his coauthors [1–5]. In particular, Ghahramani and Lau [4] presented the ﬁrst important work devoted to the study of the second duals of measure algebras. Among other things, they have conjectured that:

4 H. Javanshiri

• The Banach algebra M(G) is strongly Arens irregular. • The second dual of M(G) determines G in the category of all locally compact groups. Later on, the second dual of the measure algebras has been studied in a series of papers. In particular, Dales, Lau and Strauss [3] published the second important work devoted to the study of M(G)∗∗, the second dual space of M(G), where most of the known results about these Banach algebras up to 2012 can be found in it. Recall from [3] that M(G)∗, the first dual space of M(G), as the second dual of the ∗ ∗ e C -algebra C0(G) is a commutative unital C -algebra. Therefore, if G denotes the hyper-Stonean envelope of G, then we can recognize M(G)∗ as C(Ge), the space of all bounded complex-valued continuous functions on Ge. It follows that M(G)∗∗ M(Ge), where denotes the isometric algebra isomorphism and many authors, up to the year 2012, have used a type of this identification as a tool for the study of M(G)∗∗. In 2009, my PhD supervisor, professor Rasoul Nasr-Isfahani, brought to my attention the theory of generalised functions and pointed out the link between GL(G) and M(G)∗. Later on, I have spent a good part of the last 6 years thinking about the nature of GL(G) and GL(G)∗ and their relationship with M(G)∗ and M(G)∗∗, respectively. This talk is about the study of M(G)∗∗ using the theory of generalised functions. Particularly, I will survey what we know, what we don’t know, and what we should know. The talk is based on the following seven joint papers with Nasr-Isfahani from Isfahan and Esslamzadeh from Shiraz: • G. H. Esslamzadeh, H. Javanshiri and R. Nasr-Isfahani, Locally convex algebras which determine a locally compact group, Studia Math. 233 (2016), 197–207. • G. H. Esslamzadeh, H. Javanshiri and R. Nasr-Isfahani, Bipositive isomorphisms between the second duals of measure algebras of locally compact groups, pp. 9, preprint. • H. Javanshiri and R. Nasr-Isfahani, More on the locally convex space (M(X), β(X)) of a locally compact Hausdorff space X, Bull. Belg. Math. Soc. Simon Stevin. 23 (2016), 191–201. • H. Javanshiri and R. Nasr-Isfahani, Introverted subspaces of the duals of measure algebras, pp. 17, preprint. • H. Javanshiri and R. Nasr-Isfahani, The strict topology for the space of Radon measures on a locally compact Hausdorff space, Topology Appl. 160 (2013), 887–895. • H. Javanshiri and R. Nasr-Isfahani, The strong dual of measure algebras with certain locally convex topologies, Bull. Aust. Math. Soc. 87 (2013), 353–365. • H. Javanshiri and R. Nasr-Isfahani, Some properties of the topological centres of M(G)∗∗ in terms of generalised functions, pp. 13, in preparation.

5 State of the art in the study of the second duals of measure algebras

References [1] H. G. Dales, A. T.-M. Lau, The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177 (2005), 1–191. [2] H. G. Dales, A. T.-M. Lau and D. Strauss, Banach Algebras on Semigroups and on Their Compactiﬁcations, Dissertationes Math. 205 (2010), 1–165. [3] H. G. Dales, A. T.-M. Lau and D. Strauss, Second Duals of Measure Algebras, Dissertationes Math. 481 (2012), 1–121. [4] F. Ghahramani and A. T.-M. Lau, Multipliers and ideal in second conjugate algebra related to locally compact groups, J. Funct. Anal. 132 (1995), 170–191. [5] A. T.-M. Lau and J. Pym, Concerning the second dual of the group algebra of a locally compact group, J. London Math. Soc. 41 (1990), 445–460.

Hossein Javanshiri, Department of Mathematics Yazd University Yazd, Iran e-mail: [email protected]

6 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

Plenary Presentation

WAVELET ANALYSIS

Rajab Ali Kamyabi-Gol

Abstract Wavelet analysis is a particular time-scale presentation of signals which has found a wide range of applications in physics, mathematics and engineering. We will review the principal aspects of this technique, both from the theoretical and the practical points of view. with particular emphasis on the continuous wavelet transform (CWT).

2010 Mathematics subject classiﬁcation: Primary 42C40. Keywords and phrases: Continuous wavelet transform (CWT), signal analysis, Fourier analysis.

1. Introduction It is the fact that most real life signals, are non-stationary and usually cover a wide range of frequencies. Many signals contain transient components. Also, frequently (but not always) a direct correlation exists between the characteristic frequency of a given segment of the signal and the time duration of that segment. Clearly, the standard Fourier analysis is inadequate for treating such signals. Strictly speaking, it implies only to stationary signals and loses all information about the time localization of a given frequency components. Fourier analysis is highly unstable with respect perturbation. Facing these problems, signal analysts turn to time-frequency representations. Two time-frequency transforms stand out as particularly simple and eﬃcient. the windowed Fourier transform and the wavelet transform. In this talk we we focus on the wavelet transform. Actually there are three kinds of wavelet transform, the continuous wavelet transform (CWT), the descritized Wavelet transform and the discrete wavelet transform (DWT). The CWT plays the same role as the Fourier transform and it is mostly used for analysis and future detection in signals, where the DWT is the analogous of the discrete Fourier transform and is more appropriate for data compression and signal reconstruction.

References [1] S. T. Ali, J. P Antoine, J. P. Gazeau, Coherent States, Wavelets, and Their Generalizations, Spriger, (1999).

7 R. A. Kamyabi-Gol

Rajab Ali Kamyabi-Gol, Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran e-mail: [email protected]

8 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

Plenary Presentation

HARMONIC ANALYSIS ON HOMOGENEOUS SPACES

Narguess Tavallaei

Abstract Considering a topological group G and a closed subgroup H, we give some results on the function spaces and the Banach space of all bounded Borel measures on the homogenous space G/H. We focus on two aspects, when H is compact and when G is the semidirect product of H and one of its normal subgroups.

2010 Mathematics subject classiﬁcation: Primary 43A85, 43A10, Secondary 46H25. Keywords and phrases: Homogeneous space, measure algebra, involution, bounded approximate identity, amenability.

1. Introduction Within the past decades, wavelet transforms and other multiscale transforms have been intensively used in both applied and pure mathematics. The constructions of these transforms can be entirely based on an abstract group theoretic and representation theoretic approach. In details, a wavelet transform is deﬁned by a unitary representation of a topological group G on the Hilbert space of the square-integrable functions on a quotient space of G. Mostly, G is the semidirect product of two topological Lie groups; each of them may be considered as a homogeneous space. Due to the recent theoretical and computational work, it is valuable to study the well- known spaces built on homogenous spaces.

The function spaces and the Banach space of bounded Borel measures on a locally compact Hausdorff topological group may possess special structures and properties which may fail to hold on the spaces related to a locally compact Hausdorff space. When G is a locally compact Hausdorff topological group, for every closed subgroup H of G, the space G/H is a locally compact Hausdorff topological space on which G acts transitively from the left. The term homogeneous space means a transitive G-space which is topologically isomorphic to G/H, for some closed subgroup H of G. It has been shown that if G is σ-compact, then every transitive G-space is homeomorphic to a quotient space G/H for some closed subgroup H (cf. [3, Subsection 2.6]). We discuss recent results obtained by figuring out some relations between the

9 N. Tavallaei function spaces and bounded Borel measures on a homogeneous space and those on the corresponding topological group. One may refer to [4], [5], [8], and [9] in which, it has been introduced and investigated Fourier algebra A(G/H) and Fourier–Stieltjes algebra B(G/H), where H is compact.

2. Main results

Let G = K ⋊ H for some closed subgroups K and H. Then ρ(kh) = ∆H(h)/∆G(h), k ∈ K, h ∈ H, is a homomorphism rho-function for (G, H). When G/H is attached to the relatively invariant measure µ arising from ρ, for all 1 ≤ p < +∞, the Banach space Lp(G/H) is isometrically isomorphic to Lp(K). The Banach space M(G/H) can also be identiﬁed with M(K) and hence we may consider M(G/H) as a unital Banach ∗-algebra.

When H is a compact subgroup of a locally compact group G, considering G/H equipped with a strongly quasi-invariant Radon measure, we introduce a bounded p p surjective linear map T p : L (G) → L (G/H), where 1 ≤ p ≤ +∞. By restricting the domain of T p to a special closed subspace

p p p L (G : H) = { f ∈ L (G): Rξ f = f in L (G), ξ ∈ H}. of Lp(G), we show that Lp(G/H) is isometrically isomorphic to Lp(G : H). Using this, we can study the structure of the Lp-spaces constructing on a homogeneous space via those created on topological groups. In particular, the mapping T2 is the orthogonal projection of L2(G) on L2(G/H), by considering L2(G/H) as a closed 2 2 2 subspace of L (G). Hence, T2 maps every frame of L (G) onto a frame of L (G/H). Also, we can consider L1(G/H) as a closed subspace of the Banach algebra L1(G) and by transferring the convolution of L1(G), make L1(G/H) into a Banach algebra.

For a closed subgroup H of G, there∫ exists a norm decreasing∫ surjective linear map → / φ = φ T : M(G) M(G H), deﬁned by G/H (xH) dTm(xH) G (xH) dm(x), where φ ∈ C0(G/H). Moreover, the Banach space M(G/H) is isometrically isomorphic to M(G)/ ker(T), equipped with the usual quotient norm. So, we may deﬁne a left action of M(G) on M(G/H) which makes it into a Banach left M(G)-module.

When H is a compact subgroup of G, the Banach space M(G/H) is isometrically ∗ isomorphic to a speciﬁc closed subalgebra M(G : H) = C0(G : H) of the Banach ∗-algebra M(G). More precisely, the linear map M(G : H) → M(G/H), m 7→ Tm, is an isometry isomorphism. By transferring the convolution of Banach algebra M(G : H) to M(G/H), we make M(G/H) into a Banach algebra. In this case, one may consider L1(G/H) as a Banach subalgebra of M(G/H). Moreover, if H is a compact normal subgroup of G, then G/H is a locally compact topological group and the convolution on M(G/H), deﬁned as above, coincides with the well-known

10 Harmonic Analysis on Homogeneous Spaces convolution on the measure algebra of G/H. In this case, M(G/H) is a unital Banach ∗-algebra whose identity is δeH. Obviously, when H is a compact subgroup of G, the Banach algebra M(G/H) has a right identity δeH. We deduce M(G/H) has an identity only if H is a normal subgroup. This happens just when some approximate identity exists on each of L1(G/H) and M(G/H), or equivalently, L1(G/H) and M(G/H) are ∗-subalgebras of M(G). Moreover, each of the Banach algebras L1(G/H) and M(G/H) is amenable if and only if H is normal and G is amenable.

References [1] A. A. Arefijamal and R. A. Kamyabi-Gol, A characterization of square integrable representations associated with CWT, J. Sci. I. R. Iran 18(2) (2007) 159–166. [2] M. Fashandi, R. A. Kamyabi-Gol and A. Niknam, M. A. Pourabdollah, Continuous wavelet transform on a special homogeneous space, J. Math. Phys. 44(9) (2003) 4260-4266. [3] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, 1995. [4] B. E. Forrest, Fourier analysis on coset space, Journal of Mathematics 28(1) (1998). [5] B. E. Forrest, E. Samei and N. Spronk, Convolutions on compact groups and Fourier algebras of coset spaces, Studia Mathematica 196(3) (2010) 223–249. [6] F. Ghahramani, A. T. M. Lau and V. Losert, Isometric isomorphisms between Banach algebras related to locally compact groups, Trans. Amer. Math. Soc. 321(1) (1990) 273–283. [7] H. Javanshiri and N. Tavallaei, Measure algebras on homogeneous spaces, arXiv:1606.08773 [math.CA] (2016). [8] E. Kaniuth, Weak spectral synthesis in Fourier algebras of coset spaces, Studia Mathematica 197 (2010), no. 3, 229–246. [9] K. Parthasarathy and N. Shravan Kumar, Fourier algebras on homogeneous spaces, Bulletin des Sciences Mathematiques 135(2) (2011) 187–205. [10] H. Reiter and J. D. Stegeman, Classical Harmonic Analysis, 2nd Ed., Oxford University Press, New York, 2000. [11] N. Tavallaei, M. Ramezanpour and B. Olfatian Gilan, Structural transition between Lp(G) and Lp(G/H), Banach J. Math. Anal. 9(3) (2015) 194–205.

Narguess Tavallaei, Department of Mathematics, Damghan University, Damghan, Iran e-mail: [email protected]

11 Short Presentations The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

KURATOWSKI DENSITY IN THE STONE-Cˇ ECH COMPACTIFICATION

Mohammad Akbari Tootkaboni∗ and Jasieh Keshavarzian

Abstract C. Kuratowski introduced a measure of non-compactness for subsets of a complete metric space. Although this measure is real valued and monotone, but is not a measure in the measure-theoretical sense. Let S be a discrete semigroup, then S is embedded in l∞(S )∗. Since l∞(S )∗ is an inﬁnite Banach space , we can deﬁne the Kuratowski measure on l∞(S )∗ and so on S . In this paper, we investigate some properties of Kuratowski measure on a discrete semigroup.

2010 Mathematics subject classiﬁcation: 11B05, 54D35. Keywords and phrases: Density, Syndetic, piecewise syndetic, Thick.

1. Introduction If the set of the natural numbers is partitioned into finitely many cells, the one of the cells contains arbitrarily long arithmetic progressions. This theorem as an open problem was proved by Van der waerden in 1927. It is perhaps the result of Ramsey theory. In fact, F. Ramsey in 1930 proved a similar theorem about finite colorings. Erdos and Turan stated a conjecture about Van der Warden’s Theorem. The first, they |E∩{1,2,...,N}| defined density of a subset E of the natural numbers is defined as limN→∞ N , and asked: Any subset of the natural numbers which fails to contain an arithmetic progression of length k ∈ N must be of zero density. Szemeredi’s Theorem(1975) says whenever A is subset of N with positive upper density, A contains arbitrarily long arithmetic progressions. Szemeredi’s Theorem original proof was very long and complicated. So H. Furestenberg provided a shorter proof of this result with using dynamical system. In this area,∑ as an open problem that it had been raised by Erdos:˝ N 1 if A be subset of such that n∈A n be divergent, does A contain arbitrarily long arithmetic progressions? There is two natural definitions of density, upper density and Banach density. The first is defined by counting measure on the natural numbers, and the second, is defined bye counting measure and algebraic structure of the natural number. We define a density on a semigroup respect to norm on l∞(S )∗.

∗ speaker

13 M. Abari Tootkaboni, J. Keshavarzian

Definition 1.1. The Kuratowski measure of noncompactness λ(Ω) of the set Ω is the infinmum of the numbers d > 0 such that Ω admits a finite covering by sets of diameter smaller than d.

Now we list below some the properties of the MNC( Measure noncompactness) . 1- regularity: λ(Ω) = 0 if and only if Ω is totally bounded; 2- nonsingularity: λ is equal to zero on every one-element set; 3- monotonicity: Ω1 ⊆ Ω2 implies λ(Ω1) ≤ λ(Ω2); 4- semi-additivity: λ(Ω1 ∪ Ω2) = max{λ(Ω1), λ(Ω2)}; 5- Lipschitzianity: |λ(Ω1) − λ(Ω2)| ≤ ρ(Ω1, Ω2), where ρ denotes the Hausdorﬀ metric; 6-continuity: for any Ω ⊆ E and any ε > 0 there is a δ > 0 such that

|λ(Ω) − λ(Ω1)| < ε, for all Ω1 satisfying ρ(Ω, Ω1) < δ; 7- semi-homogeneity: λ(tΩ) = |t|λ(Ω) for any number t; 8- algebraic semi-additivity: λ(Ω1 + Ω2) ≤ λ(Ω1) + λ(Ω); 9- invariant under translation: λ(Ω + x◦) = λ(Ω) for any x◦ ∈ E;

2. Density determined by nets Let S be a discrete semigroup and suppose that λ is the Kuratowski measure on ∗ l∞(S ) . The collection of all bounded subsets of S with positive Kuratowski measure is denoted by B(S ).

Definition 2.1. Let = {Fn}n∈D be a net in B(S ), and let A ⊂ eqS . (a) dF (A) = sup{α :(∃m ∈ D)(∀n ≥ m)(λ(A ∩ Fn) ≥ α · λ(Fn))}. (b) dF (A) = sup{α :(∀m ∈ D)(∃n ≥ m)(λ(A ∩ Fn) ≥ α · λ(Fn))}. ∗ = {α ∀ ∈ ∃ ≥ ∃ ∈ ∪ { } λ ∩ ≥ α · λ } (c) dF (A) sup :( m D)( n m)( x S 1 )( (A (Fn x)) (Fn)) . ∗ Now we define DF and DF determined by these densities. Definition 2.2. Let F be a net in B(S ). (a) DF = {p ∈:(∀A ∈ p)(dF (A) > 0)}. ∗ = { ∈ ∀ ∈ ∗ > } (b) DF p :( A p)(dF (A) 0) .

Lemma 2.3. Let be a net in B(S ). If A and B are subsets of S , then dF (A ∪ B) ≤ { , } ∗ ∪ ≤ { ∗ , ∗ } ⊆ max dF (A) dF (B) and dF (A B) max dF (A) dF (B) . Consequently if A S and > ∩ , ∅ ⊆ ∗ > ∩ ∗ , ∅ dF (A) 0, then A DF and if A S and dF (A) 0, then A DF . Proof. It is obvious. □ Now we deﬁne the notions k−thick and k−piecewise syndetic. Recall that A ⊆ S ∈ ∈ + ⊆ is thick if and only if for every F P f (S ) there exists x S such∪ that F x A, A ∈ = − + is syndetic if and only if there exists H P f (S ) such that S ∪t∈H t A and A is ∈ − + piecewise syndetic if and only if there exists H P f (S ) such that t∈H t A is thick.

14 Kuratowski Density

Deﬁnition 2.4. Let A ⊆ S . − ∈ ∈ ⊆ (a) A is k thick if and only if for every F B(S ), there exists x S such that∪ Fx A. − ∈ −1 (b) A is k piecewise syndetic if and only if there exists H (S ) such that t∈H t A is k−thick.

Remark 2.5. Let F = {Fn}n∈D be a net in B(S ). If A is a k−thick subset of S , then ∗ = dF (A) 1. The three requirements in the following deﬁnition will guarantee certain properties of densities determined by nets of inﬁnite subsets of S .

Deﬁnition 2.6. Let F = {Fn}n∈D be a net in B(S ). Following are three properties that might satisfy. (∗)(∀ϵ > 0)(∀t ∈ S )(∃c ∈ N)(∃m ∈ D)(∀n ≥ m)(∃k ≥ n)(∃z ∈ S ) (λ(tFn \ Fkz) < ϵλ(Fn)) and (λ(Fk) ≤ cλ(Fn)). ′ (∗ )(∀ϵ > 0)(∀t ∈ S )(∃c ∈ N)(∃m ∈ D)(∀n ≥ m)(∃k ≥ n)(λ(tFn \ Fk) < ϵλ λ ≤ λ . (Fn)) and ( (Fk) c (Fn)) ∩ ∗∗ ∀ ∈ ∃ ∈ N ∃ ∈ ∀ ≥ λ ≤ λ −1 . ( )( H P f (S ))( c )( m D)( n m)( (Fn) c ( a∈H a Fn))

Theorem 2.7. Let F = {Fn}n∈D be a net in B(S ). If F satisﬁes (∗), B ⊆ S, t ∈ S , and ∗ −1 > ∗ > ∗ β dF (t B) 0, then dF (B) 0. In particular DF is a left ideal of S . roof α > ∗ −1 > α ϵ = ∗ −1 − α ∈ N P . Pick 0 such that dF (t B) and let dF (t B) . Pick c ∈ ∗ ϵ/ ∗ ≥ α/ and m D as guaranteed by ( ) for 2 and t. We claim that dF (B) c. To this −1 end, let r ∈ D. Pick n ∈ D and x ∈ N such that n ≥ r, n ≥ m, and λ(t B ∩ Fn x) ≥ (α + ϵ/2) · λ(Fn). Pick k ≥ n and z ∈ N such that λ(tFn \ Fkz) < (ϵ/2)λ(Fn) and λ(Fk) ≤ cλ(Fn). Since B ∩ tFn x ⊆ (B ∩ Fkzx) ∪ (tFn x \ Fkzx) so

λ(B ∩ tFn x) ≤ λ((B ∩ Fkzx) ∪ (tFn x \ Fkzx))

≤ max{λ(B ∩ Fkzx), λ(tFn x \ Fkzx)}

= λ(B ∩ Fkzx) + λ(tFn \ Fkz) and hence

λ(B ∩ Fkzx) ≥ λ(B ∩ tFn x) − λ(tFn \ Fkz) −1 ≥ λ(t B ∩ Fn x) − (ϵ/2)λ(Fn)

≥ αλ(Fn)

≥ (α/c)λ(Fk). ∗ β ∈ ∗ ∈ β ∈ To see that DF is a left ideal of S , let p DF , q S and B qp. So { ∈ −1 ∈ } ∈ −1 ∈ ∗ −1 > t S : t B p q so pick t such that t B p. Then dF (t B) 0 so ∗ > □ dF (B) 0. ′ Theorem 2.8. Let F = {Fn}n∈D be a net in B(S ). If satisﬁes (∗ ), B ⊆ S, t ∈ S , and −1 dF (t B) > 0, then dF (B) > 0. In particular DF is a left ideal of βS .

15 M. Abari Tootkaboni, J. Keshavarzian

Proof. The proof is the same of Theorem 2.7. □

Theorem 2.9. Let F = {Fn}n∈D be a net in B(S ). If satisﬁes (∗) and B is a k−piecewise ∗ > syndetic subset of S , then dF (B) 0. ∪ roof ∈ −1 − ∈ ∈ P . Pick H P∪f (S ) such that ∪t∈H t B is k thick. Given m D, pick x S ⊆ −1 λ −1 ∩ = λ such∪ that Fm x t∈H t B, so ( t∈H t B Fm x) (Fm). We conclude that ∗ −1 = ∈ ∗ −1 > dF ( t∈H t B) 1. Thus by Lemma 2.3 there is some t H such that dF (t B) 0, ∗ > □ so by Theorem 2.7 dF (B) 0.

Theorem 2.10. Let F = {Fn}n∈D be a net in B(S ). If F satisﬁes (∗∗) and B is a syndetic subset of S , then dF (B) > 0. ∪ roof ∈ = −1 ∈ N ∈ P . Pick H P f (S ) such that S t∈H t B. Pick c and m D as guaranteed ∗∗ = | | ≥ λ ∩ ≥ 1 λ by ( ) for H. Let k H ∩and let n m. We show that (B Fn) ( ck ) (Fn) and ≥ 1 = −1 λ ≤ λ ∈ ∈ thus dF (B) ck . Let G1 a∈L a Fn, so that (Fn) ∪c (G1). For s G1 pick t H ∈ ∈ , ∈ ⊆ −1 ∩ such that ts B. Since s G1 ts Fn. So G1 t∈H t (B Fn). This implies λ(G1) ≤ kλ(B ∩ Fn) as required. □ F = { } ∗∗ ∗ = Theorem 2.11. Let Fn n∈D be a net in B(S ). If satisﬁes ( ) and dF (B) 1, then B is thick. roof \ ∈ P ∪. Suppose that B is not thick, so that S B is syndetic. Pick H P f (S ) such that = −1 \ ∈ N ∈ ∗∗ = | | S t∈H t (S B) and pick c and m D as guaranteed by ( ) for H. Let k H . d∗ B > − 1 x ∈ S n ≥ m λ B ∩ F x > − 1 λ F Since ∩F ( ) 1 ck , pick and such that ( n ) (1 ck∪) ( n). Let = −1 ∈ = { ∈ ∈ \ } = G1 a∈H a Fn and for t H, let Et s G1 : tsx S B . G1 t∈H Et and λ ≥ 1 λ ∈ λ ≥ 1 λ ⊆ \ ∩ (G1) c (Fn) so pick t H such that (Et) ck (Fn). Now tEt (S B) Fn x, λ ∩ ≤ λ − λ ≤ λ − λ ≤ − 1 λ therefore (B Fn x) (Fn x) (tEt x) (Fn) (Et) (1 ck ) (Fn), a contradiction. □

References [1] R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskii Measure of Noncompactness and Condensing Operators, Berlin, (1992). [2] N. Hindman and D. Strauss, Density in Arbitrary Semigroups, Semigroup Forum. 73 (2006), 273-300.

Mohammad Akbari Tootkaboni, Department of Mathematics, University of Shahed, Tehran, Iran e-mail: [email protected]

Jasieh Keshavarzian, Department of Mathematics,

16 Kuratowski Density

University of Shahed, Tehran, Iran e-mail:

17 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

ON THE SQUARE-INTEGRABLE IRREDUCIBLE REPRESENTATIONS ON SEMI-DIRECT PRODUCT GROUPS

Farshad Afkhami

Abstract Higher dimensional analogs of the classical continuous wavelet transform are developed for Euclidean spaces whose dimension is a perfect square.The space of all n × n real matrices can be identified with Rn2 as an additive abelian group.The group of invertible n×n real matrices naturally acts on this abelian group by matrix multiplication.The resulting semidirect product group forms a group of affine transformations of Rn2 which may be viewed as a generalization of the affine transformations of Rn2 whose unique square-integrable representation underlies the classical one-dimensional continuous wavelet transform. A continuous wavelet transform for Rn2 is derived and its specific details are worked out for R4 resulting in a 4D continuous wavelet transform.We provide three equivalent versions of the irreducible representation of G=A⋊H for A=M(n,R) and H=GL(n,R) underlies the continuous wavelet transform and then three versions of the irreducible representation of G=A⋊H such that A is locally compact abelian group and H is locally compact group.

2010 Mathematics subject classiﬁcation: Primary 42C40; Secondary 43A65. Keywords and phrases: : representation, semi-direct product, wavelet transform .

References

Farshad Afkhami, Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran e-mail: [email protected]

18 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

DERIVATIONS ON THE BANACH ALGEBRA L∞(G)∗ OF A LOCALLY COMPACT GROUP

Mohammad Hossein Ahmadi Gandomani∗ and Mohammad Javad Mehdipour

Abstract In this paper, we present some results concerning derivations and Jordan derivations on the noncommutative Banach algebra L∞(G)∗. We show that the range of any (skew-) centralizing derivation on L∞(G)∗ is embedded into M(L∞, C(G)), and the zero map is the only skew-commuting derivation on L∞(G)∗. We ∞ ∗ ∞ ∗ also prove that the subspace Annr(L (G) ) of L (G) is invariant under Jordan derivations.

2010 Mathematics subject classiﬁcation: Primary 43A15, Secondary 16W25. Keywords and phrases: Locally compact group, derivation, Jordan derivation, centralizing derivation, skew centralizing derivation .

1. Introduction Let G be a locally compact group with a fixed left Haar measure λ. The Banach space of complex-valued integrable functions with respect to λ is denoted by L1(G). With the norm ∥ · ∥ and with convolution 1 ∫ ϕ ∗ ψ(x) = ϕ(y)ψ(y−1 x) dλ(y)(x ∈ G) G as product, L1(G) becomes a Banach algebra. Let L∞(G), the usual Lebesgue space as defined in [2] equipped with the essential supremum norm ∥ · ∥∞. It is well-known that the dual of L∞(G), represented by L∞(G)∗, is a Banach algebra with the first Arens product “ · ” defined by the formula ⟨m · n, f ⟩ = ⟨m, n f ⟩, where ⟨n f, ϕ⟩ = ⟨n, f ϕ⟩, and ⟨ f ϕ, ψ⟩ = ⟨ f, ϕ ∗ ψ⟩ for all m, n ∈ L∞(G)∗, f ∈ L∞(G) and ϕ, ψ ∈ L1(G); for example see [3]. Note that we may consider ϕ ∈ L1(G) as a linear functional in L∞(G) by the formula ∫ ⟨ϕ, f ⟩ = f (x)ϕ(x) dλ(x) G ∗ speaker

19 M. H. Ahmadi Gandomani and M. J. Mehdipour for all f ∈ L∞(G). Let Λ(G) denote the set of all weak∗-cluster points of an approximate identity in L1(G) bounded by one. It is easy to see that if u ∈ Λ(G), then m · u = m and u · ϕ = ϕ for all m ∈ L∞(G)∗ and ϕ ∈ L1(G). The right annihilator of L∞(G)∗ is denoted by ∞ ∗ ∞ ∗ ∞ ∗ Annr(L (G) ) and is the set of all r ∈ L (G) such that L (G) · r = {0}. It is well- known from [4] that L∞(G)∗ is the direct sum of the norm closed right ideal u · L∞(G)∗ ∞ ∗ and Annr(L (G) ) for all u ∈ Λ(G). Let M(L∞, C(G)) = {Φ : Φ is a continuous linear map from L∞(G) to C(G) and Φ(ψ ∗ f ) = ψ ∗ Φ( f ) for all ψ ∈ L1(G), f ∈ L∞(G)} where C(G) is the space of all complex-valued bounded continuous functions on G. One can show that ∞ ∞ ∗ M(L , C(G)) = ∩u∈Λ(G)u.L (G) ; see [4]. A derivation on L∞(G)∗ is a linear mapping d on L∞(G)∗ satisfying d(m · n) = d(m) · n + m · d(n). A derivation d on L∞(G)∗ is called centralizing if [d(m), m] ∈ Z(L∞(G)∗) for all m ∈ L∞(G)∗. Also, derivation d is called skew-centralizing if d(m) · m + m · d(m) ∈ Z(L∞(G)∗), for all m ∈ L∞(G)∗, where Z(L∞(G)∗) is the center of L∞(G)∗, the set of all m ∈ L∞(G)∗ such that m · n = n · m for all n ∈ L∞(G)∗ and [m, n]:= m · n − n · m for all m, n ∈ L∞(G)∗. Centralizing derivations have been studied by Posner [5]. He showed that the zero map is the only centralizing derivation on a noncommutative prime ring. In [1] Bresar proved that there is no nonzero additive mapping in a prime ring R of characteristic diﬀerent from 2 which is skew-commuting on R. Let us remark that if G is a nondiscrete group, then Hahn-Banach theorem shows ∞ ∗ that there is a nonzero element r ∈ Annr(L (G) ). Hence for every u ∈ Λ(G), we have u · r = 0 and r · u = r and r · L∞(G)∗ · r = {0}. So L∞(G)∗ is a noncommutative Banach algebra. This also implies that L∞(G)∗ is not a semiprime ring. Hence we cannot apply the well-known results concerning derivations of commutative Banach algebras and derivations of prime rings to L∞(G)∗. In this paper we investigate the truth of these results for L∞(G)∗.

20 Derivations on L∞(G)∗

In this paper, we investigate (skew-) centralizing derivations on L∞(G)∗ and prove that the range of any (skew-) centralizing derivation on L∞(G)∗ is embedded into M(L∞, C(G)). We also prove that the zero map is the only skew-commuting derivation ∞ ∗ ∞ ∗ ∞ ∗ on L (G) . Finally, we show that the subspace Annr(L (G) ) of L (G) is invariant under Jordan derivations.

2. Main Results

We commence this section with the main result of the paper.

Theorem 2.1. Let G be a locally compact group and d be a derivation on L∞(G)∗ such that [[d(m), m], m] = 0 for all m ∈ L∞(G)∗. Then the range of d is embedded into M(L∞, C(G)).

As an immediate consequence of Theorem 2.1 we have the following result.

Corollary 2.2. Let G be a locally compact group and d be a centralizing derivation on L∞(G)∗. Then the range of d is embedded into M(L∞, C(G)).

Now, we prove a result concerning skew-centralizing derivations.

Theorem 2.3. Let G be a locally compact group and d be a skew-centralizing derivation on L∞(G)∗. Then the range of d is embedded into M(L∞, C(G)).

A derivation d on L∞(G)∗ is called skew-commuting if

d(m) · m + m · d(m) = 0, for all m ∈ L∞(G)∗.

Theorem 2.4. The zero map is the only skew-commuting derivation on L∞(G)∗.

A linear mapping d on L∞(G)∗ is said to be a Jordan derivation if

d(m2) = d(m) · m + m · d(m) for all m ∈ L∞(G)∗.

Theorem 2.5. Let G be a locally compact group and d be a Jordan derivation on L∞(G)∗. Then the following statements hold. ∞ ∗ ∞ ∗ (a) d maps Annr(L (G) ) into Annr(L (G) ). (b) d(u · m) = d(u) · m + u · d(m) for all m ∈ L∞(G)∗ and u ∈ Λ(G). Moreover, d ∞ ∗ maps Λ(G) into Annr(L (G) ).

21 M. H. Ahmadi Gandomani and M. J. Mehdipour

References [1] M. Bresar M., On skew-commuting mappings of rings, Bull. Austral. Math. Soc. 47 (1993) 291- 296. [2] E. Hewitt and K. Ross, Abstract Harmonic Analysis I, Springer-Verlag, New York, 1970. [3] E. Kaniuth, A Course in Commutative Banach Algebras, Springer-Verlag, New York, 2009. [4] A. T. Lau and J. Pym, Concerning the second dual of the group algebra of a locally compact group, J. London Math. Soc. 41 (1990) 445-460. [5] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957) 1093-1100.

Mohammad Hossein Ahmadi Gandomani, Department of Mathematics, Ghirokarzin Branch, Islamic Azad University, Ghirokarzin, Iran e-mail: [email protected]

Mohammad Javad Mehdipour, Department of Mahematics, Shiraz University of Technology, Shiraz, Iran e-mail: [email protected]

22 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

SOME CHARACTERIZATIONS OF TRIVOLUTIVE BANACH ALGEBRAS

Ahmad Alinejad∗ and Mehdi Rostami

Abstract In this paper, we present a characterization of the closed ideal of a trivolutive Banach algebra (A, τ). The main results are: a) the closed ideal I = Ker τ of a unital Banach algebra A is principle. b) For ∗∗ a commutative trivolutive algebra A, there is an idempotent u ∈ A such that Ker τ = Iu in which Iu = {a ∈ A; au = 0}. After that we investigate trivolutions on the Fourier-Stieltjes algebra B(G). Moreover, we study trivolutions on the algebra A ×θ B with θ-Lau product in relation to the corresponding one of A and B.

2010 Mathematics subject classiﬁcation: Primary 46K05, Secondary 43A20. Keywords and phrases: Trivolution, conditional expectation, group algebra.

1. Introduction A famous result of Civin and Yood [2, Theorem 6.2], shows that a necessary and sufficient condition for a Banach ∗-algebra A to be Arens regular is that its involution ∗, can be naturally extended to an involution on A∗∗, called the canonical extension. In particular, the canonical involution on L1(G) has canonical extension to an involution on L1(G)∗∗ if and only if G is finite. Moreover, Grosser in [4, Theorem 1] has proved that if A is a Banach algebra with a right bounded approximate identity, then A∗∗ with the first Arens product is a Banach ∗-algebra only if A∗ · A = A∗. Applying to the group algebra L1(G) (respectively the Fourier algebra A(G)) we conclude that if L1(G)∗∗ (respectivelly A(G)∗∗) is a Banach ∗-algebra then G is discrete (respectively compact). The question of existence of involutions on L1(G)∗∗ was originally given attention by Duncan and Hosseiniun. More recently, the first author in [1] prove that if G is an amenable group, then the natural involution of A(G) has the canonical extension to A(G)∗∗ if and only if G is finite. Furthermore, they studied involutions on the second dual of algebras related to locally compact topological semigroups. The concept of trivolution as a generalization of involution has been introduced and studied by Filali, Monfared, and Singh in [3]. A trivolutive algebra is a pair (A, τ) where A is a complex algebra and τ is a continuous conjugate-linear anti-homomorphism on A such that τ3 = τ, i.e. τ is a generalized inverse of itself. Slightly more generally, they obtained

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23 A. Alinejad, M. Rostami some fundamental characterizations of trivolutions. Trivolutive algebras have been studied in various situations, for example, a well-known characterization of trivolution states that there exists a projection p : A → B, with p an algebra homomorphism, a two sided ideal I of A, and an involution ρ on subalgebra B such that A = I ⊕ B, and τ = ρ ◦ p, in which, p = τ ◦ τ, B = τ(A) = p(A), I = ker p = ker τ and ρ = τ|B. They studied trivolutions on the unitized algebra A♯ and proved that unlike involutions, every trivolution on A has a canonical extension to two trivolution on A♯. Moreover, they obtained several results on the existence or non-existence of involutions on the dual of a topologically introverted subspace. Also they investigated conditions under which the dual of a topologically introverted subspace admits trivolutions. Afterwards in [1], the authors considered trivolutions on algebras associated to locally compact topological semigroups and locally compact groups. As an interesting result, it is shown that for a locally compact group G, the second dual of the Fourier algebra always admits a trivolution with range in Bρ(G), unlike involutions. This paper has been organized as follows. In section 2, using the notion of splitting exact sequnce, for a unital Banach algebra A, the kernel of any trivolution is a principle ideal. Also, the kernel of a trivolution τ on A admits a bounded approximate identity when A has a bounded approximate identity. By this result, we characterize the kernel of any trivolution on a commutative Banach algebra A with bounded approximate identity. ∗∗ Actually we show that Ker τ = Iu = {a ∈ A; au = 0} for an idempotent u ∈ A . Moreover, we studied trivolution on the Fourier-Stieltjes algebra, B(G) with range A(G). In section 3, we strengthen [3, Theorem 2.10] and we give a characterization of the existence of trivolutions on A ×θ B, with θ-Lau product in relation to the corresponding one of A and B.

2. Trivolutive algebra Let (A, τ) be a trivolutive algebra. To follow [3], we shall recall the canonical decomposition of a trivolutive algebra . In the algebraic settings, for a trivolutive algebra (A, τ), there exist a projection p : A → B with p an algebra homomorphism, a two sided ideal I of A, and an involution ρ on subalgebra B such that

A = I ⊕ B, τ = ρ ◦ p. (1)

Moreover, it should be noted that in the canonical decompositions of a trivolutive algebra (A, τ) we have, p := τ2 = τ ◦ τ, B = p(A) = τ(A), I = Ker p = Ker τ and ρ = τ|B. Moreover, every trivolutive Banach algebra is a strongly splitting extension of an involutive Banach algebra. If A is a trivolutive Banach algebra, then the operator p is a contractive projection. Recal the deﬁnition of a conditional expectation. Deﬁnition 2.1. Let A be a Banach algebra and B be a closed subalgebra of A. A surjective projection Q : A → B is called a conditional expectation if

Q(axb) = aQ(x)b, (a, b ∈ B, x ∈ A).

24 Some characterizations of trivolutive Banach algebras

We start with a good observation of trivolutive algebras. The following theorem deal with the relation between the notion of trivolution and conditional expectation. Theorem 2.2. Let (A, τ) be a trivolutive Banach algebra. Then the projection p obtained above is a conditional expectation. In the following results our concerning will be the charactrizations of the ideal I = ker τ in the trivolutive algebra (A, τ). In the next theorem we show that if A is a unital Banach algebra with a trivolution τ, then ker τ is a principle ideal. Theorem 2.3. Let (A, τ) be a unital trivolutive Banach algebra. Then there is a central idempotent e in I = Ker τ such that I = Ae. Corollary 2.4. Let τ be a trivolution on a unital Banach algebra A such that Ker τ = Rad(A). Then A is semisimple. Theorem 2.5. Let (A, τ) be a trivolutive Banach algebra with a bounded approximate identity. Then the ideal I admits a bounded approximate identity. Remark 2.6. Let A be a commutative Banach algebra with a bounded approximate identity. We equip A∗∗ with the (ﬁrst) Arens multiplication. To each idempotent element ∗∗ u of A we associate the closed ideal Iu = {a ∈ A; au = 0} in A. Lau and Ülger in [5] have shown that a closed ideal I of A has a bounded approximate identity iﬀ there ∗∗ is an idempotent u ∈ A such that I = Iu. Now, let (A, τ) be a commutative trivolutive Banach algebra with a bounded approximate identity. Then by the above discision ∗∗ there is an idempotent u ∈ A such that Ker τ = Iu. Moreover, the quotient algebra B = Im τ = A/I is isomorphic to an involutive subalgebra of A.

3. Third Section Let A and B be Banach algebras for which the character space ∆(B) , ∅ and let θ ∈ ∆(B). Then the θ-Lau product A ×θ B is the Cartesian product A × B equipped with the algebra multiplication

(a, b) ×θ (c, d) = (ac + θ(b)c + θ(d)a, bd) and the norm ∥(a, b)∥ = ∥a∥ + ∥b∥. Then A ×θ B is a Banach algebra.

Remark 3.1. If we let B = C and θ : C → C is the identity map, then A ×θ C coincides with A♯. The general deﬁnition and intensive study of these algebras was initially given attention by Monfared. In the following we show that if A and B are trivolutive Banach algebras then so is A ×θ B and we obtain some trivolutions on A ×θ B.

Theorem 3.2. Let (A, τ1) and (B, τ2) be trivolutive algebras. Then the map τ : A ×θ B −→ A ×θ B deﬁned by

τ(a, b) = (τ1(a) + θ(τ2(b))x0, τ2(b)), ∈ 2 = − τ = τ = τ = θ ◦ τ = θ, where x0 A with x0 x0 and 1(A)x0 x0 1(A) 0 and 1(x0) 0 and 2 is a trivolution on A ×θ B.

25 A. Alinejad, M. Rostami

Theorem 3.3. Let (A, τ1) and (B, τ2) be trivolutive algebras. Then the map τ deﬁned by τ(a, b) = (τ1(a) + θ(τ2(b))x0, 0), where x0 ∈ τ1(A) is the identity of τ(A) and θ ◦ τ2 = θ, is a trivolution on A ×θ B.

References [1] A. Alinejad and A. Ghaﬀari, Involutions and Trivolutions on Second Dual of Algebras Related to Locally Compact Groups And Topological Semigroups, Proceedings Mathematical Sciences, Accepted. [2] P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pac. J. Math. 11, (1961), 847-870. [3] M. Filali, M. Sangani Monfared and A. I. Singh, Involutions and trivolutions in algebras related to second duals of group algebras, Illinois Journal of Mathematics, Vol. 57, Number 3, (2013), 755–773. [4] M. Grosser, Algebra involutions on the bidual of a Banach algebra, Manuscripta Math. 48, (1984), 291-295. [5] A. T.-M. Lau and A. Ülger, Characterization of closed ideals with bounded approximate identities in commutative Banach algebras , complemented subspaces of the group von Neumann algebras and applications, Trans. Amer. Math. Soc. 366 (2014), no. 8, 4151–4171.

Ahmad Alinejad, College of Farabi, University of Tehran, Tehran, Iran e-mail: [email protected]

Mehdi Rostami, Faculty of Mathematical and Computer Science, Amirkabir University of Technology, Tehran, Iran e-mail: [email protected]

26 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

SOLVING THE WAVE EQUATION WITH SHEARLET FRAMES

Mojgan Amin khah∗, Ataollah Askari Hemmat and Reihaneh Raisi Tousi

Abstract

In this paper, we apply parseval shearlet frames to solve the wave equation. To this end, using the Plancherel’s theorem, we calculate the shearlet coeﬃcients.

2010 Mathematics subject classiﬁcation: Primary 42C15, Secondary 42C40. Keywords and phrases: Shearlet system, wave equation, tight frame.

1. Introduction

In the context of seismic oil and gas exploration, the wave equation is the important factor to establish the link between the earth properties and the observed data at the surface. In the past, several traditional methods are proposed for solving the wave equation, including the finite difference method [1], the pseudospectral method [2], the finite element method. In recent years, wavelets are used to solve the wave equation. Compared with wavelets, it seams that shearlets are more efficient for solving the wave equation [5]. A discrete shearlet system associated with ψ ∈ L2(R2) is defined by

− 3 4 j 2 {ψ j,k,m = a ψ(S kA − j · −m): j, k ∈ Z, m ∈ Z }, a0 > 0, (1) 0 a0

where the parabolic scaling matrices Aa0 and the shearing matrix S k are given by   [ ] a   0 0  1 k Aa =  1  , S k = . (2) 0 2 0 1 0 a0

The discrete shearlet transform of f ∈ L2(R2) is the mapping deﬁned by

2 f 7→ SH ψ f ( j, k, m) = ⟨ f, ψ j,k,m⟩, ( j, k, m) ∈ Z × Z × Z .

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27 M. Amin khah, A. Askari Hemmat and R. Raisi Tousi

2. Main result We start with the construction of a tight frame of shearlets for L2(R2). Let ϕ be the 2 scaling function of a Meyer wavelet. For ξ = (ξ1, ξ2) ∈ R , let Φˆ (ξ) = Φˆ (ξ1, ξ2) = ϕˆ(ξ1)ϕˆ(ξ2) and √ 2 −2 −2 2 W1(ξ) = W1(ξ1, ξ2) = Φˆ (2 ξ1, 2 ξ2) − Φˆ (ξ1, ξ2). It follows that ∑ Φˆ 2 ξ , ξ − 2 −2 jξ , −2 jξ = , ξ , ξ ∈ R2. ( 1 2) W1 (2 1 2 2) 1 for ( 1 2) j≥0 Notice that W2 = W2(2−2 j·). In particular, the functions W2 , j ≥ 0, produce a smooth 1 j 1 ∑ 1 j 2 −2 jξ = , ξ ∈ tiling of the frequency plane into Cartesian coronae: j≥0 W1 (2 ) 1 for R2\ − 1 , 1 2 ⊂ R2. ∈ ∞ R ⊂ , [ 8 8 ] Next, let W2 C ( ) be chosen so that suppW2 [1 1] and 2 2 2 |W2(ζ − 1)| + |W2(ζ)| + |W2(ζ + 1)| = 1, for |ζ| ≤ 1. = (n) = ≥ . In addition, we will assume that W2(0) 1 and that W2 (0) 0 for all n 1 Using this notation we state the following definition [3, 4]. 2 2 2 Definition 2.1. For ξ = (ξ1, ξ2) ∈ R , the shearlet system for L (R ) is defined as the countable collection of functions j j 2 {ψ j,l,k : j ≥ 0, −2 ≤ l ≤ 2 , k ∈ Z }, (3) where − − j −2 j − j −l 2πiξA jS −lk ψˆ ξ = | | 2 ξ ξ 4 1 , j,l,k( ) detA4 W1(2 )W2( A4 S 1 )e and A4, S 1 are given by (2). Using the above observation it can be shown that the shearlet system (3) is a tight frame [3]. Theorem 2.2. The shearlet system (3) is a tight frame for L2(R2). Because of the shearlet system (3) forms a tight frame, we can expand a function f ∈ L2(R2) as a series of shearlet ∑ f = ⟨ f, ψ j,l,k⟩ψ j,l,k, (4) j,l,k

We denote the shearlet coeﬃcients ⟨ f, ψ j,l,k⟩ by C j,l,k. According to the Plancherel’s theorem, we have

C , , = ⟨ f, ψ , , ⟩ = ⟨ fˆ, ψˆ , , ⟩ j l k j l∫k j l k 1 = fˆ(ξ)ψˆ , , (ξ)dξ π 2 j l k (5) (2 ) ∫R2 − 1 − 3 j − − j −l 2πiξA jS −lk = fˆ ξ 2 W 2 jξ W ξA S e 4 1 dξ 2 ( )2 1(2 ) 2( 4 1 ) (2π) R2

28 Solving the wave equation with shearlet frames

Now with the aid of shearlet coefficients we solve the wave equation. We consider the wave equation as follows ∂2u(x, t) = c2∆u(x, t), ∂t2 ∂u(x, t) u(x, t)| = = u (x), | = = u (x), t 0 1 ∂t t 0 2 = , ∈ R2 ∆ , = ∂2u(x,t) + ∂2u(x,t) . ffi where x (x1 x2) and u(x t) ∂ 2 ∂ 2 The shearlet coe cients (5) for x1 x2 u are defined by ∫ 1 C , , = u ξ ψˆ , , ξ dξ, j l k 2 ˆ( ) j l k( ) (2π) R2 ∫ ∆ 1 c C = ∆u ξ ψˆ , , ξ dξ. j,l,k 2 ( ) j l k( ) (2π) R2 By the properties of Fourier transform for derivative, we have

c 2 2 2 ∆u(ξ1, ξ2) = ((iξ1) + (iξ2) )û(ξ) = −|ξ| uˆ(ξ). (6) Finally, setting (6) in (5) and applying a change of variables, we obtain the coefficient shearlet associated to u. substituting the coefficient shearlet in (4), we obtain the desirable result.

References [1] R. M. Alford, K. R. Kelly and D. M. Boore, Accuracy of finite difference modeling of the acoustic wave equation, Geophysics 39 (1974) 834-842. [2] J. Gazdag, Modeling of the acoustic wave equation with transform methods, Geophysics 46.6 (1981) 854-859. [3] K. Guo and D. Labate, The construction of smooth Parseval frames of shearlets, Mathematical Modelling of Natural Phenomena 8.1 (2013) 82-105. [4] S. Hauser¨ and G. Steidl, Fast finite shearlet transform, arXiv preprint arXiv:1202.1773 – (2012) [5] G. Kutyniok and W. Q. Lim, Compactly supported shearlets are optimally sparse, Journal of Approximation Theory 163.11 (2011) 1564-1589.

Mojgan Amin khah, Department of Mathematics, Faculty of Sciences and new Technologies, Graduate University of Advanced Technology, Kerman, Iran e-mail: [email protected]

Ataollah Askari Hemmat, Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar Uninersity of Kerman, Kerman, Iran e-mail: [email protected]

29 M. Amin khah, A. Askari Hemmat and R. Raisi Tousi

Reihaneh Raisi Tousi, Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran e-mail: [email protected]

30 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

SHEARLET FRAME APPLICATIONS IN SOLVING SOME 2-D NUMERICAL ANALYSIS PROBLEMS

Rama Amiri∗, Mohammad Zarebnia and Reihaneh Raisi Tousi

Abstract In numerical analysis specially in finding numerical solution for PDEs and integral equations, different basis functions are used. Frames have not been used in the mentioned methods, because of the lack of independency between their elements. In this paper, first we introduce a shearlet frame then we extract a finite basis from it.

2010 Mathematics subject classiﬁcation: Primary 42C15, Secondary 42C40. Keywords and phrases: Shearlet frame, Fredholm integral equations, collocation method.

1. Introduction In the recent years some efficient methods have been introduced which apply shearlet frames and transforms in image processing. This idea draw us in spiration from it for numerical analysis purposes. In this section, we introduce shearlet systems and we state some useful theorems. ∫ |ψˆ , |2 ψ ∈ 2 R2 (w1 w2) < ∞ Definition 1.1. If L ( ) satisfies R2 2 dw2dw1 , it is called an w1 admissible shearlet.

2 2 w2 Definition 1.2. Let ψ ∈ L (R ) be defined by ψˆ(w) = ψˆ(w1, w2) = ψˆ 1(w1)ψˆ 2( ) where w1 2 2 ψˆ 1 ∈ L (R) is a discrete wavelet and ψˆ 2 ∈ L (R) is a bump function [1]. Then ψ is called a classical shearlet. Definition 1.3. A finite discrete system of shearlets is defined as a collection of functions of the form

3 j −1 −1 m1 m2 Ψ(ψ) = {ψ , , (x) = 2 2 ψ(A S (x − t )) : t = ( , ) , j, k ∈ Z}, (1) j k m j k, j m m M N [ ] [ ] − j − j 4 0 1 k2 j j where A = , S , = , −2 ≤ k ≤ 2 . M, N are the number of j 0 2− j k j 0 1 partitions of the x, y interval, respectively.

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31 R. Amiri, M. Zarebnia and R. Raisi Tousi

Theorem 1.4. [2] Let ψ ∈ L2(R2) be a classical shearlet, then the shearlet system Ψ(ψ) ∪ Ψ˜ (ψ˜) ∪ Φ(ϕ) is a Parseval frame for L2(R2). Where

2 Φ(ϕ) = {ϕm = ϕ(x − m), m ∈ Z } is called scalling function and Ψ˜ (ψ˜) has the same deﬁnition as ψ, where the shearlet Ψ˜ can be chosen likewise with the roles of x, y in x = (x, y) (see [2]). It can be easily shown that the system

Ψˆ (ψˆ) ∪ Ψ˜ˆ (ψ˜ˆ) ∪ Φˆ (ϕˆ) (2) is also a parseval frame for L2(R). Here, Ψˆ (ψ) is deﬁned by

3 j w1 − j − j Ψˆ (ψˆ) = {ψˆ , , (x) = 2 2 exp (−2ϕi⟨w, t ⟩)ψˆ( , 2 (k2 w + w ): j k m m 4 j 1 2 m m (3) t = ( 1 , 2 ) , j, k ∈ Z}, m M N

2. Main results For constructing a shearlet frame we need the shearlet ψ, ψ˜ and the scaling function w2 ϕ. First we construct a classical shearlet ψˆ(w1, w2) = ψˆ 1(w1)ψˆ 2( ). Several choices w1 exist for ψˆ 1 and ψˆ 2. For example for ψˆ 1(w) we can choose meyer wavelet which deﬁned by on auxiliary function.   o , x < 0   2x2 , 0 ≤ x < 1 v(x) =  2  1 − 2(1 − x)2 , 1 ≤ x < 1  2 1 , x > 1 and for ψˆ 2(w) we deﬁne { √ v(1 + w) , w ≤ 0 ψˆ (w) = √ 2 v(1 − w) , w ≥ 0 Second, we consider scaling function ϕ as follows:

ϕ(w) = φ(w1)φ(w2) where   , |w| ≤ 1  1 2 φ(w) =  cos( π v(2|w| − 1)) , 1 < |w| < 1 .  2 2 0 , o.w Having the shearlet frame, we can represent every f ∈ L2(R2) as a linear combination of elements of (3). In the most numerical method for solving PDEs or 2-D integral equations, discretisizing the domain of problem in to ﬁnite grids is considered as an initial step. For instance, for domain D = [a, b] × [a, b] one can discretisize x and y

32 Shearlet frame applications in solving some 2-D numerical analysis problems domains in to M and N partitions, respectively. In this case it can be shown that the unknown function in these problems can be generally represented. As follows,

j j ∑j0−1 2∑−1 ∑j0−1 2∑−1 = ψˆ + ˜ ψ˜ˆ f Cm1m2 j,k,m1,m2 Cm1m2 j,k,m1,m2 j=∑0 k=−∑2 j+1 j=0 k=−2 j+1 (4) + ϕˆ , dm1m2 m1,m2 m1 m2 = , , ··· , − , = , , ··· , − = { , } . where m1 0 1 M 1 m2 0 1 N 1 and j0 [log2 max M N ] Now it is an important question how one can choose elements so that the system in (3) is linearly independent. We answer to this question by property choosing j, k, m. Here we give an example. Example 2.1. Consider the following 2-D Fredholm integral equation ∫ ∫ 2 2 5 f (x, y) = xyz f (t, z)dtdz = xy. 1 1 2 Let M = N = 4, by using (3) we can represent f as follows,

j j ∑j0−1 2∑−1 ∑j0−1 2∑−1 ∑ = ψˆ + ˜ ψ + ϕˆ . f Cm1m2 j,k,m Cm1m2 j,k,m dm1m2 m (5) j=0 k=−2 j+1 j=0 k=−2 j+1 m∈I In orther to solving the mentioned problem using collocation method, we need a basis. Let ψ be a classical shearlet that we has constructed. we consider the approximation space as follows

S approx = span{{ψ0,−1,0,0, ψ0,0,0,0, ψ0,1,0,0, ψ1,−2,0,0, ψ1,−1,0,0, ψ1,0,0,0, ψ1,1,0,0, ψ1,2,0,0}

∪ {ψ˜ 0,−1,0,0, ψ˜ 0,0,0,0, ··· , ψ˜ 1,2,0,0} ∪ {ϕ0,0, ϕ1,1, ϕ0,2, ϕ0,3}}. In this example, the approximation space has 16 elements, where the functions are chosen to be independent. We reach this aim by choosing appropriate indices. By replacing these basis functions in (5) and applying the collocation method with collocation points k π k π (x , y ) = (cos 1 , cos 2 ), M = N = 4, k , K = 0, 1, 2, 3. k1 k2 M N 1 2 we can obtain an approximate solution.

References [1] S. Hauser and G. Steidel, Convex Multiclass Segmentation with Shearlet Regularization, Inter- national Journal of Computer Mathematics 90 (2013) 62-81. [2] G. Kutyniok and W. Q. Lim, Compactly supported shearlets are optimally sparse, Journal of Approximation Theory, (2011) 1564-1589.

33 R. Amiri, M. Zarebnia and R. Raisi Tousi

Rama Amiri, Department of Mathematics, Faculty of Mathematica and Statstics, University of Mohaghegh Ardabili, Ardabil, Iran e-mail: [email protected]

Mohammad Zarebnia, Department of Mathematics, Faculty of Mathematica and Statstics, University of Mohaghegh Ardabili, Ardabil, Iran e-mail: [email protected]

Reihaneh Raisi Tousi, Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran e-mail: [email protected]

34 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

ANALYSIS OF INPAINTING VIA UNIVERSAL SHEARLET SYSTEMS

Zahra Amiri∗ and Rajab Ali Kamyabi-Gol

Abstract

Currently, thresholding and compressed sensing in combination with both wavelet and shearlet transforms have been very successful in inpaiting tasks. However, numerical results demonstrate that shearlets outperform wavelets in the problem of image inpainting. In this paper we set up a particular model by inspired seismic data and a box mask to model missing data. The challenge is to ﬁll in the box that is attended in corrupted images.

2010 Mathematics subject classiﬁcation: Primary 42C, Secondary 65T60 . Keywords and phrases: Co-Sparsity, compressed Sensing, ℓ1 minimization.

1. Introduction

Reconstructing missing data is a popular challenge in both the analog and digital area. Also known as inpainting, this activity is the process of filling in the missing region or techniques for making undetectable modifications to images, modifying the corrupted ones which are not familiar with the original images. Applications of inpainting range from restoring of missing blocks in video data to removal of occlusions such as text from images and repairing of scratched photos. Due to the vast interest in this topic, there exist several excellent reports on inpainting via compressed sensing which is a fundamental method to recover sparsified data by ℓ1 minimization [3]. The work done in those reports focused on analyzing the concept of clustered sparsity which leads to theoretical bounds and results. Currently, the directional representation systems such as shearlets have been shown to outperform not only wavelets, but also most other directional systems [4]. In addition, superiority of shearlets over wavelets for a basic thresholding algorithm can be found in [3]. In [3], Kutyniok introduced the more flexible universal shearlet systems, which are associated with an arbitrary scaling sequence. We investigate The performance for inpainting of this novel construction shearlet systems.

∗ speaker

35 Please supply the list of authors

2. Abstract Inpainting Framework We start by analyzing the abstract Hilbert space, which is considered later on. Let x0 be a signal in separable Hilbert Space H. We assume that H can be decomposed into a direct sum of two closed subspaces, namely, a subspace HM which is associated 0 with the missing part of x and a subspace HK which is related to the known part of the signal. Hence, H = HK ⊕ HM = PKH ⊕ PMH,where PM and PK denote the orthogonal projections onto those subspaces, respectively. Note that, we will try 0 to ﬁnd the missing part PM x , so the problem of data recovery can be formulated as 0 0 follows: Given a corrupt signal PK x , recover the missing part PM x . Depending on the dimension of the given model, we consider H = L2(RD), D ∈ N. If the measurable D 2 subset M ⊆ R is the missing area of the image, we may set HM = L (M). Now, we present the methods for recovering a signal which will be useful in the sequel. In fact, one of the fundamental methodologies for spare recovery is ℓ1 minimization, which recovers the original signal by the following recovery algorithm 1 [3]:

Algorithm 1 Inpainting via ℓ1 Minimization Input: Incomplete signal PK x0 ∈ HK. Parseval frame Φ = (ϕi)i∈I. ℓ1 ⋆ = ∥ ∥ 0 = Compute: ( -INP) x argminx∈H TΦ x ℓ1(I) subject to PK x PK x 2 where TΦ is analysis operator respect Φ (TΦ : H → ℓ (I), x → (⟨x, ϕi⟩)i∈I). Output: recovered signal x⋆ ∈ H.

Since all Parseval frames are not basis, there are many solutions such c which ⋆ x = TΦc, only the speciﬁc solution TΦ x produces the desired numerical stabilities. Further, the assumption sparsity signal x0 by Φ provides a good recovery which is expected to occur. Now, in order to analyze the optimization problem by inpainting algorithm, we need to introduce two important notions, δ-clustered sparsity and cluster coherence. These notions were applied to study the geometric separation problem and sparsity [1]. Deﬁnition 2.1. [3]. Fix δ > 0. A signal x ∈ H is called δ-clustered sparse in a Parseval frame Φ (with respect to Λ ⊆ I) if

∥1Λc TΦ x∥ℓ1 ≤ δ. (1) In this case, Λ is said to be δ-cluster for x in Φ. The δ-clustered sparsity elucidates that coeﬃcients outside of Λ are small. In fact, the cluster sparsity depends on the chosen set of indices Λ, enlarging Λ leads to smaller δ in (1). 0 Cluster coherence introduced in [3] to investigate the missing part of signal x on HM looks as follows:

36 Inpainting

Λ ⊆ µ Λ, Φ Definition 2.2. [3]. Let I. The cluster coherence c( PM ∑) of Parseval frame Φ H Λ µ Λ, Φ = |⟨ ϕ , ϕ ⟩|, with respect to M and is defined by c( PM ) max j∈J i∈Λ PM i PM j where PMΦ = (PMϕi)i∈I. In order to clarify the significance of universal shearlet systems, let us recall the main idea of classical shearlet systems . For generator ψ ∈ L2(R2), a system of shearlet 3 j l j 2 is defined by {ψ j,l,k = 2 2 ψ(S A [.] − k): j ∈ Z, l ∈ Z, k ∈ Z }, where ( ) ( ) 22 0 1 1 A = , S = 0 2 0 1 denote the parabolic scaling matrix and shearing matrix, respectively. This approach is enhanced by cone-adapted shearlet systems. The( universal) shearlet systems were 2 j 2 0 introduced with associated scaling matrix Aα = α , where (α ) ⊆ (−∞, 2) j 0 2 j j j α ⊆ R to produce more flexibility in each scale. A sequence ( j) j∈N0 is called a scaling m −2 −1 1 2 1 sequence if α ∈ A = { |m ∈ Z, m ≤ 2 j − 1} = {..., , , 0, , ,..., 2 − }, for j j j j j j j j j ≥ 1 and α0 = 0. α Definition 2.3. Let ( j) j∈N0 be a scaling sequence. Then universal-scaling shearlet system or universal shearlet system is defined by

SH(ϕ, v, (α j) j) = SHLow(ϕ) ∪ SHInt(ϕ, v, (α j) j) ∪ SHBound(ϕ, v, (α j) j). The next Theorem shows that universal shearlet systems are a frame for L2(R2). Theorem 2.4. With notations as above, the universal shearlet system is a Parseval frame for L2(R2).

3. Inpainting with specific Image model on L2(R2) The general approach in this section is the same as in the previous one. We 1 investigate the inpainting results of ℓ minimization by chosen proper index set Λ j and estimate the relative sparsity and cluster coherence respect to this set. At the first, we would like to analyze a specific mathematical model which is the model of corrupted line segments. Let w ∈ C∞(R2) be a function that is supported in [−ρ, ρ] × [−η, η] where ρ, η > 0. A whole sequence of models (w j) j≥0 is given by 2 w j(x) = w ∗ F j(x) = ⟨w, F j(x − [·])⟩, x ∈ R , where filters F j are defined by the inverse Fourier transform of the corona functions in [3]. Now, we define the mask of a missing part of image as follows. The mask Mh is the intersection of a small vertical strip around the x2-axis and a small horizontal strip 2 x1 x2 around the x1-axis which is given by Mh = {(x1, x2) ∈ R : |x1| ≤ h , |x2| ≤ h }. For fix some ε > 0, we define the clusters ε j 2 Λ j = {( j, l, k, α j, d): |l| < 1, |k2| < 2 , k ∈ Z , d = 1, 2}, j ≥ 0.

37 Please supply the list of authors

Figure 1. Sketch of the corrupted modeling image.

We may determine the relative sparsity of the shearlet coeﬃcient with respect to the Λ1. cluster j Now, we can present the error estimate of Theorem to show the success of image inpainting with special image model. Inpainting result for shearlets and wavelets in special cases can be found in [3].

Theorem 3.1. Let (α j) j be a scaling sequence and Ψ = SH(ϕ, υ, (α j) j) be a universal α > ε > = x1 × x2 ∈ shearlet system. If lim inf j→∞ j 0 and for a ﬁxed 0, h j (h j h j ) j − +α +ε o(2 (2 j ) j). Then ∗ ∥w − w j∥ ,Ψ j 1 −N j ( ) j ∈ o(2 ), as j → ∞ ∥w j∥1,Ψ ∈ N ∗ for every N 0, where the recover provided by Algorithm 1 is denoted by w j.

Acknowledgement The ﬁrst author would like to thank Professor Gitta Kutyniok for stimulating discussions and pointing out various references.

References [1] D. Donoho and G. Kutyniok, Microlocal analysis of the geometric separation problem, Comm. Pure Appl. Math., 66 (2013), 1-47. [2] S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing, Birkhauser, 2013. [3] E. J. King, G. Kutyniok, and X. Zhuang, Analysis of inpainting via clustered sparsity and microlocal analysis, J. Math. Imaging Vis. 48 (2014), 205-234. [4] G. Kutyniok, W.-Q Lim, and R. Reisenhofer, Shearlab 3D: Faithful digital shearlet transforms based on compactly supported shearlets, ACM Transactions on Mathematical Software, 42 ( 1), 2015.

Zahra Amiri, Department of Mathematics, Ferdowsi University of Mashhad, Iran e-mail: za−[email protected]

Rajab Ali Kamyabi-Gol, Department of Mathematics, Ferdowsi University of Mashhad, Iran e-mail: [email protected]

38 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

DUALS OF K-FUSION FRAMES

Fahimeh Arabyani Neyshaburi

Abstract In this paper, we consider the notion of K-fusion frames in Hilbert spaces, which is a new generalization of fusion frames. Our main purpose is to reconstruct the elements from the range of the bounded operator K on a Hilbert space H by using a family of closed subspaces in H. For this end, we introduce the notion of duality for K-fusion frames and characterize duals of some K-fusion frames. We also survey the robustness of K-fusion frames under some perturbations.

2010 Mathematics subject classiﬁcation: Primary 42C15, Secondary 42C40. Keywords and phrases: K-fusion frame, K-dual, perturbation.

1. Introduction Let H be a separable Hilbert space and I be a countable index set, a sequence F := { fi}i∈I ⊆ H is called a K-frame for H, if there exist constants A, B > 0 such that ∑ ∗ 2 2 2 A∥K f ∥ ≤ |⟨ f, fi⟩| ≤ B∥ f ∥ , ( f ∈ H). (1) i∈I K-frames [5] were introduced to reconstruct elements from the range of a bounded linear operator K ∈ B(H). Clearly, if K = IH , then F is an ordinary frame and so K-frames arise as a generalization of the ordinary frames. Authors in [1] introduced the notion of duality for K-frames and presented some approaches for construction and characterization of K-duals. Fusion frame theory is a natural generalization of frame theory in separable Hilbert spaces, which is introduced by P. Casazza and G. Kutyniok in [2]. Fusion frames are applied in signal and image processing, sampling theory, ﬁlter banks and a variety of applications that cannot be modeled by discrete frames. In the following, we review basic deﬁnitions and results of fusion frames. Let {Wi}i∈I be a family of closed subspaces of H and {ωi}i∈I a family of weights, i.e. ωi > 0, i ∈ I. Then {(Wi, ωi)}i∈I is called a fusion frame for H if there exist the constants 0 < A ≤ B < ∞ such that ∑ ∥ ∥2 ≤ ω2∥π ∥2 ≤ ∥ ∥2, ∈ H , A f i Wi f B f ( f ) (2) i∈I

39 F. Arabyani

π H where Wi denotes the orthogonal projection from Hilbert space onto a closed subspace Wi. The constants A and B are called the fusion frame bounds. If we only have the upper bound in (2) we call {(Wi, ωi)}i∈I a Bessel fusion sequence. For each sequence {Wi}i∈I of closed subspaces in H, the space   ∑  ∑  ⊕ = { } ∈ , ∥ ∥2 < ∞ , Wi  fi i∈I : fi Wi fi  i∈I i∈I ∑ ⟨{ } , { } ⟩ = ⟨ , ⟩ with the inner product fi i∈I gi i∈I i∈I fi gi is a Hilbert space.∑ For every Bessel = { , ω } H ⊕ → H fusion sequence W : (W∑i i) i∈I of , the synthesis∑ operator TW : i∈I Wi is { } = ω { } ∈ ⊕ deﬁned by TW ( fi i∈I{) i∈I } i fi, for each fi i∈I i∈I Wi. The analysis operator, ∗ = ω π ∈ H H → H is given by TW ( f ) ∑i Wi ( f ) i∈I, f and the fusion frame operator S W : = ω2π is deﬁned by S W f i∈I i Wi f , which is a bounded, invertible and positive operator on H [2]. Throughout this paper, we suppose I is a countable index set and IH is the identity operator on H. For two Hilbert spaces H1 and H2 we denote by B(H1, H2) the collection of all bounded linear operators between H1 and H2, and we abbreviate B(H, H) by B(H). Also we denote the range of K ∈ B(H) by R(K), and the orthogonal projection of H onto a closed subspace V ⊆ H is denoted by πV .

2. Main results In this section, we introduce the notion of dual for K-fusion frames in Hilbert spaces. Then we, try to identify and characterize duals of some K-fusion frames.

Definition 2.1. Let {Wi}i∈I be a family of closed subspaces of H and {ωi}i∈I be a family of weights, i.e. ωi > 0, i ∈ I. We call W = {(Wi, ωi)}i∈I a K-fusion frame for H, if there exist positive constants 0 < A, B < ∞ such that ∑ ∥ ∗ ∥2 ≤ ω2∥π ∥2 ≤ ∥ ∥2, ∈ H . A K f i Wi f B f ( f ) (3) i∈I The constants A and B in (3) are called lower and upper bounds of W, respectively. ∩ = { } We call W a minimal K-fusion frame, whenever Wi span j∈I, j,iW j 0 . Obviously, a K-fusion frame is a Bessel fusion sequence and so the synthesis operator, the analysis operator and the frame operator of W are defined similar to fusion frames, however for a K-fusion frame, the synthesis operator is not onto and the frame operator is not invertible, in general. = { , ω } = { , υ } Now, for a K∑-fusion frame∑W (Wi i) i∈I and a Bessel sequence V (Vi i) i∈I, ϕ ⊕ → ⊕ we define vw : i∈I Vi i∈I Wi by ϕ { } = {π −1 ∗ } . vw fi i∈I Wi (S W ) K fi i∈I

Clearly ϕvw is a bounded linear operator, and we will have the following deﬁnition, where is a generalization of duality introduced by Gavru¸tain˘ [4].

40 Duals of K-fusion frames

Deﬁnition 2.2. Let W = {(Wi, ωi)}i∈I be a K-fusion frame. A Bessel sequence {(Vi, υi)}i∈I is called a K-dual of W if ∑ = ϕ ∗ = ω υ π −1 ∗ π , ∈ H . K f TW vwTV f i i Wi (S W ) K Vi f ( f ) (4) i∈I

e ∗ −1 = { π , ω } ∈ If W : (K S W S W (R(K))Wi i) i I is a Bessel sequence, then we can easily see that We is a K-dual for W; it is called the canonical K-dual of W. However, in the next example we construct a K-fusion frame for which not only We is not a Bessel fusion sequence, but also it has no K-dual. H = 2 { }∞ Example 2.3. Let l with the standard orthonormal basis en n=1. Deﬁne  ∑  ∞ 1  = e2m−1 i = 1,  m 1 m2 Ke =  i  0 i = 2,  e8m i = m + 2, (m ∈ N).

Then K ∈ B(H), and if we consider the subspaces W2 = span{e2 + e4}, W4 = span{e2 + e4}, and Wn = span{en}, for all n , 2, 4. Then {(Wn, 1)} is a K-fusion = { , υ }∞ frame. Moreover, a direct calculation shows that every sequence V (Vi i) i=1 satisﬁes (4) is not a Bessel fusion sequence, i.e., W has no K-dual. { π , ω } It is worth noticing that, when ( S W (R(K))Wi i) i∈I is a Bessel fusion sequence, then We is also a Bessel fusion sequence. In the next proposition we show that for some K-fusion frames many K-duals may be exist. Proposition 2.4. Suppose that K ∈ B(H) is a closed range operator and W = e {(Wi, ωi)}i∈I is a K-fusion frame. If W is a Bessel fusion sequence, then W has at least a K-dual diﬀerent from the canonical K-dual. The next result characterize all K-duals of minimal K-fusion frames. e Theorem 2.5. Let W = {(Wi, ωi)}i∈I be a minimal K-fusion frame for H and W be a Bessel fusion sequence. Then a Bessel fusion sequence V = {(Vi, ωi)}i∈I is a K-dual of ∗ −1π ⊆ ∈ W if and only if K S W S W (R(K))Wi Vi, for all i I.

3. Perturbation of K-fusion frames Stability of fusion frames have been considered in [3]. In this section, we study robustness of K-fusion frames under some perturbations.

Theorem 3.1. Let W = {(Wi, ωi)}i∈I be a K-fusion frame for H with bounds A and B, respectively. Also, let Z = {(Zi, zi)}i∈I be a (λ1, λ2, ε)-perturbation of W for some 0 < λ1, λ2 < 1 and ε > 0, i.e., ∥ ω π − π ∥ ≤ λ ∥ω π ∥ + λ ∥ π ∥ + εω ∥ ∗ ∥, ( i Wi zi Zi ) f 1 i Wi f 2 zi Zi i K f

41 F. Arabyani for all i ∈ I and f ∈ H, such that √ (1 − λ ) A ε < ∑ 1 . ∥ ∥ ω2 1/2 K ( i∈I i ) Then Z is a K-fusion frame for H. Finally, in the following theorem we show that under some small perturbations, K-duals of a K-fusion frame turn to the approximate K-dual for perturbed K-fusion frame.

Theorem 3.2. Let W = {(Wi, ωi)}i∈I be a K-fusion frame for H with bounds A and B, respectively. Also, let Zi ⊆ H be closed subspace of H for all i. If ε > 0 such that ∥ ∗ − ∗ ∥ < ε∥ ∗ ∥. (TW TZ) f K f √ √ (i) If 0 <√ ε < A, then Z = {(Zi, ωi)}i∈I is a K-fusion frame with bounds ( A − ε) and ( B + ε∥K∥), respectively. (ii) If ε > 0 is suﬃciently small, then every K-dual V = {(Vi, υi)}i∈I of W satisﬁes ∥ − ϕ ∗ ∥ < K TZ vzTV 1.

References [1] F. Arabyani Neyshaburi, A. Arefijamaal, Some constructions of K-frames and their duals, To appear in Rocky Mountain. Math. [2] P. G. Casazza, G. Kutyniok, Frames of subspaces, Contemp. Math 345 (2004), 87-114. [3] P. G. Casazza, G. Kutyniok and S. Li, Fusion frames and distributed processing, Appl. Comput. Harmon. Anal 25 (1) (2008), 114-132. [4] P. Gavru˘ ta¸ , On the duality of fusion frames, J. Math. Anal Appl. 333 (2)(2007) , 871-879. [5] L. Gavru˘ ta¸ , Frames for operators, Appl. Comp. Harm. Anal 32 (2012), 139-144.

Fahimeh Arabyani Neyshaburi, Department of Mathematics and Computer Sciences, University of Hakim Sabzevari, Sabzevar, Iran e-mail: [email protected]

42 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

DISTRIBUTION FUNCTIONS, DECREASING REARRANGEMENTS AND LP, α SPACES

Omid Baghani

Abstract

In this work, we introduce some useful spaces for example the Orlicz, Lorentz and Lp,α spaces. Finally some properties of these spaces are stated.

2010 Mathematics subject classiﬁcation: Primary 46L99, Secondary 47B47.

Keywords and phrases: Orlicz space, Lorentz space, Lp,α space.

1. Introduction Although the Lebesgue spaces Lp, 0 ≤ p ≤ ∞ play a primary role in many areas of mathematical analysis, there are other classes of Banach spaces of measurable functions that are also of interest. The larger classes of Orlicz spaces and Lorentz spaces, for example, are of intrinsic importance.

2. Distribution functions and decreasing rearrangements As in the previous chapter, (R, µ) denotes a totally σ−ﬁnite measure space.

Deﬁnition 2.1. The distribution function µ f of a function f in M0 = M0(R, µ) is given by

µ f (λ) = µ{x ∈ R : | f (x)| > λ}, λ ≥ 0. (1)

Observe that µ f depends only on the absolute value | f | of the function f , and µ f may assume the value +∞.

Deﬁnition 2.2. Suppose f belongs to M0. The decreasing rearrangement of f is the function f ∗ deﬁned on [0, ∞) by

∗ f (t) = inf{λ : µ f (λ) ≤ t}, t ≥ 0. (2)

The next result gives alternative descriptions of the Lp−norm in terms of the distribution function and the decreasing rearrangement. ( See [2].)

43 O. Baghani

Proposition 2.3. Let f ∈ M0. If 0 < p < ∞, then ∫ ∫ ∞ ∫ ∞ p p−1 ∗ p | f | dµ = p λ µ f (λ)dλ = ( f ) (t)dt. (3) R 0 0 Furthermore, in the case p = ∞ ∗ ess sup | f | = inf{λ : µ f (λ) = 0} = f (0). (4) x∈R Proof. □

3. Lorentz spaces Deﬁnition 3.1. Let (R, µ) be a totally σ−ﬁnite measure space and suppose 0 < p, q < ∞. The Lorentz space Lp,q = Lp,q(R, µ) consists of all f in M0(R, µ) for which the quantity  ∫  ∞ / ∗ /  { (t1 p f (t))q dt }1 q, 0 < q < ∞,  0 t  (5)   1/p ∗ , = ∞, sup0

It is clear from Proposition 2.3 that the Lorentz space Lp,p, (0 < p ≤ ∞), coincides with the Lebesgue space Lp, and

∥ f ∥p,p = ∥ f ∥p, f ∈ Lp.

Note also that the space L∞,q, for ﬁnite q is trivial, in the sense that it contains only the zero function. The next result shows that, for any ﬁxed p, the Lorentz spaces Lp,q increase as the secondary exponent q increases. (See [2]). Proposition 3.2. Suppose 0 < p ≤ ∞ and 0 < q ≤ r ≤ ∞, then

∥ f ∥p,r ≤ c∥ f ∥p,q (6)

for all f in M0(R, µ), where c is a constant depending only on p, q, and r. In particular, Lp,q ⊂ Lp,r. Proof. □

4. Orlicz spaces We conclude this chapter with a brief look at another interesting class of rearrangement- invariant spaces: the Orlicz spaces. Definition 4.1. Let ϕ : [0, ∞) → [0, ∞) be increasing and left-continuous, with ϕ(0) = 0. Suppose on (0, ∞) that ϕ is neither identically zero nor identically infinite. Then the function Φ defined by ∫ s Φ(s) = ϕ(u)du, s ≥ 0, 0 is said to be a Young’s function.

44 Distribution functions, decreasing rearrangements and Lp, α spaces

Note that a Young’s function is convex on the interval where it is finite. Definition 4.2. Let Φ be a Young’s function. The Orlicz class P(Φ) consists of all measurable functions f on the interval [0, a] for which the functional ∫ a MΦ( f ) = Φ(| f (x)|)dx 0 is finite.

5. Lp,α spaces

In this section we introduce a new class of normed spaces, called Lp,α spaces. As we see the norm in such spaces is defined in the term of fractional integrals. Next, we illustrate briefly some properties of these spaces [1, 4, 5]. Definition 5.1. Let 1 ≤ p < ∞ and 0 < α < 1 be fixed. We say that a measurable n n function f : [0, 1] → R belongs to Lp,α([0, 1], R ) if and only if the quantity ∫ ( t | |p ) / ∥ ∥ = f (s) 1 p, f p,α : sup α ds 0≤t≤1 0 (t − s) is finite. We close our discussion of Lp,α spaces, by introducing appropriate spaces corresponding to the limiting case p = ∞. If f is a measurable function on [0, 1], we define ∥ f ∥∞ ∥ f ∥ ,α = , p → ∞, (7) p (1 − α)1/p where ∥ f ∥∞ stands for the essential supremum of f . See [3] for more details about spaces equipped with such norms. Note that the results discussed until now can be extended to the case p = ∞.

Lemma 5.2. If 0 < p < q ≤ ∞ and 0 < α < 1, then Lq,α ⊆ Lp,α and ( )(1/p)−(1/q) ∥ ∥ ≤ ∥ ∥ 1 f p,α f q,α 1−α . Proof. If q = ∞, this is obvious ∫ ∫ ( t | |p ) / t ∥ ∥ ∥ ∥ = f (s) 1 p ≤ ∥ ∥ ds ≤ f ∞ . f p,α sup α ds f ∞ sup α 0≤t≤1 0 (t − s) 0≤t≤1 0 (t − s) 1 − α If q < ∞, we use Hölder’s inequality with the conjugate exponents q/p and q/(q − p),

p p p ∥ f ∥p,α = ∥| f | .1∥1,α ≤ ∥| f | ∥q/p,α∥1∥q/(q−p),α. Taking supprimum and pth roots, we obtain ( ) 1 (1/p)−(1/q) ∥ f ∥ ,α ≤ ∥ f ∥ ,α . p q 1 − α □

45 O. Baghani

≤ ≤ ∞ < α < Lemma∑ 5.3. For 1 p ∪ and 0 1, the set of simple functions = n α χ n = , s(x) j=1 j E j (x), where j=1 E j [0 1], is dense in Lp,α.

Proof. Clearly such functions are in Lp,α. Let f ∈ Lp,α. Since f is a bounded , { }∞ function on [0 1] a.e., there exists a sequence sn n=1 of simple functions which is uniformly convergence to f on [0, 1] a.e.. Therefore

∥sn − f ∥p,α ≤ ∥sn − f ∥∞ → 0, as n → ∞. □

References [1] O. Baghani, On fractional Langevin equation involving two fractional orders, Commun. Nonlinear Sci. Numer. Simulat., 42 (2017) 675-681. [2] C. Bennett and R. Sharpley, Interpolation of Operators , Academic Press, Inc. 1988. [3] G. B. Folland, Real Analysis: Modern Techniques and Their Applications, John Wiley, second ed., 1999. [4] A. Kufner, L. Maligranda and L. E. Persson, The prehistory of the Hardy inequality, Amer. Math . Monthly 113 (2006) 715-732. [5] D. R. Owen and K. Wang, Weakly Lipschitzian mappings and restricted uniqueness of solutions of ordinary diﬀerential equations,J. Diﬀer. Equ., 95 (1992) 385–398.

Omid Baghani, Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran e-mail: [email protected]

46 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

THE RELATION BETWEEN EXISTENCE OF ADMISSIBLE VECTORS AND COMPACTNESS OF A LOCALLY COMPACT GROUP

alireza bagheri salec∗ and javad saadatmandan

Abstract Let G be a locally compact group and π : G −→ U(Hπ) be a unitary representation. In this article we will study on the existence of an admissible vector for irreducible representations and the compactness of G. In fact, it will be shown that if G is a compact group then every irreducible representation of G has an admissible vector, and also has a bounded cyclic vector. Conversely if G has property (T) and a ﬁnite irreducible representation of G has an admissible vector, then G is a compact group. Since containment of an irreducible representation π in the left regular representation λG is a necessary and suﬃcient condition for existence of admissible vector, hence it seems a natural object of weakly containment of these representation and therefore property (T) is peered.

2010 Mathematics subject classiﬁcation: Primary 42C40, Secondary 43A65. Keywords and phrases: Admissible vector, property (T), compact group.

1. Introduction There are interesting connections between admissible vectors and the compactness of the group G. Some results for containment of representations can be obtain by property (T). A locally compact group G has the property (T) if there exist a compact subset Q and a real number ϵ > 0 such that, whenever π is a continuous unitary representation of G on a Hilbert space H for which there exists a vector η ∈ H of norm 1 with ||π η − η|| < ϵ ξ , supx∈Q (x) , then there exists an invariant vector, namely a vector 0 in H such that π(x)ξ = ξ for all x ∈ G. The vector η with above property is called (Q, ϵ)−invariant. If for every compact subset Q and a real number ϵ > 0 there exist a (Q, ϵ)−invariant vector, it is written 1G ≺ π and if there exist invariant vectors, it is written 1G < π. If a group G has the property (T), then it is called a Kazhdan group. The trivial example of Kazhdan groups are compact groups. Compact groups can be characterised as the locally compact groups which are amenable and which have property (T). If a group G be compact, then every irreducible representation on G has the admissible vectors. In this article we have investigated conditions that conﬁrm the converse statements. For example if a locally compact group G has a representation that has a invariant vector and a admissible vector, then G is compact. ∗ speaker

47 alireza bagheri salec and javad saadatmandan

2. Definitions and Preliminaries A unitary, strongly continuous representation, or simply a representation, of a locally compact group G is a group homomorphism π : G −→ U(Hπ), which is continuous when the right hand side is endowed with the strong operator topology, that is x −→ π(x)η is continuous from G to Hπ for each η ∈ Hπ. Therefore, all coefficient functions of the type G ∋ x −→ ⟨ξ, π(x)η⟩ ∈ C are continuous, where ξ, η ∈ Hπ. This is because the weak and strong operator topologies coincide on U(Hπ). See [4] for more detailes. The representation of G is called irreducible if Hπ and {0} are the only subspace of Hπ that invariant under π. We use the notation π1 < π2 when π1 is unitary equivalent with some subrepresentation of π2, and we say that π1 is contained in π2. 2 Let G be a locally compact group. The left regular representation λG acts on L (G) by −1 2 (λG(x) f )(y) = f (x y). Clearly this representation is unitary and for f, g ∈ L (G), we ∗ ∗ −1 have (g ∗ f )(x) = ⟨g, λG(x) f ⟩, where f (x) = f (x ). Let (π1, H) and (ρ, K) be unitary representations of the topological group G.A representation π is called weakly contained in ρ, if every function of positive type associated to π can be approximated, uniformly on compact subsets of G, by finite sums of functions of positive type associated to ρ. This means for every ξ in H, every compact subset Q of G and every ϵ > 0, there exist η1, ..., ηn in K such that, for all xQ, ∑n | < π(x)ξ, ξ > − < ρ(x)ηi, ηi > | < ϵ. i=1 We write for this π ≺ ρ. By theorem 1.2.1 in [2], a topological group G has the property (T), if and only if " 1G ≺ λG then 1G < λG ". Let (π, Hπ) denote a strongly continuous unitary representation of a locally compact group G. We endow G with it’s left Haar measure. For each η ∈ Hπ associate the coeffitient operator Vη : Hπ −→ Cb(G) defined by Vηξ(x) = ⟨ξ, π(x)η⟩. The vec- 2 tor η ∈ Hπ is called admissible if Vη : Hπ −→ L (G) is an isometery. In this case 2 Vη : Hπ −→ L (G) is called the (generalized) continuous wavelet transform.

3. Main Results We concentrated on existance of admissible vectors and in particular on solving this question: Is the compactness of a locally compact group G is equivalent to, "every irredusible representation of G has an admissible vector"?

The main theorem for existence of admissible vectors is the following theorem, Theorem 2.25 in [5]. Theorem 3.1. Let π be an irreducible representation of G. π has admissible vectors iﬀ π < λG. The following theorem that ﬁned in[1], [2] and [3] shows the relation between amenability of a locally compact.

48 Admidssiblity and compactness

Theorem 3.2. Let G be a locally compact group. The following are equivalent:

1. G is amenable, 2. Every irreducible, unitary representation of G is weakly contained in λG, 3. The trivial representation of G on C (the one that maps every element of G to 1) is weakly contained in λG, 4. λG is amenable. Corollary 3.3. Let G be a locally compact group. If every irreducible representation of G has an admissible vector then G is amenable. Proof. If π is an arbitrary irreducible representation which has admissible vectors, then by Theorem 3.1 π < λG . Therefore, π is weakly contained in λG and by Theorem 3.2, G is amenable. □ Remark 3.4. Note that the converse of 3.3 is not hold in general. In the other word amenability condition generally does not imply the existence of admissible vector for all irreducible representations. For an example for instance let G = R with the ordinary operation +,.. Since G is an abelian group, it is amenable. But λG has no 2 irreducible subrepresentation, where λG is representation of R on L (R) diﬁned by [λG(x) f ](t) = f (t − x) (see [4] page 72 for more detailes). It is seen that the condition ”existence of admissible vector for all irreducible representation” is more stronger than amenability, and so we need a stronger condition such as compactness. In this context, the following theorem is a useful case. Theorem 3.5. Let G be a locally compact group. The following are equivalent: (i) G is compact; (ii) 1G is contained in λG; Proof. See Theorem A.5.1 in [1] for detailes. □ Corollary 3.6. Let G be a locally compact group. If there exists a representation of G such that has a non zero vector that is admissible and invariant, then G is compact.

Proof. Let π is a representation of G such that has a non zero vector η ∈ Hπ that is admissible and invariant vector. then ∫ ∫ 2 2 2 2 2 ||η|| = ||Vη|| = | < η, π(x)η > | dµ(x) = | < η, η > | dµ(x) = ||η|| µ(G) G G therefore µ is ﬁnite and then G is compact. □ Proposition 3.7. Let G be a compact group. Then every irreducible representation of G has an admissible vector. Proof. It follows from Peter-weyl Theorem(iii) [[4], Theorem 5.12], and Theorem 3.1. □

49 alireza bagheri salec and javad saadatmandan

Theorem 3.8. Let G be a locally compact group. The following are equivalent: (i) G is compact; (ii) G is amenable and has the property (T) . Proof. See Theorem 1.1.6 in [1] for detailes. □ Proposition 3.9. Let G be a compacrt group and let π and ρ be unitary representations of G. Then π ≺ ρ if and only if every irreducible subrepresentation of π is contained in ρ. Therefore if π is irreducible then π ≺ ρ is equivalent to π < ρ. Proof. See Theorem F.1.8 in [2] for detailes. Proposition 3.10. Let G be a locally compact group and has property (T). If there exist a finite irreduscible unitary representation of G that has an admissible vector, then G is amenable and therefore, by theorem 3.8 G is compact. Proof. If π be a finite irreduscible unitary representation of G that has a admissible vector then by 3.1, π < λG. Therfore by [2] Corollarry 5.9 λG is amenable, then by theorem 3.2, G is amenable and by theorem 3.5, G is compact. □ The following theorem explain the relation between existennce of admissible vectors and compactness in particular case. See [5], Theorem 2.35. Theorem 3.11. Let G be a SIN-group, i.e., every neighborhood of unity contains a conjugation-invariant neighborhood. If G has a discrete series representation, i.e., there exists irreducible representation that has admissible vectors then G is compact. In particular, if G is discrete and has a discrete series representation, then G is finite. If G is abelian and has a discrete series representation, then G is compact.

References [1] B. Bekka, Amenable unitary representations of locally compact groups, Invent. Math, 100, (1990) 383-401. [2] B. Bekka, P.Harpe and A.Valette, Kazhdan’s property (T), Cambridge University Press, New York, 2008. [3] A. Hulanicki, Groups whose regular representation weakly contains all unitary representations , Studia Math. 24 (1964) 37-59. [4] G.B. Folland, A course in abstract harmonic analysis, CRC Press, Boca Raton, 1995. [5] H. Fuhr, Abstract Harmonic Analysis of Continuous Wavelet Transforms, Springer-Verlag Berlin Heidelberg, 2005. alireza bagheri salec, Department of Mathematics, University of Qom, Iran. e-mail: alireza−bagheri−[email protected] javad saadatmandan, Department of Mathematics,

50 Admidssiblity and compactness

University of Qom, Iran. e-mail: [email protected]

51 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

RECENT PROGRESSES IN ZERO PRODUCT PRESERVING MAPS ON CERTAIN BANACH ALGEBRAS

Mohammad Ali Bahmani∗ and Amir Hossein Mokhtari

Abstract In this talk we present some recent results on zero preserving maps on certain Banach algebras, including C∗-algebras, group convolution algebras, matrix like algebras, etc.

2010 Mathematics subject classiﬁcation: Primary 43A20, Secondary 46H05, 47B48. Keywords and phrases: Zero product preserving map, C∗-algebra, group convolution algebra.

1. Introduction Let A be a Banach algebra. A bounded bilinear map Φ : A × A −→ X, where X is a Banach space, is said to be zero product preserving if Φ(x, y) = 0 whenever xy = 0, (x, y ∈ A). A zero product preserving map Φ is said to be proper if there exists a bounded linear map T : A → X such that Φ(x, y) = T(xy) for all x, y ∈ A. A Banach algebra A is called zero product determined if every zero product preserving map Φ : A × A −→ X is proper. If we replace in this deﬁnition the ordinary product xy by the Lie product given by [x, y] = xy − yx (resp. Jordan product given by x ◦ y = xy + yx), then we say that A is zero Lie (resp. Jordan) product determined. The question of whether a zero preserving map for certain algebras is proper has been widely studied, see for examples [2–5] and references there in.

2. Main Results We commence with the following elementary proposition which gives an equivalent condition for zero preserving bilinear maps to be proper. Proposition 2.1. For a zero preserving bilinear map Φ : A × A −→ X on a unital algebra X the following assertions are equivalent. (i) Φ is proper. (ii)Φ(x, y) = Φ(xy, 1) for all x, y ∈ X. We also present the next basic elementary property of a zero product preserving bilinear map, a more general version of that has been proved in [1, Theorem 2.1]. ∗ speaker

52 M.A. Bahmani, A.H. Mokhtari

Proposition 2.2. Let A be a unital algebra and let X be a module. If Φ : A × A −→ X is a zero product preserving map then

Φ(xe, y) = Φ(x, ey) for all x, y ∈ A, e ∈ I(A), (1) where I(A) denotes the subalgebra of A generated by all idempotents of A. In particular,

Φ(x, e) = Φ(xe, 1) and Φ(e, x) = Φ(1, ex) for all x ∈ A, e ∈ I(A). (2)

In particular, if A = I(A) then A is zero product determined. Following [3], we say that a Banach algebra A has property B if for every bounded bilinear map Φ : A × A −→ X, the condition

a, b ∈ A, ab = 0 =⇒ Φ(a, b) = 0, implies the condition

Φ(ab, c) = Φ(a, bc), (a, b, c ∈ A).

The group algebra L1(G), all C∗-algebras and every Banach algebra generated by idempotents are examples of Banach algebras saisfying the property B. As it has been investigated in [3], it is worthwhile mentioning that the amenability of a Banach algebra A plays a crucial role for properness of zero preserving maps on A. Let us recall that a Banach algebra A is said be amenable if every bounded derivation from A into each dual Banach A-module is inner. The group algebra L1(G) for a amenable locally compact, a commutative C∗-algebra and the algebra of all compact operators on a Hilbert space are the most famous examples of amenable Banach algebras. In the realm of amenable Banach algebra we quote and discuss the following result from [3]. Theorem 2.3. Let A be an amenable Banach algebra with property (B), let X be a Banach space,and let Φ : A × A → X be a bounded bilinear map satisfying the condition:

a, b ∈ A, ab = ba = 0 =⇒ Φ(a, b) = 0.

Then there exist continuous linear operators ϕ :[A, A] → X and ψ : A → X such that

Φ(a, b) = ϕ([a, b]) + ψ(a ◦ b), (a, b ∈ A); where [A, A] is the linear span of all [a, b] with a, b ∈ A We give some discussions on the condition "A is amenable ” in the above mentioned theorem. We conclude with some results on zero preserving maps on a generalized matrix algebra and discuss some problems, to the best of our knowledge, seem to be open.

53 Recent Progresses in Zero Product Preserving Maps on Certain Banach Algebras

References [1] H. Bierwirth, M. Brešar and M. Grašic,ˇ On maps determined by zero products, Comm. Algebra 40 (2012), 2081-2090. [2] J. Alaminos, M. Bresarˇ , J.Extremera and A.R. Villena, Maps preserving zero products, Studia Math. 193 (2009), 131-159 [3] J. Alaminos, M. Bresarˇ , J. Extremera and A.R. Villena, Zero Lie product determined Banach algebras, arXive:1610.03638v1. [4] J. Alaminos, J. Extremera, A.R. Villena, Disjointness-preserving linear maps on Banach function algebras associated to a locally compact group, Ann. Funct. Anal. 7 (2016), 442-451. [5] M.A. Bahmani, A.H. Mokhtari and H.R. Ebrahimi Vishki, Zero product preserving maps on some matrix like algebras, in preparation.

Mohammad Ali Bahmani, Department of Pure Mathematics, Ferdowsi University of Mashhad, Iran e-mail: [email protected]

Amir Hossein Mokhtari, Technical Faculty of Ferdows, University of Birjand, Ferdows, Iran e-mail: [email protected]

54 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

COMPARISON OF THE TOPOLOGICAL CENTERS OF A BILINEAR MAPPING AND ITS THIRD ADJOINT

Sedigheh Barootkoob

Abstract Let f : X × Y → Z be a bilinear mapping on normed spaces. In this paper we investigate that are the topological centers of f , w∗−dense in the corresponding topological centers of its extensions f ∗∗∗ and f t∗∗∗t? we show that although it has positive answer on some special cases but this is not true in general.

2010 Mathematics subject classiﬁcation: Primary 46H20; Secondary 46H25. Keywords and phrases: bilinear mapping, topological center, Arens regular.

1. Introduction According to [1] and [2] for every bounded bilinear mapping f : X × Y → Z (on normed spaces X, Y and Z) we have two natural extensions from X∗∗ × Y∗∗ to Z∗∗. Also the definition of regularity of bilinear mappings mentioned in [1] and [2]. First of all We recall these definitions. For a bounded bilinear mapping f : X × Y → Z we define the adjoint f ∗ : Z∗ × X → Y∗ of f by

⟨ f ∗(z∗, x), y⟩ = ⟨z∗, f (x, y)⟩ (x ∈ X, y ∈ Y and z∗ ∈ Z∗).

Also this process may be repeated to deﬁne f ∗∗ = ( f ∗)∗ : Y∗∗ × Z∗ → X∗ and f ∗∗∗ = ( f ∗∗)∗ : X∗∗ × Y∗∗ → Z∗∗. It can readily veriﬁed that f ∗∗∗ is the unique extension of f for which the maps

· 7→ f ∗∗∗(·, y∗∗), · 7→ f ∗∗∗(x, ·)(x ∈ X, y∗∗ ∈ Y∗∗), are w∗ − w∗−separately continuous. Let f t be the transpose of f , that is the bounded bilinear mapping f t : Y × X −→ Z deﬁned by f t(y, x) = f (x, y)(x ∈ X, y ∈ Y). If we continue the latter process with f t instead of f , we come to the bounded bilinear mapping f t∗∗∗t : X∗∗ × Y∗∗ → Z∗∗, that is the unique extension of f for which the maps

· 7→ f t∗∗∗t(x∗∗, ·), · 7→ f t∗∗∗t(·, y)(y ∈ Y, x∗∗ ∈ X∗∗),

55 S. Barootkoob are w∗ − w∗− continuous. We deﬁne the left topological center Zℓ( f ) by ∗∗ ∗∗ ∗∗ ∗∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗ Zℓ( f ) = {x ∈ X ; y −→ f (x , y ): Y −→ Z is w − continuous} = {x∗∗ ∈ X∗∗; f ∗∗∗(x∗∗, y∗∗) = f t∗∗∗t(x∗∗, y∗∗) for every y∗∗ ∈ Y∗∗}, and the right topological center Zr( f ) of f by

∗∗ ∗∗ ∗∗ t∗∗∗t ∗∗ ∗∗ ∗∗ ∗∗ ∗ Zr( f ) = {y ∈ Y ; x −→ f (x , y ): X −→ Z is w − continuous} = {y∗∗ ∈ Y∗∗; f ∗∗∗(x∗∗, y∗∗) = f t∗∗∗t(x∗∗, y∗∗) for every x∗∗ ∈ X∗∗}.

t Clearly, X ⊆ Zℓ( f ), Y ⊆ Zr( f ) and Zr( f ) = Zℓ( f ). A bounded bilinear mapping f is said to be Arens regular if f ∗∗∗ = f t∗∗∗t. This is ∗∗ ∗∗ equivalent to Zℓ( f ) = X as well as Zr( f ) = Y . The mapping f is said to be left (resp. right) strongly Arens irregular if Zℓ( f ) = X (resp. Zr( f ) = Y). ∗∗ ∗∗∗ ∗∗∗∗ ∗∗ We know that X ⊆ Zℓ( f ) ⊆ X ⊆ Zℓ( f ) ⊆ X and Y ⊆ Z ( f ) ⊆ Y ⊆ r ∗ ∗∗∗ ∗∗∗∗ w Zr( f ) ⊆ Y in general. In this paper we investigate the relationship of Zℓ( f ) ∗∗∗ t∗∗∗t with Zℓ( f ) and Zℓ( f ) and similarly for the right topological centers.

2. Main results ∗ ∗ w ∗∗∗ w t∗∗∗t Theorem 2.1. (i) Zℓ( f ) ⊆ Zℓ( f ) if and only if Zℓ( f ) ⊆ Zℓ( f ) ∗ ∗ w ∗∗∗ w t∗∗∗t (ii) Zr( f ) ⊆ Zr( f ) if and only if Zr( f ) ⊆ Zr( f ) Corollary 2.2. If f is Arens regular then f ∗∗∗ is Arens regular if and only if f t∗∗∗t is Arens regular. ∗ ∗∗∗ w t∗∗∗t t∗∗∗t Corollary 2.3. If f is Arens regular then Zℓ( f ) ⊆ Zℓ( f ), and if f is Arens ∗ w ∗∗∗ regular then Zℓ( f ) ⊆ Zℓ( f ). Theorem 2.1 says that it is sufficient to investigate only the relationship of the topological centers f ∗∗∗ and w∗−cluster of the topological centers of f . ∗ w ∗∗∗ Also it is easy to see that if X is reflexive then Zℓ( f ) = X = Zℓ( f ) and if Y is ∗ w ∗∗∗ reflexive then Zr( f ) = X = Zr( f ). So we assume that X and Y are not reflexive. On the other hand in [3] it is shown that there is an Arens regular bilinear mapping f ∗ ∗∗∗ ∗∗∗ w such that f is not Arens regular. Therefore in this case Zℓ( f ) ⊊ Zℓ( f ) and the equality are not valid in general. ∗ ∗ w ∗∗∗ w In the sequel we investigate the relationship Zℓ( f ) with Zℓ( f ) and Zr( f ) with ∗∗∗ Zr( f ) in special cases. The following theorem has a proof similar to the proof of theorem 2.1. ∗ ∗∗∗∗ w ∗∗∗∗ ∗∗∗∗ Theorem 2.4. (i) For each x ∈ Zℓ( f ) and y ∈ Y ,

f ∗∗∗∗∗∗(x∗∗∗∗, y∗∗∗∗) = f t∗∗∗t∗∗∗(x∗∗∗∗, y∗∗∗∗)

56 Comparison of the topological centers of a bilinear mapping and its third adjoint and f t∗∗∗∗∗∗t(x∗∗∗∗, y∗∗∗∗) = f ∗∗∗t∗∗∗t(x∗∗∗∗, y∗∗∗∗). ∗∗∗∗ ∗∗∗ ∗∗∗∗ ∗∗∗∗ (ii) For each x ∈ Zℓ( f ) and y ∈ Y ,

f ∗∗∗∗∗∗(x∗∗∗∗, y∗∗∗∗) = f ∗∗∗t∗∗∗t(x∗∗∗∗, y∗∗∗∗).

∗ ∗∗∗∗ w ∗∗∗∗ ∗∗∗∗ (iii) For each y ∈ Zr( f ) and x ∈ X , f ∗∗∗∗∗∗(x∗∗∗∗, y∗∗∗∗) = f t∗∗∗t∗∗∗(x∗∗∗∗, y∗∗∗∗) and f t∗∗∗∗∗∗t(x∗∗∗∗, y∗∗∗∗) = f ∗∗∗t∗∗∗t(x∗∗∗∗, y∗∗∗∗) ∗∗∗∗ ∗∗∗ ∗∗∗∗ ∗∗∗∗ (iv) For each y ∈ Zr( f ) and x ∈ X , f ∗∗∗∗∗∗(x∗∗∗∗, y∗∗∗∗) = f ∗∗∗t∗∗∗t(x∗∗∗∗, y∗∗∗∗).

∗ ∗∗∗∗∗∗ ∗∗∗∗∗∗ w | ∗ = t t| ∗ ⊆ Corollary 2.5. (i) f w ∗∗∗∗ f w ∗∗∗∗ if and only if Zℓ( f ) Zℓ( f ) ×Y Zℓ( f ) ×Y ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ t t| ∗ = t t | ∗ Zℓ( f ) if and only if f w ∗∗∗∗ f w ∗∗∗∗ . Zℓ( f ) ×Y Zℓ( f ) ×Y ∗ ∗∗∗∗∗∗ t∗∗∗∗∗∗t w ∗∗∗ (ii) f | w∗ = f | w∗ if and only if Z ( f ) ⊆ Z ( f ) if and X∗∗∗∗×Z ( f ) X∗∗∗∗×Z ( f ) r r ∗∗∗ ∗∗∗ r ∗∗∗ ∗∗∗ r t t| ∗ = t t | ∗ only if f ∗∗∗∗ w f ∗∗∗∗ w . X ×Zr( f ) X ×Zr( f ) ∗ ∗∗∗ w ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ⊆ t t| ∗∗∗ ∗∗∗∗ = t t | ∗∗∗ ∗∗∗∗ (iii) If Zℓ( f ) Zℓ( f ) then f Zℓ( f )×Y f Zℓ( f )×Y and ∗∗∗∗∗∗ ∗∗∗∗∗∗ | ∗∗∗ ∗∗∗∗ = t t| ∗∗∗ ∗∗∗∗ . f Zℓ( f )×Y f Zℓ( f )×Y

∗ ∗∗∗ w ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ⊆ t t| ∗∗∗∗ ∗∗∗ = t t | ∗∗∗∗ ∗∗∗ (iv) If Zr( f ) Zr( f ) then f X ×Zr( f ) f X ×Zr( f ) and ∗∗∗∗∗∗ ∗∗∗∗∗∗ | ∗∗∗∗ ∗∗∗ = t t| ∗∗∗∗ ∗∗∗ . f X ×Zr( f ) f X ×Zr( f )

∗ ∗ t∗∗∗t∗∗∗ ∗∗∗t∗∗∗t w t∗∗∗t w t∗∗∗t Corollary 2.6. If f = f , then Zℓ( f ) ⊆ Zℓ( f ) and Zr( f ) ⊆ Zr( f ) on the other hand by two routine w∗−limit, we have the following proposition. ∗ ∗ ∗∗ w ∗∗∗ w Proposition 2.7. If X ⊆ Zℓ( f ) then Zℓ( f ) ⊆ Zℓ( f ) . ∗ ∗∗∗ w Note that if f is Arens regular, then Zℓ( f ) ⊆ Zℓ( f ) and if it is strongly Arens ∗ ∗∗∗ w irregular then it maybe Zℓ( f ) ⊈ Zℓ( f ) .

References [1] R. Arens, Operations induced in function classes, Monatsh. Math. 55 (1951) 1-19. [2] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951) 839-848, 1951. [3] S. Barootkoob and H. R. Ebrahimi Vishki, Non-Arens regularity of the third adjoint of certain module operations, The 45th Annual Iranian Mathematics conference August 26-29 2014, 256– 258.

57 S. Barootkoob

Sedigheh Barootkoob, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord, Iran e-mail: [email protected]

58 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

VARIOUS NOTIONS OF AMENABILITY ON MODULE EXTENSION BANACH ALGEBRAS

Sedigheh Barootkoob

Abstract We extend the notion of semiweak amenability of Banach algebras for all positive odd integers. Indeed we study the notion of (2n+1)−semiweak amenability of Banach algebras and investigate the relationship between (2n + 1)−semiweak amenability and supper weak amenability of a Banach algebra with (2n + 1)−weak amenability of those Banach algebra. Finally we characterize the equivalent conditions for supper weak amenability and (2n + 1)−semiweak amenability of module extension Banach algebras. Our results lead to some equivalent conditions for (2n + 1)−weak amenability of unital module extension Banach algebras.

2010 Mathematics subject classiﬁcation: Primary 46H20; Secondary 46H25. Keywords and phrases: n−weak amenability, supper weak amenability, (2n + 1)−semiweak amenability.

1. Introduction A derivation from a Banach algebra A to a Banach A-module X is a bounded linear mapping D : A → X such that D(ab) = D(a)b + aD(b) for all a, b ∈ A. For each x ∈ X the mapping dx : a → ax−xa, (a ∈ A) is a derivation, called the inner derivation implemented by x. The concept of n-weak amenability was introduced and intensively studied by Dales, Ghahramani and Gronbæk [1]. A Banach algebra A is said to be n-weakly amenable (n ∈ N) if every derivation from A into A(n) is inner, where A(n) is the nth−dual module of A; we also write A(0) = A. A is called permanently weakly amenable if it is n−weakly amenable for each natural number n. C∗− algebras and the group algebra L1(G), for each locally compact group G, are the main examples of permanently weakly amenable Banach algebras (for a proof see [1], [4]). Let A be a Banach algebra and X be a nonzero Banach A−module. The ℓ1−direct sum of A and X with the norm and algebra product deﬁned by

∥(a, x)} = ∥a∥ + ∥b∥, (a, x)(b, y) = (ab, ay + xb)(a, b ∈ A x, y ∈ X) is denoted by A ⊕ X and is called a module extension Banach algebra. The equivalent conditions for n−weak amenability of module extension Banach algebras are investigated in [3].

59 S. Barootkoob

If A and B are Banach algebras and φ : A → B be a Banach algebra homomorphism, then we consider B as an A−module by the following module actions

ab = φ(a)b and ba = bφ(a)(a ∈ A, b ∈ B).

We denote the above A-module by Bφ. A Banach algebra A is called supper weakly amenable if for every Banach algebra B and each continues homomorphism φ : A → B and every bounded derivation ∗ dφ : A → Bφ we have ′ ′ ′ ⟨dφ(a), φ(a )⟩ + ⟨dφ(a ), φ(a)⟩ = 0 (a, a ∈ A).

For more details see [2]. The notion semiweak amenability of Banach algebras is also deﬁned in [2]. In this paper we extend this notion for every odd integer 2n + 1. Then we investigate the equivalent conditions for supper weak amenability and (2n+1)−semiweak amenability of a module extension Banach algebra A ⊕ X. The results lead to some equivalent conditions for n−weak amenability of module extension Banach algebra A ⊕ X, when A is unital. However there are some almost similar equivalent conditions for n−weak amenability of module extension Banach algebras in general [3].

2. Main results Deﬁnition 2.1. Let n ∈ N ∪ {0}. A Banach algebra A is Called (2n + 1)−semiweakly amenable if every derivation D : A → A(2n+1) that

⟨D(a), b⟩ + ⟨D(b), a⟩ = 0 (a, b ∈ A), is inner. Maybe it seems that we can similarly extend the deﬁnition of supper weak amenability to (2n + 1)−supper weak amenability of Banach algebras by changing ∗ (2n+1) Bφ to Bφ . But we can show that (2n + 1)−supper weak amenability is no thing else than supper weak amenability of Banach algebras. Proposition 2.2. If a Banach algebra A is (2n + 1)−semiweakly amenable, then it is (2n − 1)−semiweakly amenable. The following theorem is an extension of Theorem 3.3 of [2]. Theorem 2.3. A is (2n + 1)−weakly amenable if and only if A is supper weakly amenable and (2n + 1)−semiweakly amenable. Note that the latter proposition and theorem imply that if a Banach algebra A is (2n + 1)−weakly amenable, then it is (2n − 1)−weakly amenable [1]. Now in the sequel we characterize (2n + 1)−semiweakly amenable module extension Banach algebras and supper weakly amenable module extension Banach algebras.

60 Various notions of amenability

Theorem 2.4. The module extension Banach algebra A ⊕ X is (2n + 1)−semiweakly amenable if and only if (i) A is (2n + 1)−semiweakly amenable. (ii) The only A−module morphism S : X → X(2n+1), for which ⟨S (x), y⟩ + ⟨S (y), x⟩ = 0 (x, y ∈ X). is zero. (iii) For each A−module morphism T : X → A(2n+1) with ⟨T(x), a⟩ = 0, (a ∈ A, x ∈ X), there exists Φ ∈ X(2n+1) such that for each a ∈ A, aΦ = Φa and for each x ∈ X, T(x) = xΦ − Φx. (iv) For each bounded derivation D : A → X(2n+1) with ⟨D(a), x⟩ = 0, (a ∈ A, x ∈ X), (2n+1) there exists Ψ ∈ X such that for each x ∈ X, xΨ = Ψx and D = δΨ. Theorem 2.5. If A is unital, then the module extension Banach algebra A ⊕ X is supper weakly amenable if and only if (i) A is supper weakly amenable. (ii) If T : X → A∗ is a linear mapping for which there exists a derivation D : A → X∗ such that T(ax) = aT(x) + D(a)x, T(xa) = T(x)a + xD(a)(a ∈ A, x ∈ X), then for every x ∈ X we have ⟨T(x), 1A⟩ = 0. Now one can conclude from Theorem 2.3, Theorem 2.4 and Theorem 2.5 that for every unital Banach algebra A, A ⊕ X is (2n + 1)−weakly amenable if and only if (i) A is (2n + 1)−weakly amenable. (ii) The only A−module morphism S : X → X(2n+1), for which ⟨S (x), y⟩ + ⟨S (y), x⟩ = 0 (x, y ∈ X). is zero. (iii) For each A−module morphism T : X → A(2n+1) there exists Φ ∈ X(2n+1) such that for each a ∈ A, aΦ = Φa and for each x ∈ X, T(x) = xΦ − Φx. (iv) For each bounded derivation D : A → X(2n+1), there exists Ψ ∈ X(2n+1) such that for each x ∈ X, xΨ = Ψx and D = δΨ. However there are similar equivalent conditions for (2n + 1)−weak amenability of module extension Banach algebras in general case, which maybe A is not unital [3].

References [1] H.G. Dales, F. Ghahramani and N. Grønbæk, Derivations into iterated duals of Banach algebras, Studia Math. 128 (1) (1998), 19–54. [2] M. Eshagi Gordji, Homomorphisms, Amenability and weak amenability of Banach algebras, Math. FA (1) (2006), 1–8. [3] Y. Zhang, Weak amenability of module extensions of Banach algebras, Trans. Amer. Math. Soc. 354 (10) (2002), 4131–4151. [4] Y. Zhang, 2m−Weak amenability of group algebras, J. Math. Anal. Appl. 396 (2012), 412–416.

61 S. Barootkoob

Sedigheh Barootkoob, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord, Iran e-mail: [email protected]

62 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

THE X-RAY TRANSFORM AND ITS APPLICATION IN CRYSTALLOGRAPHY

Tajadin Derikvand

Abstract In this article, a review on deﬁnition of the X- ray transform is presented. We shall show that the X- ray transform is a special case of the Radon transform on homogeneous spaces when the topological group E(n) -the Euclidean group- acts on R2 transitively. Then its application in crystallography is brieﬂy studied.

2010 Mathematics subject classiﬁcation: 20H15. Keywords and phrases: Crystallography, homogeneous spaces, topological group, X-ray transform..

1. Introduction The X-ray transform as a special case of the Radon transform on homogeneous spaces appears in crystallography and in material sciences. More recent applications to X-Ray technology and tomography have widened interest in the subject. A well- known problem in material sciences is the determination of material properties. This method is also known as texture analysis. The analysis of crystallographic preferred orientations by means of orientation density functions and pole density functions is a widely used method in texture analysis. One can see more in this regard in [1–3].

2. Radon transform and X-ray transform In this section, first we describe the action of isomery group and its relation to the definition of the Radon transform on homogeneous spaces, then it will be specialized to the X-ray transform. Definition 2.1. Let G be a locally compact group and S a locally compact Hausdorff space. A left action of G on S is a continuous map (x, s) → xs from G × S to S such that (i)s → xs is homeomorphism of S for each x ∈ G, and x(ys) = (xy)s for all x, y ∈ G and s ∈ S . A space equipped with an action of is called a space. A space is called transitive if for every there exists such that . Let X denotes the plane of R2 viewed as a subset of {(x, y, 1) : x, y ∈ R}. Consider the isometry group G := M(2) of matrices in the form of

63 T. Derikvand

   α β   x  α, β, γ, η , =  γ η  ∈ , R , ( ; x y):  y  GL(3 ) 0 0 1 [ ] α β α > ∈ where 0, and γ η O(2). Then G acts transitively on X with the action

(α, β, γ, η; x, y)□(a, b) = (αa + βb + x, γa + ηb + y).

The isotropy group of X0 := (0, 0) is K = O(2). Let Y := P2 be the space of all lines in the plane of R2. If we denote the line v = mu + h by [(m, h)] then the action of G on X induces a transitive action of G on Y: γ + ηm γ + ηm (α, β, γ, η; x, y)♢[(m, h)] = [( , − (βh + x) + ηh + y)], α + βm α + βm Indeed, we have (α, β, γ, η; x, y)□(u, mu + h) = (αu + β(mu + h) + x, γu + β(mu + h) + y).

Now, the isotropy group of the x-axis (the line [(0, 0)]) is H = Z2 M(1) Thus the group L := K ∩ H = Z2 and H/L = M(1) = R has a Haar measure ∫ π 1 2 iθ dh , , , , = ds K/L = T f e dθ (1 0 0 1;s 0) . and also has a normalized Haar measure 2π 0 ( ) for all f ∈ Cc(K/L). ′ ′ ′ Also, if g = (α, β, γ, η; x, y), h = (1, 0, 0, 1; s, 0) and k = (α′ , β , γ , η ; 0, 0), then So the Radon transform is ∫ ∫ +∞ RK,H f (gH) = f (ghK)dh = f (αs + x, γs + y)ds. H/L −∞ Now, putting x := p cos(φ), y := p sin(φ), γ := cos(φ) and α := − sin(φ), we have ∫ +∞ RK,H f (p, φ) = f (pcosϕ − ssinϕ, psinϕ + scosϕ)ds. −∞   1 − x2 − y2 for x2 + y2 < 1 Let f (gK) = f (x, y) =  . Then f ∈ Cc(G/K) and 0 for x2 + y2 ≥ 1 = , φ = 4 − 2 3/2. RK,H f (gH) RK,H f (p ) 3 (1 p )

3. X-ray transform as a line integral X-ray imaging relies on the principle that an object will absorb or scatter X- rays of a particular energy in a manner dependent on its composition, quantified by the attenuation coefficient. When a beam of X-ray photons emits on texture of homogeneous material, the beam intensity decreases according to the equation −µx I = I0e , where I and I0 are input and output intensities. The attenuation coefficient µ depends on density of the material ρ and the nuclear composition characterized by

64 The X-ray transform and its application in crystallography the atomic number z. The distance passing a beam through the material is denoted ff ff by x. If it passes a xi’s distance∑ through di erent textures with di erent attenuation − µi xi coefficient µi’s then I = I0e i . Thus ∫ I p := −log( ) = µ(ρ, z)ds, I0 L indicates inside quality of the texture thus it is called a single projection. The line integral depends on distance of path from the origin and the angle ϕ that it turns left or right. Moving the source and detector together yields a vector of values called profile and various values of p and ϕ yeild sample matrix. Definition 3.1. Let f be a function∫ on some domain D ⊆ R2, and let L be a line in the = , plane, the line integral R f L f (x y)ds is said the Radon transform of f . The first theorem in Radons’ 1917 paper asserts that the above integral is well- defined.

4. X-ray transform and Crystallography Functional properties of various materials used in different areas such as life sciences, electronics, mechanics, mining engineering, food industry and etc. depend on their atomic and molecular structure. Knowledge of the internal structure and texture analysis of materials give us applicable technical properties. Usage of the various methods such as magnetic resonance imaging (MRI), computerized tomography scan (CT-scan) and X-ray diffraction that in the crystalline materials analysis is very com- mon. Crystallography can be referred as the study of theatomic positions via measur- ing the distribution of diffracted X-rays. The first experiment to identify the nature of the crystalline by using distribution of the diffracted radiations was conducted by Laue and showed that when radiation strikes with a crystal, they disperse in different directions and at various intensities. Stereographic projection technique has been found to measure and collect data of all intensities, it also help us to read out the angular relationships between different planes and directions in the crystal or unit cell in a lattice, this data set is called pole distribution density function (P. D. F). Consider two fixed coordinate systems on specimen (Ks) and a on crystal (Kc) in it The crystallographic orientation of an individual crystal gives us an orthogonal transformation g = g (ϕ1, ϕ, ϕ2) ∈ O(3) which brings into coincidence with (See [1, 4]). The set of all symmetries of the extended atom lattice forms a group, the so called space group of the crystal. Here and are the orthogonal group and the group of all translations in. The orthogonal part of the space group is called point group or isotropy group of the crystal and it is denoted by S point. A crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind, for a periodic crystal. A direction specified by coordinates relative to a crystal coordinate system is called crystal direction, we denote it by h .Two crystal directions

65 T. Derikvand h1, h2 are called crystallographically equivalent if there exists a symmetry g ∈ S space 2 of the crystal such that h1 = gh2 . We denote by S /S space the set of all classes of crystallographically equivalent directions. It is called orientation space. A direction specified by its coordinate vector relative to the specimen coordinate system is called specimen direction and is usually denoted by the letter r ∈ S2. With this notation a crystal direction and a specimen direction represent the same physical direction if andonly if r = gh. The class gS point is called crystal orientation Consider a one type, polycrystalline specimen, i.e. a compound of identical crystals all possessing the same point group. Next we assume that each crystal has a well defined crystal orientation relative to the specimen thus neglecting e.g. internal crystal defects. Then, if one denotes dv the volume elements of the sample which possess the orientation g within the element of orientation dg , and denotes the v the total sample volume ∆ = dv/v v, then, an orientation distribution function f (g) , is defined by f (g) dg where 2 dg = 1/8π sin ϕdϕ1dϕdϕ2. More abstractly we define Theorem 4.1. The function f , which depends on the orientation g , can be developed in a series of generalized spherical harmonics:

∑∞ ∑l ∑l −i(mϕ1+nϕ2) wlmnzlmn(ϕ)e i=0 n=−l m=−l

Where wlmn are the coeﬃcients of the series of generalized spherical harmonics. zlmn(ϕ) are the certain generalizations of the associated Jacob function.

References [1] C. Hammond, The basic of crystallography and diﬀraction, Oxford science publications, New Yourk, 2009. [2] J. Imhof, Determination of the Orientation Distribution Function from One Pole-ﬁgure, Texture and Microstructures 5 (1982) 7386. [3] A. A. Jafarpour , Linear processing of X-ray scattering patterns with missing pixels, arXive preprint arXive (2014). [4] R. O. Williams, Analytical Methods for Representing Complex Textures by Biaxial Pole Figures, Journal of Applied Physics 39 (1968) 4329-4335.

Tajadin Derikvand, International Campus, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran. e-mail: [email protected]

66 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

TRANSITIVITY PROPERTY AND EXTENDED WEIL FORMULA

Fatemeh Esmaeelzadeh

Abstract

For a locally compact group G and two closed subgroups H, N of G, we are going to write µG/H as the integral, with respect to µG/N of a family of measures on G/H indexed by the points of G/N, in which µG/H and µG/N are measures on quotient spaces G/H and G/N, respectively.

2010 Mathematics subject classiﬁcation: Primary 43A05, Secondary 43A85. Keywords and phrases: Quotient spaces, canonical mapping, integrable function, Weil formula.

1. Introduction Let G be a locally compact group and H be a closed subgroup of G with left Haar measures λ and λH, respectively. Consider G/H as a quotient space on which G acts from the left and a Radon measure µ on G/H is said to be G-invariant if µx(yH) = µ(yH) for all x, y ∈ G, where µx is deﬁned by µx(E) = µ(xE) (for Borel subsets E of G/H). It is well known that there is a G-invariant Radon measure µ on G/H if and only if ∆G|H = ∆H, where ∆G, ∆H are the modular functions on G and H, respectively. In this case we have, ∫ ∫ ∫ ∫

f (x)dλ(x) = P f (xH)dµ(xH) = f (xh)dλH(h)dµ(xH), (1) G G/H G/H H ∫ = λ in which P f (xH) H f (xh)d H(h) is a continuous linear map from Cc(G) onto Cc(G/H). The formula (1) is called Weil formula. It has been shown that if λ be a positive Radon measure on G such that dλ(xh) = ∆H(h)dλ(x), for h ∈ H, one can then form a unique positive Radon measure µG/H on quotient space G/H ( For more details see [2, 4, 5]).

2. Main result Let G be a locally compact group and H, N be two closed subgroups of G such that H ⊆ N and ∆N|H = ∆H. We denote by qN, qH, p, the canonical mappings of

67 F. Esmaeelzadeh

G onto G/N, of G onto G/H and of N onto N/H. Let λN and λH be the left Haar measures on N and H, respectively. Since ∆N|H = ∆H, then there exists a G-invariant measure µN/H on N/H . On the otherhand, if λ is a left Haar measure on G such that dλ(xh) = ∆H(h)dλ(x), one can find Radon measures µG/N, µG/H on quotient spaces G/N and G/H. Then it is easy to check that, the mapping (x, n) 7→ qH(xn) of G × N into G/H is continuous. Since qH(xnh) = qH(xn), for all h ∈ H, this mapping defines a continuous mapping of G × (N/H) into G/H. Whence for each fixed x ∈ G, the mapping ψx of N into G such that ψx(n) = xn, defines a mapping ωx of N/H into G/H in which ωx(p(n)) = qH(ψx(n)) = qH(xn).

It is easy to show that ψxn = ψxoϱN(n), thererfore that ωxn = ωxoϱN/H(n), for all n ∈ N, ′ ′ in which ϱN(n)(n ) = nn . The following lemma shows that the map ωx is proper. Lemma 2.1. Let E be a compact subset of G/H and K be a compact subset of G. Then ∪ ω−1 / ∪ ω−1 x∈K x (E) is relatively compact in N H. In particular, x∈K x (E) is contained in a compact subset of N/H.

Proof. Let F be a compact subset of G such that qH(F) = E. Let L be theH set of n ∈ N such that Kn intersects F. Then L is compact (see [1], ChapterIII, 4.5, ∈ ∈ ∪ ω−1 ∈ Theorem1). Let n N, such that p(n) x∈K x (E). Thus there exists x K such that ωx(p(n)) ∈ E. i.e. qH(xn) ∈ E and since qH(F) = E, there exists h ∈ H, xnh ∈ F. ∈ = ∈ ∪ ω−1 ⊆ □ Then nh L. So p(nh) p(n) p(L). That is x∈K x (E) p(L).

Let M(N/H) and M(G/H) be complex measure spaces on quotient spaces N/H and G/H, respectively, as introduced in [5]. Lemma 2.1 shows that the mapping ωx is proper. Then ωx extends continuously to a map from M(N/H) into M(G/H) ([3], Section 4.5). Now let φ ∈ Cc(G/H). Deﬁne the function Ψ of G into M(G/H) such that ∫

Ψ(x) = ⟨φ, ωx(µN/H)⟩ = φ(ωx(p(n))dµN/H(p(n)). N/H

The function Ψ is continuous and compact support. Moreover, since the measure µN/H is G-invariant, we have Ψ(xn) = ⟨φ, ωxn(µN/H)⟩

= ⟨φ, ω oϱ / (n)(µ / )⟩ ∫ x N H N H ′ ′ = φ(ω oϱ / (n)(p(n )))dµ / (p(n )) ∫N/H x N H N H ′ ′ = φ(ω (nn H))dµ / (p(n )) ∫N/H x N H ′ ′ = φ ω µ / N/H ( x(n H))d N H(p(n )) = ⟨φ, ωx(µN/H)⟩ = Ψ(x),

68 Transitivity Property and Extended Weil Formula

for n ∈ N. Then the mapping Ψ˜ of G/N into M(G/H) in which

Ψ˜ (qN(x)) = ⟨φ, ωx(µN/H)⟩, is continuous with compact support, for all φ ∈ Cc(G/H). Proposition 2.2. Let φ ∈ C (G/H). Then ∫ c ∫

⟨φ, ωx(µN/H)⟩dµG/N(qN(x)) = φ(qH(x))dµG/H(qH(x)). G/N G/H Proof. By (1), for φ ∈ C (G/H) we have ∫ c ∫

φ(qH(x))dµG/H(qH(x)) = f (x)dλ(x), G/H G where∫ φ = P f and f ∈ Cc(G). Also, ⟨φ, ω (µ / )⟩dµ / (q (x)) = ∫G/N ∫ x N H G N N (P f (ω (p(n))dµ / (p(n))dµ / (q (x)) = ∫G/N ∫N/H x N H G N N (P f (q (xn))dµ / (p(n))dµ / (q (x)) = ∫G/N ∫N/H ∫ H N H G N N L f (nh)dλ (h)dµ / (p(n))dµ / (q (x)) = ∫G/N ∫N/H H x H N H G N N f (xn)dλ (n)dµ / (q (x)) = ∫G/N N N G N N λ , G f (x)d (x) in which Lx f (n) = f (xn). □

Corollary 2.3. (i) Let φ be a µG/H-integrable function on G/H. There exists a µG/N-negligible subset E of G/N having the following property: if x ∈ G is such that∫ qN(x) < E, then the function φoωx on N/H is µN/H-integrable. The integral φ(ωx(p(n))dµN/H(p(n) is a µG/N-integrable function and N∫/H ∫ ∫

φ(qH(x))dµG/H(qH(x)) = dµG/N(qN(x)) φ(ωx(p(n))dµN/H(p(n)). (2) G/H G/N N/H

(ii) suppose that there exists a bounded positive measure µG/H on quotient space G/H. Then there exists a bounded positive measure on quotient space N/H. Proof. (i) By proposition 2.2 we have, ∫ φ(q (x))dµ / (q (x)) = ∫G/H H G H H ⟨φ, ω (µ / )⟩dµ / (q (x)) = ∫G/N ∫ x N H G N N φ ω µ / µ / . G/N N/H( ( x(p(n))d N H(p(n))d G N(qN(x))

(ii) The function 1 on G/H is µG/H- integrable. By the part (i), the function 1 on N/H is µN/H-integrable. Thus µN/H is bounded. □ Remark 2.4. If H = {e}, then the Weil formula can be concluded from (2).

69 F. Esmaeelzadeh

References [1] N. Bourbaki, Elements of Mathematics, General Topology, springer, Verlag Berlin, New york, (1971). [2] F. Esmaeelzadeh and R. A. Kamyabi Gol, Homogeneous Spaces and Square Integrable Represen- tations, Ann. Funct. Anal. 7 (2016). [3] G. B. Folland, Real Analysis , Modern techniques and their applications. 2nd ed. Wiley, New York, (1999). [4] . B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, FL, (1995). [5] H. Reiter and J. Stegeman, Classical Harmonic Analysis and Locally compact Group, Claredon press, (2000).

Fatemeh Esmaeelzadeh, Department of Mathematics, Bojnourd Branch, Islamic Azad University, Bojnourd, Iran e-mail: [email protected]

70 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

SECOND TRANSPOSE OF A DUAL VALUED DERIVATION

Morteza Essmaili

Abstract Let A be a Banach algebra and X be an arbitrary Banach A-bimodule. In this paper, we study second transpose a derivation with value in dual Banach A-module X∗. Indeed, for a continuous derivation D : A −→ X∗ we obtain a necessary and suﬃcient condition such that the bounded linear map Λ ◦ D′′ : A∗∗ −→ X∗∗∗ to be a derivation, where Λ is composition of restriction and canonical injection maps.

2010 Mathematics subject classiﬁcation: Primary 46H20, Secondary 46H25. Keywords and phrases: Derivation, second transpose, Banach module, module actions.

1. Introduction The second transpose a derivation from a Banach algebra A into dual module A∗ has been discussed in some papers, see for example [3], [4] and [5]. Dales, Rodriguez and Velasco in [3] studied second transpose a A∗-valued derivation D : A −→ A∗ and obtained conditions under which the second transpose D′′ : A∗∗ −→ A∗∗∗ is a derivation. Indeed, it is shown that D′′ is a derivation if and only if D′′(A∗∗) · A∗∗ ⊆ A∗ [3, Theorem 7.1]. Also in [5], the authors investigated the second transpose of a derivation D on Banach algebra A with value in X∗, where X is an arbitrary Banach A-bimodule. Indeed, they obtained a necessary and sufficient condition under which D′′ : A∗∗ −→ X∗∗∗ be a derivation and generalized some results in [3]. Weak amenability of second dual of Banach algebras has been studied with a different approach by Ghahramani, Loy and Willis in [4]. They considered some conditions under which the map Λ ◦ D′′ : A∗∗ −→ A∗∗∗ to be a derivation, where Λ is composition of restriction and canonical injection maps as defined in Section 2. Indeed, they claimed that Λ ◦ D′′ is a derivation in the case where A is a left ideal of A or D is weakly compact, see also [2]. They used the fact that for each F, G ∈ A∗∗, the equality Λ(D′′(F) · G) = (Λ ◦ D′′(F)) · G holds on A∗∗ if and only if it holds just on A. Recently, the authors in [1] obtained a necessary and sufficient condition such that the map Λ ◦ D′′ : A∗∗ −→ A∗∗∗ be a derivation as follows and showed that part of the proof of [4, Theorem 2.3] and [2, Lemma 4(ii)] have the same computational gap.

71 M. Essmaili

Theorem 1.1. ([1, Theorem 3.1]) Let A be a Banach algebra and D : A −→ A∗ be a continuous derivation. Then the following are equivalent: (i) Λ ◦ D′′ : A∗∗ −→ A∗∗∗ is a derivation. (ii) For every F, G ∈ A∗∗, Λ(D′′(F) · G) = (Λ ◦ D′′(F)) · G on A and

Λ ◦ D′′(A∗∗) ⊆ WAP(A).

In this paper, our aim is to study of the map Λ ◦ D′′ in general case which is an extension of some results in [1]. Indeed, for an arbitrary Banach A-bimodule X and continuous derivation D : A −→ X∗, we obtain a necessary and suﬃcient condition under which the map Λ◦D′′ : A∗∗ −→ X∗∗∗ to be a derivation. As a consequence, in the case where D is a weakly compact derivation, we obtain a simple condition equivalent to the map Λ ◦ D′′ be a derivation.

2. Main results Throughout the paper, X denotes a Banach space and X∗ , X∗∗ and X∗∗∗ are ﬁrst, ∗∗ second and third dual of X, respectively. The canonical embedding map κX : X −→ X is deﬁned by ∗ κX(x)( f ) = f (x)(x ∈ X, f ∈ X ).

We also use the notation κX(x) = bx. Furthermore, we consider the restriction map R : X∗∗∗ −→ X∗ by the equation

R(ψ)(x) = ψ(bx)(ψ ∈ X∗∗∗, x ∈ X).

Definition 2.1. Let X be a Banach space. We define the bounded linear map Λ : X∗∗∗ −→ X∗∗∗ by ∗∗∗ Λ(ψ) = κX∗ ◦ R(ψ)(ψ ∈ X ). Definition 2.2. Let X, Y and Z be Banach spaces and π : X × Y −→ Z be a bounded bilinear map. The flip map πr : Y × X −→ Z is regarded by the equation

πr(y, x) = π(x, y)(x ∈ X, y ∈ Y).

We also deﬁne the transpose of π∗ : Z∗ × X −→ Y∗ by

π∗(z∗, x)(y):= z∗(π(x, y)) (x ∈ X, y ∈ Y, z∗ ∈ Z∗).

Let A be a Banach algebra and X be a Banach A-bimodule with the left module action π1 : A × X −→ X and the right module action π2 : X × A −→ X. Similar to [5], we denote this module structure by the triple (π1, X, π2). It is straightforward to check ∗ πr∗r π∗, that X is a Banach A-module where the left and right module actions are 2 and 1 respectively. Moreover, X∗∗ is a Banach (A∗∗, □)-module with the left and right module π∗∗∗ π∗∗∗. π∗∗∗r∗r, ∗∗∗, π∗∗∗∗ actions 1 and 2 By this fact, we conclude that the triple ( 2 X 1 ) is a Banach (A∗∗, □)-module.

72 Second transpose of a dual valued derivation

Let A be a Banach algebra and (π1, X, π2) be a Banach A-module. A bounded linear map D : A −→ X is a derivation if

D(ab) = π1(a, D(b)) + π2(D(a), b)(a, b ∈ A). πr∗r, ∗, π∗ −→ ∗ Now, consider the A-module structure ( 2 X 1) and suppose that D : A X is a derivation. In this section, we obtain a necessary and suﬃcient condition such that the map Λ ◦ D′′ : A∗∗ −→ X∗∗∗ be a derivation by regarding the A∗∗-module structure π∗∗∗r∗r, ∗∗∗, π∗∗∗∗ . ( 2 X 1 )

Theorem 2.3. Let A be a Banach algebra and (π1, X, π2) be a Banach A-module. If D : A −→ X∗ is a continuous derivation, then for each F, G ∈ A∗∗, Λ ◦ ′′ □ = Λ π∗∗∗∗ ′′ , + Λ πr∗r∗∗∗ , ′′ . D (F G) ( 1 (D (F) G)) ( 2 (F D (G))) We recall that in throughout this section, X∗ is regarded as a A-module with πr∗r, ∗, π∗ ∗∗∗ ∗∗ the structure ( 2 X 1) and moreover X is a A -module with the structure π∗∗∗r∗r, ∗∗∗, π∗∗∗∗ . ( 2 X 1 ) The following theorem gives a necessary and suﬃcient condition such that the map Λ ◦ D′′ : A∗∗ −→ X∗∗∗ be a derivation.

Theorem 2.4. Let A be a Banach algebra and (π1, X, π2) be a Banach A-module. If D : A −→ X∗ is a continuous derivation, then the following are equivalent: (i) Λ ◦ D′′ : A∗∗ −→ X∗∗∗ is a derivation. (ii) For each F, G ∈ A∗∗ we have Λ π∗∗∗∗ ′′ , = π∗∗∗∗ Λ ◦ ′′ , . ( 1 (D (F) G)) 1 ( D (F) G)

Deﬁnition 2.5. Let A be a Banach algebra and (π1, X, π2) be a Banach A-module. We deﬁne set W by W = { ∈ ∗ ∗∗ −→ ⟨π∗∗∗ , ∗∗ , ⟩ ∗ f X : the map x 1 (G x ) f is weak continuous on X∗∗ , for each G ∈ A∗∗},

In particular, let A be a Banach algebra X = A and π1 = π2 = m, where m is the multiplication map A. it is easy to see that W is exactly the set of all almost periodic functionals on A. By this fact, the following theorem may be regarded as a generalization of [1, Theorem 3.1].

Theorem 2.6. Let A be a Banach algebra and (π1, X, π2) be a Banach A-module. If D : A −→ X∗ is a continuous derivation, then the following are equivalent: (i) Λ ◦ D′′ : A∗∗ −→ X∗∗∗ is a derivation. , ∈ ∗∗, Λ π∗∗∗∗ ′′ , = π∗∗∗∗ Λ ◦ ′′ , (ii) For each F G A the equality ( 1 (D (F) G)) 1 ( D (F) G) holds on X and moreover Λ ◦ D′′(A∗∗) ⊆ W. As a consequence, we have the following characterization for weakly compact derivations.

73 M. Essmaili

Corollary 2.7. Let A be a Banach algebra, (π1, X, π2) be a Banach A-module and D : A −→ X∗ be a weakly compact derivation. Then Λ ◦ D′′ : A∗∗ −→ X∗∗∗ is a derivation if and only if D′′(A∗∗) ⊆ W.

References [1] M. Amini, M. Essmaili and M. Filali, The second transpose of a derivation and weak amenability of the second dual Banach algebras, New York Journal Math. 22, 265-275 (2016). [2] S. Barootkoob, H. R. E. Vishki, Lifting derivations and n-weak amenability of the second dual of a Banach algebra, Bull. Aust. Math. Soc. 83, 122-129 (2011). [3] H. G. Dales, A. Rodriguez and M. V. Velasco, The second transpose of a derivation, J. London Math. Soc. 64 (2), 707-721 (2001). [4] F. Ghahramani, R. J. Loy and G. A. Willis, Amenability and weak amenability of second conjugate Banach algebras, Proc. Amer. Math. Soc. 124, 1489-1497 (1996). [5] S. Mohammadzadeh and H. R. E. Vishki, Arens regularity of module actions and the second adjoint of a derivation, Bull. Aust. Math. Soc. 77, 465-476 (2008).

Morteza Essmaili, Department of Mathematics, Kharazmi University, Tehran, Iran e-mail: [email protected]

74 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

WEAK ALMOST PERIODICITY AND UNIFORM CONTINUITY ON A LOCALLY COMPACT QUANTUM GROUP

Ramin Faal and Hamid Reza Ebrahimi Vishki∗

Abstract First we present some useful characterizations for the function spaces AP(G), WAP(G) and LUC(G) on a locally compact quantum group G and then we study some conditions under which AP(G) and WAP(G) are C∗-algebras. We also investigate the equality LUC(G) = WAP(G).

2010 Mathematics subject classiﬁcation: Primary 47L10, Secondary 46H25. Keywords and phrases: Locally compact quantum group, coamenable, (weak) almost periodicity, left uniform continuity.

1. Introduction Let G be a locally compact group. It is known that the function spaces AP(G), ∗ WAP(G) and LUC(G) are C -subalgebras of Cb(G). Moreover, they enjoy the inclusion relations AP(G) ⊆ WAP(G) ⊆ LUC(G), where the equality hold under some additional conditions. For example, it has been known that WAP(G) = LUC(G) if and only if G is compact. From the operator theory point of view, it is known that WAP(G) can be identified with WAP(L1(G)). Also there is a characterization of left uniform continuity, namely LUC(G) = L∞(G) · L1(G). For complete information with more details on these funtion spaces and their inclusion relashionships one may refer to [1]. We refer to [4] for the definitions and basic facts on Hopf von Neumann algebras and locally compact quantum groups. In [2] Daws showed that for an abelian Hopf von Neumann algebra (M, Γ), WAP(M∗) = {x ∈ M : Lx : M∗ → M is weakly compact} ∗ and AP(M∗) = {x ∈ M : Lx : M∗ → M is compact} are C -subalgebras of M. In [3] he defined WAP(M, Γ) for an arbitrary Hopf von Neumann algebra (M, Γ) and proved that ∗ it is the largest C -algebra contained in WAP(M∗). However, in contrast to the situation for group, it was not known whether the equality WAP(M∗) = WAP(M, Γ) holds. In this talk we show that for a coamenable locally compact quantum group G, the 1 1 ∗ spaces WAP(L (G)) and AP(L (G)) are C -algebras and that WAP(M∗) = WAP(M, Γ).

∗ speaker

75 Ramin Faal,Hamid Reza Ebrahimi Vishki

We also deﬁne the notation of completely weakly compact operator for an arbitrary Hopf von Neumann algebra and we give a charaterization of WAP(M, Γ) in terms of complete weak compactness of left translation operators. On the other hand Runde [5] showed that for certain locally compact quantum groups G, LUC(G) is a C∗-algebra. In this talk we aslo improve Runde’s result.

2. Main Results

For a Hopf von Neumann algebra M we deﬁne CWAP(M∗) and we show the following result. Theorem 2.1. Let M be a Hopf von Neumann algebra. Then ∗ (i) CWAP(M∗) is equal to WAP(M, Γ), and it is the largest C -algebra in WAP(M∗). ∗ (ii) If M is abelian, then WAP(M∗) = CWAP(M∗), so WAP(M∗) is a C -subalgebra of M. We give some new characterizations for WAP(L1(G)), AP(L1(G)) and LUC(G) from which we present the following results. Theorem 2.2. Let G be a coamenable locally compact quantum group. Then WAP(L1(G)) and WAP(L1(G)) are a C∗-algebras. Theorem 2.3. Let G be a locally compact quantum group such that L1(G) is strongly Arens irregular. Then WAP(L1(G)) = LUC(G) implies that G is compact.

References [1] J. Berglund, H. Junghenn and P. Milnes, Analysis on Semigroups; Function Spaces, Compactiﬁ- cations, Representations, Wiley, New York, 1989. [2] M. Daws, Characterising weakly almost periodic functionals on the measure algebra, Studia Math. 204 (2011) 213-234. [3] M. Daws, Non-commutative separate continuity and weakly almost periodicity for Hopf von Neumann algebras, J. Funct. Anal. 269 (2015) 683-704. [4] J. Kustermans and S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (2003), 68–92. [5] V. Runde, Uniform continuity over locally compact quantum groups, J. London Math. Soc. 80 (2009) 55-71.

Ramin Faal, Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran e-mail: [email protected]

Hamid Reza Ebrahimi Vishki, Department of Pure Mathematics and Centre of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad,

76 Weak Almost Periodicity and Uniform Continuity on a Locally Compact Quantum Group

P. O. Box 1159, Mashhad 91775, Iran e-mail: [email protected]

77 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

CONTINUOUS WAVELET TRANSFORM ON A QUATERNIONIC HILBERT SPACE

Massoumeh Fashandi

Abstract In this paper the new concept of the quaternionic unitary representation of a locally compact group to the unitary group of a quaternionic Hilbert space is studied. A continuous wavelet transform by means of a special case of such representations is deﬁned to extend the continuous wavelet transform related to semidirect product groups.

2010 Mathematics subject classiﬁcation: Primary 46E22, 47B07, 47A67. Keywords and phrases: Quaternionic Hilbert space, unitary representation, continuous wavelet transform.

1. Introduction and Preliminaries The theory of wavelet transforms is closely related to the theory of unitary representations of locally compact groups on the unitary group of a Hilbert space. In this paper, instead of a complex Hilbert space, we use a quaternionic Hilbert space and deﬁne a quaternionic unitary representation from a locally compact group to the unitary group of the quaternionic Hilbert space. Generalizing the results of [5], we prove some new facts regarding the irreducibility of a special kind of a quaternionic representation that is introduced in this paper. Quaternionic wavelet transform is highly applicable; according to [5], the results of [5] as well as the results of this paper, could be applicable in studying stereophonic or stereoscopic signals. Also, color images can be modeled by two-dimensional quaternion valued functions, so the quaternionic wavelet transform are useed in color image processing. We refer the interested reader to [1] and [2] for more information about quaternionic Fourier and wavelet transforms. Throughout this paper, H will stand for the noncummutative ﬁeld of quaternions and ∗ H = H − {0}. A quaternion q is of the form q = x0 + x1i + x2 j + x3k, where x0, x1, x2 and x3 ∈ R and i, j and k are the imaginary units and obey the following multiplication rules: i2 = j2 = k2 = −1, i j = − ji = k, jk = −k j = i, and ki = −ik = j. = − − − The quaternionic√ conjugate of q is q x0 x1i x2 j x3k and the norm of the quaternion | | = 2 + 2 + 2 + 2 q is q x0 x1 x3 x4. It will be convenient to consider a quaternion q as a

78 M. Fashandi linear combination of two complex numbers q = (x0 + x1i) + (x2 + x3i) j. Therefore, 2 2 2 q¯ = (x0 + x1i) − (x2 + x3i) j and |q| = |x0 + ix1| + |x2 + x3i| . Let H be a linear vector space over the ﬁeld of quaternions under the right scalar multiplication. We suppose that a function ⟨.|.⟩H : H × H −→ H, which is called an inner product, exists such that for every u, v, w ∈ H and p, q ∈ H the following properties hold:

(i) ⟨u|v⟩H = ⟨v|u⟩H; (ii) ⟨u|u⟩H > 0 unless u = 0; (iii) ⟨u|vp + wq⟩H = ⟨u|v⟩H p + ⟨u|w⟩Hq. If the space H is√ a complete metric space with the metric generated by the quaternionic norm ∥u∥H = ⟨u|u⟩H, then H is called a right quaternionic Hilbert space and it shares most of the properties of a complex Hilbert space, including Cauchy- Schwartz inequality and triangle inequality (see [4]).

Example 1.1. For a measure space (X, µX), the space of all quaternion valued square 2 integrable functions on X, which is shown by LH(X, µX), is a right quaternionic Hilbert space with the inner product ∫ ∫

⟨ f |g⟩ 2 = ⟨ f |g⟩Hdµ = f (x)g(x)dµ (x), (1) LH(X,µX ) X X X X and the right scalar multiplication is deﬁned by ( f q)(x) = f (x)q, for q ∈ H.

For a right quaternionic Hilbert space H, it is said that T : H −→ H is a right linear operator if for all u, v ∈ H and p ∈ H,

T(up + v) = (Tu)p + Tv.

Such an operator is said to be bounded if there exists K ≥ 0 such that for all u ∈ H,

∥Tu∥H ≤ K∥u∥H.

The set of all bounded right linear operators on H is denoted by B(H). The norm of a bounded right linear operator T is deﬁned similar to the complex case and B(H) is a complete normed space (see [4]; Proposition 2.11, for more properties of B(H)). In what follows, by an operator we mean a bounded right linear operator. Adjoint of an operator as well as unitary and positive operators are deﬁned similar to the complex case. Many properties of the adjoint operator, are stated and proved in Theorem 2.15 ∗ and Remark 2.16 of [4], including ∥T∥B(H) = ∥T ∥B(H). It is emphasised that, the adjoint operation is not an involution on B(H), as the equality (qT)∗ = qT ∗ holds only for q ∈ R. The set of unitary operators of B(H) with composition, forms a group which is called the unitary group of H and is shown by U(H). See [4] for more properties of unitary and positive operators on H.

79 CWT on a Quaternionic Hilbert Space

2. Quaternionic Unitary Representation Let G be a locally compact group, H a right quaternionic Hilbert space and U(H) the unitary group of unitary operators on H. A quaternionic unitary representation of G is a homomorphism UH from G into U(H)– that is a map UH : G → U(H) satisfying −1 −1 ∗ UH(xy) = UH(x)UH(y) and UH(x ) = UH(x) = UH(x) – such that x → UH(x)u is continuous from G to H, for any u ∈ H. A closed subspace M of H is called an invariant subspace for UH if UH(g)(M) ⊆ M, for all g ∈ G. If UH admits a non trivial invariant subspace, then UH is called reducible, otherwise it is called irreducible. Similar to the complex case, a quaternionic unitary representation UH of G on H is called square-integrable if there exists a non zero element φ ∈ H such that ∫ 2 |⟨φ|UH(g)φ⟩H| dµG(g) < ∞. (2) G A unit vector φ satisfying (2) is said to be an admissible wavelet for UH, and the constant ∫ 2 cφ = |⟨φ|UH(g)φ⟩H| dµG(g), (3) G is called the wavelet constant associated to the admissible wavelet φ. In the following proposition by means of a unitary representation of G on the 2 complex Hilbert space L (X, µX), a quaternionic unitary representation is built. Proposition 2.1. Let π be a unitary representation of the locally compact group G into 2 2 U(L (X, µX)), for a measure space (X, µX). Deﬁne UH : G → U(LH(X, µX)), by 2 UH(g) f = π(g) f1 + π(g) f2 j, f ∈ LH(X, µX), (4) 2 where f = f1 + f2 j, for two complex valued functions f1 and f2 in L (X, µX). Then (i) UH is a quaternionic unitary representation; (ii) If π is irreducible, then so is UH; (iii) π is square-integrable if and only if UH is.

3. Quaternionic Continuous Wavelet Transform Let G = S σT be the semidirect product of two locally compact groups S and T, where S is abelian and t 7→ σt is a homomorphism from T into the group of automorphisms of S . Let (s, t) 7→ σt(s) be a continuous mapping from S × T onto S , then G is a locally compact group with the Cartesian product topology. By the transitive action of G on S deﬁned by (a, b).s = aσb(s), the group S is a homogeneous space of G. One can see [3] for related deﬁnitions. Let T˜ = {(e1, t), t ∈ T}, where e1 is the identity element of S . The rho-function for the ∆ ˜ (e , t) pair (G, T˜), that will be shown by ρ(t) during this paper, is given by ρ(t) = T 1 , ∆G(e1, t) ∆ ∆ ˜ π where T˜ and G are modular functions of T and G, respectively. If from G into U(L2(S, µ )), is the unitary representation [S ] −1/2 −1 −1/2 −1 π(a, b) f (s) = ρ(b) f ((a, b) .s) = ρ(b) f (σb−1 (a s)), (5)

80 M. Fashandi

2 the continuous wavelet transform (CWT) of f ∈ L (S, µS ) is deﬁned by

, = ⟨π , ψ| ⟩ 2 , Wψ f (s t) (s t) f L (S,µS ) 1 2 ψ ∈ ∩ , µ ∥ψ∥ 2 = in which, L L (S S ) is a wavelet. This means that L (S,µS ) 1 and there is a finite and non negative constant Cψ such that for any γ ∈ Sˆ , the dual group of S , ∫ 2 Cψ = |ψˆ(γ ◦ σt)| dµT (t). T Definition 3.1. Let π be the representation defined by (5), ψ the wavelet defined by 2 (3), and Ψ = ψ + ψ j. The quaternionic CWT of f = f1 + f2 j ∈ LH(S, µS ) at (a, b) ∈ G is defined by H W f (a, b) = ⟨UH(a, b)Ψ| f ⟩ 2 Ψ LH ,µ , (S S ) 2 where, UH from G = S σT into the unitary group of LH(S, µS ) is considered as [ ] [ UH(a, b) f (s) = π(a, b) f1 + π(a, b) f2 j ](s). In the following proposition we extend some of the well-known properties of the CWT to the quaternionic one. Proposition 3.2. By the notations of Definition 3.1, H (i) WΨ f (a, b) = [(Wψ f1 + Wψ f2) + (Wψ f2 − Wψ f1) j](a, b); H H (ii) ⟨W f |W g⟩ 2 = 2Cψ⟨ f |g⟩ 2 ; Ψ Ψ LH(G,µG) LH(S,µS ) ∥ H ∥2 = ∥ ∥2 (iii) WΨ f 2 2Cψ f 2 . LH(G,µG) LH(S,µS )

Acknowledgement The author would like to express her most sincere appreciation to the memory of Professor Syed Twareque Ali from Concordia University, for the valuable discussions.

References [1] M. Bahri, R. Ashino and R. Vaillancourt, Continuous quaternion fourier and wavelet transforms, Int. J. Wavelets Multiresolut. Inf. Process, 12, (2014). [2] E. Hitzer and S. J. Sangwine, Quaternion and Cliﬀord Fourier Transforms and Wavelets, Trends Math., (2013). [3] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Inc., (1995). [4] R. Ghiloni, V. Moretti and A. Perotti, Continuous slice functional calculus in quaternionic Hilbert spaces, Rev. Math. Phys. 25, (2013). [5] S. T. Ali, and K. Thirulogasanthar, The quaternionic aﬃne group and related continuous wavelet transforms on complex and quaternionic Hilbert spaces, J. Math. Phys., 55, (2014).

Massoumeh Fashandi, Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran e-mail: [email protected]

81 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

SOLVING A NONLINEAR MULTI-ORDER FRACTIONAL DIFFERENTIAL EQUATION USING HAAR WAVELETS

Fatemeh Ghomanjani

Abstract In this paper, the Haar wavelets is applied to obtain approximate solutions of nonlinear multi-order fractional diﬀerential equations (M-FDEs). The fractional derivative is described in the Caputo sense. Numerical example is presented to verify the eﬃciency and accuracy of the proposed algorithm. The results reveal that the method is accurate and easy to implement.

2010 Mathematics subject classiﬁcation: Primary 65K10, Secondary 26A33. Keywords and phrases: Multi-Order Fractional Diﬀerential Equations; Caputo Derivative..

1. Introduction Many phenomena in engineering physics, chemistry, and other sciences can be described very successfully by models that use mathematical tools of fractional calculus, i.e. the theory of derivatives and integrals of non-integer order. For example, they have been successfully used in modeling frequency dependent damping behavior of many viscoelastic materials. There are numerous research which demonstrate the applications of fractional derivatives in the areas of electrochemical processes, dielectric polarization, colored noise, and chaos [1]. The organization of this study is arranged as follows. In Section 2, Basic Preliminaries is presented. Properties of the Haar basis is stated in 3. An numerical example is provided in Section 4. Finally, Section 5 will give a conclusion brieﬂy.

2. Basic Preliminaries Let x :[a, b] → R be a function, α > 0 a real number, and n = α, where α denotes the smallest integer greater than or equal to α (see [2]). The left (left RLFI) and right (right RLFI) Riemann-Liouville fractional integrals are deﬁned by ∫ 1 t Iα x(t) = (t − τ)α−1 x(τ) dτ, (left RLFI), a t Γ(α) ∫ a b α = 1 τ − α−1 τ τ, , tIb x(t) ( t) x( ) d (right RLFI) Γ(α) t

82 F. Ghomanjani

The left (left RLFD) and right (right RLFD) Riemann-Liouville fractional derivatives are given according to ∫ 1 dn t Dα x(t) = (t − τ)n−α−1 x(τ) dτ, (left RLFD), a t Γ(n − α) dtn ∫ a − n n b α = ( 1) d τ − n−α−1 τ τ, , tDb x(t) n ( t) x( ) d (right RLFD) (1) Γ(n − α) dt t Moreover, the left (left CFD) and right (right CFD) Caputo fractional derivatives are defined by means of ∫ 1 t C Dα x(t) = (t − τ)n−α−1 x(n)(τ) dτ, (left CFD), a t Γ(n − α) ∫a − n b C α = ( 1) τ − n−α−1 (n) τ τ, , t Db x(t) ( t) x ( ) d (right CFD) (2) Γ(n − α) t The relation between the right RLFD and the right CFD is as follows: ∑n−1 x(k)(b) C Dα x(t) = Dα x(t) − (b − t)k−α, (3) t b t b Γ(k − α + 1) k=0 Further, it holds C α = , 0 Dt c 0 (4) where c is a constant, and { 0, for n ∈ N , and n < ⌈α⌉ C α n = 0 0 Dt t Γ(n+1) n−α, ∈ N ≥ ⌈α⌉ (5) Γ(n+1−α) t for n 0 and n where N0 = {0, 1, 2,...}. We recall that for α ∈ N the Caputo differential operator coincides with the usual differential operator of integer order. In this paper, fractional differential equation was considered ∑l α αr D x(t) = a(t)x(t) + br(t)D x(qrt), m − 1 < α ≤ m, t ∈ [0, b], r=1 (i) x (t) = µi, i = 0, 1,..., m − 1.

Here, 0 < qr < 1, 0 ≤ αr < α ≤ m, r = 1, 2,..., l; x is an unknown function; a and br, r = 1, 2,..., l, are the known functions deﬁned in [0, b].

3. Properties of the Haar basis The RH functions RH(r, t), r = 1, 2,..., are composed of three values +1, −1, 0 and can be deﬁned on the interval [0, 1) by [2] as   , ≤ ≤  1 J1 t J 1  2 RH(r, t) =  −1, J 1 ≤ t ≤ J0  2 0, otherwise

83 Bernstein polynomial approach where j − u 1 J = , u = 0, , 1. (6) u 2i 2 The value of r deﬁnes two parameters i and j via r = 2i + j − 1, i = 0, 1, 2, 3,..., j = 1, 2, 3,..., 2i. RH(0, t) is deﬁned for i = j = 0 and is given by RH(0, t) = 1, 0 ≤ t ≤ 1. (7) A set of the there RH functions is shown in Figs. 1,2,3, where, r = 3, 4, 5. A set of there RH functions is shown in Figs. 1,2,3, where, r = 3, 4, 5. The orthogonality property is given by ∫ [ 1 −i, = , , = 2 r v RH(r t)RH(v t) dt , , 0 0 r v

Figure 1. The graph of RH function

Figure 2. The graph of RH function

Figure 3. The graph of RH function

84 F. Ghomanjani

4. Numerical example In this section, a numerical example is presented to illustrate the proposed method. Example 4.1. Consider the fractional neutral pantograph diﬀerential equation (see [2])

γ 3 1 γ 1 1 γ 1 D x(t) = x(t) + x( t) + D 1 x( t) + D x( t) − t2 − t + 1, 0 < γ < γ ≤ 2, 4 2 2 2 2 1 x(0) = x′(0) = 0

Table 1. The comparision of the absolute errors on interval [0, 1] for Example 4.1

t Runge-Kutta method Present method 0.1 1 × 1−3 3.4 × 10−6 0.2 2.02 × 10−3 5.1 × 10−5 0.3 3.07 × 10−3 3.2 × 10−5 0.4 3.14 × 10−3 5.8 × 10−4 0.5 5.34 × 10−3 4.1 × 10−4

5. Conclusions Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason one may need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the approximate solution of a class of multi-order fractional differential equations. The fractional derivatives are described in the Caputo sense. Our main aim is to evaluation of fractional derivative using Haar wavelet collocation method and implementing it to solve the nonlinear multi-order fractional differential equations. Illustrative example is included to demonstrate the validity and applicability of the technique.

References [1] Y. Yang, Solving a Nonlinear Multi-Order Fractional Diﬀerential Equation Using Legendre Pseudo-Spectral Method, Applied Mathematics, (2013), 113-118. [2] P. Rahimkhani, Y. Ordokhani and E. Babolian, Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet, Journal of Computa- tional and Applied Mathematics, (2016), http://dx.doi.org/10.1016/j.cam.2016.06.005.

Fatemeh Ghomanjani, Kashmar Higher Education Institute, Kashmar, Iran e-mail: [email protected]

85 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

Wavelet Collocation Method for Solving The Model of Drug Release

Zahra Kalateh Bojdi∗ and Ataollah Askari Hemmat

Abstract In this paper we investigate wavelet collocation-ﬁnite diﬀerence method for solving two-dimensional model of drug release in the cardiovascular tissue from the stent.

2010 Mathematics subject classification: Primary 65T60, Secondary 65M06, 65M70.. Keywords and phrases: Wavelet, collocation method, finite difference method, stent, drug release.

1. Introduction Arterial diseases are among the leading causes of death in the industrialized world. They may cause a reduction of the blood flow to important organs and to muscles, because of the narrowing or occlusion of the affected arteries. Drug release depends on many factors, such as the geometry and location of the vessel, the geometry of the stent, the coating properties as its chemical composition and porosity, and drug characteristics as for example its diffusivity. Due to the involvement of so many factors, prediction of drug release represents an important issue and mathematical models are a useful tool to design an appropriate drug delivery system. The paper is organized as follows. Section 2 is devoted to the description of the model. In Section 3 we explain wavelet collocation method for solving the two- dimensional model of drug release from the stent.

2. Description of the model

The drug release system in the arterial wall Ωw can be modeled as follows [4]:

∂a Klag + uw∇a − Dw△a = 0, in Ωw, ∂t kw ∂a Dw + α(t)a = β(t)c0, on Γ, ∂nw ∂a a Dw + Pw = 0, on Γadv, ∂nw ϵwkw a = 0, on Γbl ∪ Γs. (1) ∗ speaker

86 Z. Kalateh Bojdi and A. Askari Hemmat where a are the volume averaged solid concentration of the free drug inside the arterial wall. Dw denotes the diffusion coefficient of the considered drug in the tissue, Klag denotes the decrease of convective transport due to collisions of the solid particles with the structure of the porous wall (0 ≤ Klag ≤ 1), kw is an additional partition coefficient that defines the ratio between the drug bound to the tissue matrix and that dissolved in the fluid, Pw is the permeability of the tissue and ϵw is the its porosity. kb Finally, uw = − ∇p, where kb and µb are the hydraulic permeability of the arterial µb wall and the viscosity of the blood plasma respectively and p is the pressure.

Figure 1. Stent S in contact with the vessel wall V.

3. Multiresolution analysis and wavelet collocation method Let φ be a Daubechies wavelet’s scaling function. Therefore φ is compact support and ∑N−1 φ(x) = akφ(2x − k), (2) k=0 where {ak} are the filter∫ coefficients and N is an even positive integer. Suppose φ(x) is ∞ φ = normalized such that: −∞ (x) dx 1. We introduce[1] θ(x):= (φ ∗ φ(−·))(x), (3) the function θ is called autocorrelation function of φ. Theorem 3.1. The function θ, have the following properties[1, 3]: ∑ ∑ θ = N−1 θ − = = 1 N−1−k , ≥ 1. (x) k=−N+1 ck (2x k), that ck c−k 2 i=0 aiak+i k 0, 2. supp (θ) ⊆ [−N + 1, N − 1], θ = δ , ∈ 3. (k) 0,k k Z, ∑ = δ = θ k , ∈ θ = N−1 θ k θ − 4. c2k 0,k, ck ( 2 ) k Z, therefore (x) k=−N+1 ( 2 ) (2x k), where N is an even positive integer in Daubechies wavelet, the sequence {ck}k∈Z is called the scaling filter and δ0,k is the Kronecker delta function.

2 Deﬁnition 3.2. A sequence of subspaces {V j} j∈Z in L (R) is called a Multiresolution analysis (MRA) for L2(R) with scaling function φ, if[2]: 2 1. V ⊆ V + ⊆ L (R), ∩j j 1 ∪ = { }, = 2 2. j∈Z V j 0 and j∈ZV j L (R) − j 3. f (·) ∈ V j ⇔ f (2 ·) ∈ V0, 4. f (·) ∈ V0 ⇔ f (· − n) ∈ V0, for all n ∈ Z, 5. There exist a function φ ∈ V0, called scaling function, such that {φ(· − k)}k∈Z is an orthonormal basis for V0.

87 Wavelet Collocation Method for Solving The Model of Drug Release

2 Corollary 3.3. Let {V j} j∈Z be MRA for L (R) with scaling function φ. There exist coefficients {a } ∈ such that k k Z ∑ φ(x) = akφ(2x − k). k∈Z j/2 j and for any j, k ∈ Z define φ jk(x) = 2 φ(2 x − k). Then {φ jk(x)}k∈Z is an orthonormal basis for V j [2]. 2 If {V j} j∈Z is a multiresolution anaysis for L (R) with scaling function ϕ and wavelet ψ { ′ = ⊗ } 2 2 . , then V j V j V j j∈Z is a multiresolution analysis of L (R ) We can easily show that ′ = (x) ⊗ (y) = (x) ⊕ (x) ⊗ (y) ⊕ (y) V1 V1 V1 (V0 W0 ) (V0 W0 ) (4) = (x) ⊗ (y) ⊕ (x) ⊗ (y) ⊕ (x) ⊗ (y) ⊕ (x) ⊗ (y) (V0 V0 ) (V0 W0 ) (W0 V0 ) (W0 W0 ) = ′ ⊕ ′1 ⊕ ′2 ⊕ ′3. V0 W0 W0 W0 This 2-D multiresolution analysis requires one scaling function Φ , = ϕ ϕ ∈ ′ , (x y) (y) (x) V0 and three wavelets Ψ1(x, y) = ϕ(x)ψ(y), Ψ2(x, y) = ϕ(y)ψ(x), Ψ3(x, y) = ψ(x)ψ(y), ′ where Ψi is the wavelet associated to W i for i = 1, 2, 3, respectively. j Define V j = span{θ(2 · −k), k}, that j ∈ Z. So {V j} j∈Z generates an MRA with scaling function θ [2, 3]. ∫ The derivatives of the function θ defined by θ(x) = φ(t)φ(t − x)dt are ∫ ∫ θ′(l) = − φ(t)φ′(t − l)dt, θ′′(l) = − φ′(t)φ′(t − l)dt.

− j Thus we compute derivatives of the function θ at the point xl = l2 . Let J be arbitrary. We estimate the solution for equation (1) with corresponding initial and boundray conditions using the following expansion: ∑ ∑ J J a(x, y) ≈ aklθ(2 x − k)θ(2 y − l), (5) k∈Z l∈Z = J, J J = −J J = −J where akl a(xk yl ), xk k2 and yl l2 . The ﬁrst derivative of a with respect ∂a ≃ an+1−an . to time, are estimated by ∂t ∆t J = p J = q Thus the discretization of Eq. (1) at given collocation points xp 2J and yq 2J , p, q = 1, ··· , 2J − 1, is   ∑ ∑ K k   n+1 , = lag b J∆  n θ′ − + n θ′ −  a (xp yq) 2 t p1 akq (p k) p2 apl (q l) kw µb  k∈Z l∈Z  ∑ ∑  + 2J∆  n θ′′ − + n θ′′ −  Dw2 t  akq (p k) apl (q l) k∈Z l∈Z + ∆ n , + n . tS (xp yq) apq (6)

88 Z. Kalateh Bojdi and A. Askari Hemmat

Now, we can write an+1 = Aan + Bn, where the vector Bn is generated by the boundary conditions.

4. Conclusion Wavelet collocation-ﬁnite diﬀerence approximation to the solution of the two- dimensional model of drug release from the stent is constructed. The equation (1) with corresponding initial and boundary conditions, can be solved successfully using the proposed method in this paper. The numerical results will be presented at the speech.

References [1] G. Beylkin and N. Saito, Wavelets, their autocorrelation functions, and multiresolution representation of signals, Expanded abstract in Proceedings ICASSP-92 4 (1992) 381-384. [2] I. Daubechies, Ten Lectures on Wavelets, Soc. for Indtr. Appl. Math., Philadelphia 61, 1992. [3] Z. Kalateh Bojdi and A. Askari Hemmat, Wavelet collocation methods for solving the Pennes bioheat transfer equation, In Press, International Journal for Light and Electron Optics (2016). [4] S. Minicini, Mathematical and Numerical Modeling of Controlled Drug Release, Ph.D Thesis, Politecnico di Milano, Italy, 2009.

Zahra Kalateh Bojdi, Department of Mathematics, University of Advanced Technology, City Kerman, Iran e-mail: [email protected]

Ataollah Askari Hemmat, Department of Applied Mathematics, University of Shahid Bahonar, City Kerman, Iran e-mail: [email protected]

89 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

CERTAIN SUBSPACES OF THE DUAL OF A BANACH ALGEBRA

Ali Akbar Khadem Maboudi∗

Abstract The dual of an introverted subspace of the dual of a Banach algebra enjoys two (Arens type) products. In this talk we investigate the topological centers related to these products for a general introverted subspace. Some older results on certain introverted subspaces of L∞(G) are extended to a general introverted subspace.

2010 Mathematics subject classiﬁcation: Primary: 46H25. Keywords and phrases: Arens product, topological center, introverted subspace.

1. Introduction Following [1], the second dual A∗∗ of a Banach algebra A enjoys two, in general, d multiplications ( each turning A∗∗ into a Banach algebra. In the case where these multiplications are coincide, A is said to be Arens regular. Let A be a Banach algebra, A∗ and A∗∗ be the dual and the second dual of A, respectively. We shall make A∗ into a Banach A-module under the module operations given by: ⟨ f · a, b⟩ = ⟨ f, ab⟩, ⟨a · f, b⟩ = ⟨ f, ba⟩. A subspace X of A∗ is called left (resp. right) invariant if A·X ⊆ X (resp. X·A ⊆ X). A subspace X is called invariant, if it is both left and right invariant. Let X be an invariant subspace of A∗, m ∈ X∗ and f ∈ X. We define, m□ f and f ♢m so that, for every a ∈ A ⟨m□ f, a⟩ = ⟨m, f □a⟩ and, ⟨a, f ♢m⟩ = ⟨a♢ f, m⟩. Then X is called left (resp. right) introverted, if X∗□X ⊆ X (resp. X♢X∗ ⊆ X); X is called introverted if it is both left and right introverted. If X is left (resp. right) introverted, then (X∗, □) (resp. (X∗, ♢)) is a Banach algebra under the product m□n (resp. m♢n), in which, m□n and m♢n are defined so that, ⟨m□n, f ⟩ = ⟨m, n□ f ⟩, and ⟨ f, m♢n⟩ = ⟨ f ♢m, n⟩, for all f ∈ X. The products ♢ and □ are called the first and the second Arens (type) products on X∗, respectively. An introverted subspace X of A∗ is called Arens (type) regular if m□n = m♢n for every m, n ∈ X. ∗ speaker

90 A.A. Khadem Maboudi

For a Banach algebra A it is obvious that A∗ is introverted. In this case □ and ♢ are the so-called (ﬁrst and second) Arens products on A∗∗, which makes A∗∗ into a Banach algebras under each of these products. A less trivial example of a left (resp. right) introverted subspace of A∗ is A∗□A (resp. A♢A∗), in the case where A enjoys a bounded approximate identity; [2].

If X is left introverted, then one may show that for every n ∈ X∗ the mapping m 7→ m□n is w∗ − w∗ continuous, but m 7→ n□m is not continuous in general, unless n ∗ ∗ is in A. Whence the first topological center Z1(X ) of X is defined so that, ∗ ∗ ∗ ∗ Z1(X ) = {n ∈ X ; m 7→ n□misw − w − continuous}. ∗ ∗ If X right introverted, the second topological center Z2(X ) of X is defined so that, ∗ ∗∗ ∗ ∗ Z2(X ) = {n ∈ A ; m 7→ m♢nis w − w − continuous}. ∗ ∗ ∗ ∗ Trivially, A ⊆ Z1(X ) ∩ Z2(X ); and also Z1(X ) and Z2( ) are closed subalgebras ∗ ∗ ∗∗ of (X , □) and (X , ♢), respectively. We use the notations Z1 and Z2 for Z1(A ) and ∗∗ Z2(A ), respectively. For the group algebra A = L1(G), in which G is a locally compact topological 1 ∗ group, it is known that Z1 = Z2 = L (G);[4]. Note that in this case A □A = LUC(G) ∗ and A♢A = RUC(G). For A = K(c0), the operator algebra of all compact linear operators on the sequence space c0, it has been shown that Z1 , Z2;[5].

2. The results We start with the following elementary fact on the introversion ptoprty of invariant subspaces. Proposition 2.1. Every w∗−closed invariant subspace of A∗ is introverted. ∗ A functional f ∈ A is called weakly almost periodic if f □A1 is weakly relatively compact in A∗. The set of all weakly compact elements of A∗ will denote by wap(A). It is easy to verify that wap(A) is a closed subspace of A∗. The following result describes introversion of X in terms of the inclusion X ⊆ wap(A). Theorem 2.2. For every norm closed translation invariant subspace X of A∗, the following are equivalent. (i) X ⊆ wap(A). ∗ ∗ (ii) X is left introverted and Z1(X ) = X . ∗ ∗ (iii) X is right introverted and Z2(X ) = X . As an immediate consequence we have the next corollary. Corollary 2.3. Every norm closed invariant subspace X of wap(A) is introverted, and ∗ ∗ ∗ Arens (type) regular, i.e. Z1(X ) = X = Z2(X ). In particular wap(A) is introverted. The following result is an extension of [5, Corollay 3.2].

91 Certain Subspaces of the Dual of a Banach Algebra

Theorem 2.4. Let A be a Banach algebra and X be a left introverted subspace of A∗ then ∗ (i) X♢Z1(X ) ⊆ X. ∗ (ii) If X□A = X, then A♢Z1(X ) ⊆ Z1. ∗ (iii) if X□A = A and A.A = A, then A♢Z1(X ) = A♢Z1. The next result extends [5, Theorem 3.6]) to a general left introverted subspace of A∗. Theorem 2.5. Let X be a norm closed left introverted subspace of A∗ such that X□A = X, and A.A = A. Then the following statements are equivalents. (i)X ⊆ wap(A). ∗ (ii)A♢X ⊆ Z1. ∗ ∗ (iii) A♢X ⊆ A♢Z1(X ). ∗ ∗ (iv) Z1(X ) = X .

References [1] A. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–84. [2] H.G. Dales, Banach Algebras and Automatic Continuity, London Math. Soc. Monographs 24 (Clarendon Press, Oxford, 2000). [3] A.A. Khadem-Maboudi and H.R. Ebrahimi Vishki, Strong Arens irregularity of bilinear mappings and reﬂexivity, Banach J. Math. Anal. 6 (2012), no. 1, 155–160. [4] A.T.-M. Lau and V. Losert, On the second conjugate algebra of locally compact group, J. London Math. Soc. 37(1988), 464-470. [5] A.T.-M. Lau and A. Ulger, Topological centers of certain dual algebras, Trans. Amer. Math. Soc. 348 (1996) no. 3, 1191–1212.

Ali Akbar Khadem Maboudi, Department of Biostatistics, Faculty of Paramedical sciences, Shahid Beheshti Univer- sity of Medical sciences, Tehran, Iran e-mail: [email protected]

92 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

BANACH FUNCTION ALGEBRAS GENERATED BY NORMED VECTOR SPACES AND LINEAR FUNCTIONALS

Ali Reza Khoddami

Abstract Given a non-zero normed vector space A and a contractive non-zero element φ ∈ A∗ (the dual space of φ = (0) A), we introduce the Banach function algebra C (K), where K B1 is the closed unit ball of A. We investigate and characterize some basic properties, such as idempotent elements, nilpotent elements and bounded approximate identities of the Banach function algebra Cφ(K). Finally we characterize some elements of the character space △(Cφ(K)).

2010 Mathematics subject classiﬁcation: Primary 46J10, Secondary 46H20, 46B28. Keywords and phrases: Banach function algebra, normed vector space, bounded approximate identity, idempotent element, character space.

1. Introduction Let A be a non-zero normed vector space and let φ be a non-zero contractive (∥φ∥ ≤ 1) ∗ = (0) element{ of A . Also let K B1 be the closed} unit ball of A. Suppose that C(K) = f : K −→ C, f is continuous and bounded . Clearly C(K) is a Banach space { } with respect to the norm, ∥ f ∥∞ = sup | f (x)|, x ∈ K . For f, g ∈ C(K) deﬁne,

( f · g)(x) = f (x)g(x)φ(x), x ∈ K.

We shall show that (C(K), ·) is a non-unital commutative Banach algebra that we denote it by Cφ(K). Also we characterize the idempotent, nilpotent elements and the bounded approximate identities of Cφ(K). Finally we characterize some elements of the character space of Cφ(K). Many basic properties of Cφ(K) are investigated in [1].

2. Main results In this section let A be a non-zero normed vector space and let φ be a non-zero ∥φ∥ ≤ = (0) ⊆ linear functional on A with 1. Also let K B1 A.

93 Ali Reza Khoddami

Proposition 2.1. Let A be a non-zero normed vector space and let φ ∈ A∗ be a non- zero element such that ∥φ∥ ≤ 1. Then Cφ(K) is a non-unital commutative Banach algebra. { } ∗ ∗ Set A = f : K −→ C, f ∈ A . So we can present the following result. K K φ ∈ ∗ Proposition 2.2. Let A be a non-zero normed vector space and let A be a non- ∗ φ ∗ zero element such that ∥φ∥ ≤ 1. Then A ⊆ C (K) and for each f ∈ A , ∥ f ∥ = K ∥ f ∥∞. In the following Proposition we characterize the form of idempotent elements of Cφ(K). Proposition 2.3. Let A be a non-zero normed vector space and let φ ∈ A∗ be a non- zero element such that ∥φ∥ ≤ 1. Then f ∈ Cφ(K) is idempotent if and only if   − 0 x ∈ f 1({0}) f (x) =   1 < −1 { } φ(x) x f ( 0 ) Proposition 2.4. Let A be a non-zero normed vector space and let φ ∈ A∗ be a non-zero element such that ∥φ∥ ≤ 1. Then f ∈ Cφ(K) is nilpotent if and only if f = 0. K−ker(φ) We give a result concerning bounded approximate identity. Theorem 2.5. Let A be a non-zero normed vector space and let φ ∈ A∗ be a non-zero element such that ∥φ∥ ≤ 1. Then there is no bounded approximate identity in Cφ(K). For each k ∈ K, deﬁne kˆ : Cφ(K) −→ C by kˆ( f ) = f (k), f ∈ Cφ(K). Obviously K ⊆ Cφ(K)∗. Recall that an element ψ ∈ Cφ(K)∗ is a character if ψ( f · g) = ψ( f )ψ(g), f, g ∈ Cφ(K). △ Cφ K Cφ K Let ( ( )) be the set of all characters on ( ). One can easily check that φ 0ˆ < △(C (K)). Indeed, As 1 · 1 = φ then 0(1ˆ · 1) = 0(ˆ φ ) = φ(0) = 0 , 0(1)ˆ 0(1)ˆ = 1. K K Proposition 2.6. Let A be a non-zero normed vector space and let φ ∈ A∗ be a non- zero element such that ∥φ∥ ≤ 1. Then K ∩ φ−1({1}) ⊆ △(Cφ(K)). Remark 2.7. Clearly K ∩ ker(φ) ⊈ △(Cφ(K)). Indeed if k ∈ K ∩ ker(φ) then 0 = kˆ(1 · 1) , kˆ(1)kˆ(1) = 1. In the sequel we present two questions. Question 1. Characterize the space △(Cφ(K)). Question 2. Characterize the Banach function algebra Cφ(K) in the case where A is ﬁnite dimensional.

94 Banach function algebras generated by normed vector spaces

References [1] A. R. Khoddami, On a certain class of Banach function algebras, preprint.

Ali Reza Khoddami, Department of Pure Mathematics, P. O. Box 3619995161-316, Shahrood, Iran e-mail: [email protected]

95 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

ON TERNARY AMENABLE COMMUTATIVE JB∗ TRIPLE SYSTEMS

Amin Allah Khosravi

Abstract In this paper we introduce the terms of triple approximate identity and ternary amenability for JB∗ triple systems and then we show that every ternary amenable commutative JB∗ triple system possesses an triple approximate identity.

2010 Mathematics subject classiﬁcation: Primary 46K70, Secondary 17C65. Keywords and phrases: Triple amenability, JB∗ triples, commutative JB∗ triples .

1. Introduction JB∗−triples arose in the study of bounded symmetric domains in Banach spaces. A JB∗−triple system is a complex Banach space M together with a continuous triple product {·, ·, ·} : M × M × M → M satisfying the following conditions:

(1){·, ·, ·} is symmetric and bilinear in the outer two variables and conjugate-linear in middle variable;

(2){·, ·, ·} obeys the so-called Jordan identity

{a, b, {x, y, z}} = {{a, b, x}, y, z} − {x, {b, a, y}, z} + {x, y{a, b, z}}, orequivalently,

ϕ(a, b)ϕ(x, y) − ϕ(x, y)ϕ(a, b) = ϕ(ϕ(a, b)x, y) − ϕ(x, ϕ(b, a)y), for all a, b, x, y, z ∈ M, where ϕ(a, b)(x):= {a, b, x};

(3)for each a ∈ M, the operator ϕ(a, a) from M to M is a hermitian operator with non-negative spectrum;

(4)∥{x, x, x}| = ∥x∥3 for each x ∈ M.

Note that ϕ(a, b) is sometimes denoted by a□b and operators of this form are called box operators.

96 A. A. Khosravi

A JB∗ triple system M is called commutative if the box operators a□b and x□y commute for all a, b, x, y ∈ M. It can be shown that M is commutative if and only if {a, b, {x, y, z}} = {a, {b, x, y}, z} = {{a, b, x}, y, z}, a, b, x, y, z ∈ M C∗−algebras and JB∗−algebras are the main examples of JB∗−triples. More precisely, every C∗−algebra (resp. every JB∗−algebra) is a JB∗−triple with respect to the product { , , } = 1 ∗ + ∗ { , , } = ◦ ∗ ◦ + ◦ ∗ ◦ − ◦ ◦ ∗ x y z 2 (xy z zy x) (resp. x y z (x y ) z (z y ) x (x z) y ). An example of a commutative JB∗ triple is any C∗− algebra with the triple product deﬁned by {x, y, z} = xy∗z. The basic facts about JB∗−triples can be found in [5] and some of the references therein. Deﬁnition 1.1. Following the terms used in [2], a triple M-module is a vector space X equipped with three mappings

{· · · }l : X × E × E → X, {· · · }m : E × X × E → X and {· · · }r : E × E × X → X satisfying:

1. {· · · }l is linear in first and second variable and conjugate linear in last variable, {· · · }m is conjugate trilinear and {· · · }r is linear in middle and last variable and conjugate linear in first variable; 2. {x, b, a}l = {a, b, x}r and {a, x, b}m = {b, x, a}m for every a, b ∈ M and x ∈ X; 3. {a, b, {c, d, e}} = {{a, b, c}, d, e} − {d, {b, a, d}e} + {c, d, {a, b, e}}, where {· · · } denotes any of the mapping {· · · }l, {· · · }m, {· · · }r or the triple product of M and one of the elements a, b, c, d, e is in X and the rest are in M. In [4], the triple M module defined above is called the triple M module of type I. It is called the triple M module of type II if the first item of the above definition is replaced by: {· · · }l, {· · · }m and {· · · }r are linear in outer variables and conjugate linear in middle variable. It is also shown that the dual of a triple M-module of type I (resp II) is a triple M- module of type II (resp I). We usually write the expression triple M-module, without declaring the type, whenever a statement is true for both types or whenever the type is clear from the context. Definition 1.2. Let M be a Jordan triple system and X be a triple M-module. A bounded linear operator D : M → X is said to be a ternary derivation if

D({abc}) = {D(a)bc}l + {aD(b)c}m + {abD(c)}r, a, b, c ∈ M.

A ternary derivation D : M → X is said to be inner if there exists ai ∈ M and xi ∈ X for i = 1, 2, ··· , n such that ∑n D(b) = ({xiaib}l − {bxiai}m) i=1

97 On ternary amenable commutative JB∗ triple systems

Definition 1.3. A Jordan triple system M is called ternary amenable, if every bounded derivation D : M → X∗ is inner for every Banach triple M-module X. Definition 1.4. Let M be a Jordan triple system. A left-bounded approximate identity i i i for M is a set of pair nets {(eα, uα)α∈I : i = 1, 2, ··· , n} in which for each i, {e }α and i {u }α are bounded nets in M, that satisfies ∑n i i lim( (eα□uα)) = id . α M i=1 It is called a right-bounded approximate identity if ∑n i i lim (uα□eα) = id . α M i=1 And it is called a middle-bounded approximate identity if ∑n i i lim Q(eα, uα) = id . α M i=1 i i The set of {(eα, uα)α∈I : i = 1, 2, ··· , n} is called a bounded approximate identity for M if it is a left, right and middle-bounded approximate identity for M.

2. Main results The following theorem establishes the relation between ordinary amenability of Banach algebras and ternary amenability: Theorem 2.1. Let M be a Banach algebra and X a Banach M module. M is a Jordan triple system with the canonical triple product and X∗ is a triple Banach M module. If M is ternary amenable then it is ordinary amenable. A result of Dineen [1] shows that the bidual of a JB∗-triple is again a JB∗-triple. In [3], besides Dineen’s approach, we gave an alternative construction, by mimicking Arens’ method. This method leads us to the following theorem: Theorem 2.2. Every ternary amenable commutative JB∗-triple system has a bounded triple approximate identity.

References [1] S. Dineen, The second dual of a JB∗-triple system, Complex Analysis, Functional Analysis and Approximation Theory (Ed. J. Mujica), North-Holland, Amsterdam, (1986). [2] T. Ho, A.M. Peralta and B. Russo, Ternary weakly amenable C∗-algebras and JB∗-triples, Quart. J. Math. 64 (2013), 1109-1139. [3] A. A. Khosravi and H. R. Ebrahimi Vishki, Aron-Berner extensions of trilinear maps with application to the biduals of a JB∗-triple, preprint [4] M. Niazi, M.R. Miri and H.R. Ebrahimi Vishki, Ternary Weak Amenability of the Bidual of a JB∗−Triple, to appear in Banach J. Math. Anal.

98 A. A. Khosravi

[5] B. Russo, Structure of JB∗−triples, Jordan Algebras, Proceedings of the Oberwolfach Conference 1992 (Ed. W. Kaup, K. Mc Crimmon and H. Petersson), de Gruyter, Berlin, (1994), 209-280.

Amin Allah Khosravi, Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran e-mail: [email protected]

99 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

IDEMPOTENT STATES ON QUANTUM GROUPS VIA VON NEUMANN ALGEBRAS

Fatemeh Khosravi∗ and Paweł Kasprzak

Abstract We study idempotent states on locally compact quantum groups and its dual quantum groups and ﬁgure out their relations with quantum subgroups. We establish a one to one correspondence between idempotent states on a locally compact quantum groups G and integrable coideal von Neumann subalgebras of L∞(G) that are preserved by the scaling group. We also establish a one to one correspondence between open quantum subgroups of G and central idempotent states on the dual quantum group Gb.

2010 Mathematics subject classiﬁcation: Primary 46L65, Secondary 43A05, 46L30, 60B15. Keywords and phrases: Coideals, idempotent states, locally compact quantum groups, quantum subgroups.

1. Introduction The theory of locally compact quantum groups, formulated in the language of operator algebras, is a rapidly developing ﬁeld closely related to abstract harmonic analysis. A locally compact quantum group G is a virtual object, studied via its algebras of ∗ functions: a C -algebra 0(G), playing the role of the algebra of continuous functions on G vanishing at inﬁnity, and a von Neumann algebra L∞(G), viewed as the algebra of essentially bounded measurable functions on G; both of these are equipped with coproducts, operations encoding the multiplication operation of G. It is often essential to study both of the avatars of G mentioned above (as well as the universal counterpart G ∗ u G of 0( ), the C -algebra 0( )) at the same time. In recent years we have seen an increased interest in the notion of idempotent states in locally compact quantum groups. So we study the actions of locally compact quantum groups that correspond to idempotent states. The latter can be viewed as a generalization of the quantum quotient by a compact quantum subgroup. In particular, an idempotent state ω on G gives rise to a von Neumann coideal N ⊂ L∞(G/H) which in the subgroup case is the quotient L∞(G/H) by a compact quantum subgroups H ⊂ G. We give a von Neumann characterization of N ⊂ L∞(G) corresponding to an idempotent state in terms of the integrability of the G-action on N. Finally we

∗ speaker

100 F. Khosravi, P. Kasprzak characterize subalgebras N ⊂ L∞(G) which are of the form L∞(G/H) with H ⊂ G being an open quantum subgroup.

2. Preliminaries For the theory of locally compact quantum group we refer to [5]. Let us recall that a von Neumann algebraic locally compact quantum group is a quadruple G = ∞ ∞ (L (G), ∆G, φG, ψG), where L (G) is a von Neumann algebra with a coassociative ∞ ∞ ∞ comultiplication ∆G : L (G) → L (G) ⊗¯ L (G), and φG and ψG are respectively normal semiﬁnite faithful left and right Haar weights on L∞(G). The GNS Hilbert 2 space of the right Haar weight ψG will be denoted by L (G) and the corresponding GNS map will be denoted by ηG. The antipode, the scaling group and the unitary antipode will be denoted by S , (τt)t∈R and R. The multiplicative unitary WG ∈ B(L2(G) ⊗ L2(G)) is a unique unitary operator such that G W (ηG(x) ⊗ ηG(y)) = (ηG ⊗ ηG)(∆G(x)(1 ⊗ y)) G for all x, y ∈ D(ηG). Using W , G can be recovered as follows: { } ∞ G 2 σ-weak cls L (G) = (ω ⊗ id)W ω ∈ B(L (G))∗ , G G∗ ∆G(x) =W (x ⊗ 1)W . A locally compact quantum group admits a dual object Gb. It can be described in terms b of WG { b } ∞ b G 2 σ-weak cls L (G) = (ω ⊗ id)W ω ∈ B(L (G))∗ , A von Neumann subalgebra N ⊂ L∞(G) is called ∞ • (Left) coideal if ∆G(N) ⊂ L (G) ⊗¯ N; ∗ • Normal if WG(1 ⊗ N)WG ⊂ L∞(Gb) ⊗¯ N; • Integrable if the set of integrable elements with respect to the right Haar weight + ψG is dense in N ; in other words, the restriction of ψG to N is semiﬁnite. ∗ G The -algebraic version (0(G), ∆G) of a given quantum group G is recovered from W as follows { } G 2 norm-cls 0(G) = (ω ⊗ id)W ω ∈ B(L (G))∗ ∗ A locally compact quantum group G is compact if the C -algebra 0(G) is unital. A G u G G locally compact quantum group is assigned with a universal version 0( ) of 0( ) that ∆u ∈ u G ,u G ⊗u G . is equipped with a comultiplication G Mor(0( ) 0 ( ) 0 ( )) The comultiplication ∆u u G ∗ G induces a Banach algebra structures on 0( ) . The corresponding multiplication ∗ ω ∈ u G will be denoted by . A state S (0( )) is said to be an idempotent state if ω ∗ ω = ω. G ∈ Gb ⊗ G V V G ∈ u Gb ⊗u Multiplicative unitary W M(0( ) 0 ( )) admits the universal lift M(0( ) 0 G G Gb Λ ∈ u G , G ( )). The reducing morphisms for and will be denoted by G Mor(0( ) 0 ( )) Λ ∈ u Gb , Gb Λ ⊗ Λ V GV = G and Gb Mor(0( ) 0 ( )) respectively. We have ( Gb G)( ) W . We shall G W G = Λ ⊗ V V G ∈ Gb ⊗u G also use the half-lifted version of W , ( Gb id)( ) M(0( ) 0 ( )).

101 IDEMPOTENT STATES ON QUANTUM GROUPS VIA VON NEUMANN ALGEBRAS

Following [4], any µ ∈u (G)∗ deﬁne a normal map L∞(G) → L∞(G) such that 0 ∗ x 7→ (id ⊗ µ)( W G(x ⊗ 1) W G ) for all x ∈ L∞(G). We shall use a notation ∗ µ ∗ x = (id ⊗ µ)( W G(x ⊗ 1) W G ).

3. Idempotent States and Quantum Subgroups ω ∈u G ∗ G ∞ G → ∞ G Let 0 ( ) be an idempotent state on and let E : L ( ) L ( ) be the conditional expectation assigned to ω. N = E(L∞(G)) is a von Neumann algebra in ∞ ∞ L (G) then N is an integarable coideal of L (G) which is τt-invariant. ∞ Theorem 3.1. Let N be an integrable coideal of L (G) which is τt-invariant. Then there exists a unique conditional expectation E : L∞(G) → L∞(G) onto N such that ψ = ψ ∈ ∞ G + G(x) G(E(x)) for all x L ( ) . Moreover there exists a unique{ idempotent state ω ∈u G ∗ ∈ ∞ G = ω ∗ = ∈ ∞ G ω ∗ = } 0 ( ) such that for every x L ( ), E(x) x and N x L ( ): x x . Corollary 3.2. There is a 1-1 correspondence between integrable coideals N ⊂ ∞ L (G) preserved by τt and idempotent states on G, where denoting the conditional expectation given by ω with E : L∞(G) → L∞(G), we have N = E(L∞(G)). Moreover E preserves ψG and φG. Now we define closed quantum subgroups in the sense of Vaes. There exists another definition of closed quantum subgroups defined by Woronowicz and is weaker than Vaes definition, actually integrability of action associated to a closed quantum subgroups in the sense of Vaes yields a closed quantum subgroup in the sense of Woronowicz [3]. Definition 3.3. Let H and G be locally compact quantum groups. Following [1], we say H is a closed quantum subgroup of G if there exists an injective normal unital ∗-homomorphism γ : L∞(Hb) → L∞(Gb) such that

(γ ⊗ γ) ◦ ∆Hb = ∆Gb ◦ γ Let H be a closed quantum subgroup of G, then H acts on L∞(G) on the right (in von Neumann algebraic sense) by the following formula α : L∞(G) → L∞(G) ⊗¯ L∞(H), x 7→ V(x ⊗ 1)V∗ where V = (γ ⊗ id)WH. The ﬁxed point space of α is denoted by { } L∞(G/H) = x ∈ L∞(G) α(x) = x ⊗ 1 If H is a compact quantum subgroup of G then there is a conditional expectation ∞ ∞ ∞ E : L (G) → L (G) onto L (G/H) which is deﬁned by E = (id ⊗ ψH) ◦ α where ψH is the Haar measure of H. In the next theorem we get a 1-1 correspondence between idempotent states of Haar type (i.e. the states corresponding to a Haar measure on a compact quantum subgroup H of G) and normal integrable coideals N ⊂ L∞(G) preserved by the scaling group.

102 F. Khosravi, P. Kasprzak

Theorem 3.4. Let N be a normal integrable coideal von Neumann subalgebra of ∞ L (G) which is τt-invariant. Then there exists a unique compact quantum subgroup H ⊂ G such that N = L∞(G/H). In the last part we establish a 1-1 correspondence between open quantum subgroups of G [2], and central idempotent states on Gb. Deﬁnition 3.5. Let H and G be locally compact quantum groups. Then H is an open quantum subgroup of G if there is a surjective normal ∗-homomorphism ρ : L∞(G) → L∞(H) such that ∆H ◦ ρ = (ρ ⊗ ρ) ◦ ∆G. Theorem 3.6. Let H be an open quantum subgroup of a locally compact quantum group G. Then there exists a conditional expectation E : L∞(Gb) → L∞(Hb) such that (id⊗ E)◦∆Gb = ∆Gb ◦ E = (E ⊗id)◦∆Gb. Conversely for a von Neumann subalgebra N of L∞(Gb) equipped with a conditional expectation E : L∞(Gb) → L∞(Gb) onto N satisfying

(id ⊗ E) ◦ ∆Gb = ∆Gb ◦ E = (E ⊗ id) ◦ ∆Gb (1) there exists a unique open quantum subgroup H of G such that N = L∞(Hb). Corollary 3.7. Let G be a locally compact quantum group. There is a 1-1 correspondence between open quantum subgroups of G and central idempotent states ω on Gb, ω ∈u Gb ∗ ω ∗ µ = µ ∗ ω µ ∈u Gb ∗ i.e. idempotent states 0 ( ) such that for all 0 ( ) . References [1] M. Daws, P. Kasprzak, A. Skalski, and P. Sołtan, Closed quantum subgroups of locally compact quantum groups. Adv. Math. 231 (2012), 3473–3501. [2] M. Kalantar, P. Kasprzak, and A. Skalski, Open quantum subgroups of locally compact quantum groups. Adv. Math. 303 (2016), 322–359. [3] P. Kasprzak, F. Khosravi, and P.M. Sołtan, Integrable actions and quantum subgroups, Preprint arXiv:1603.06084 [math.OA]. [4] J. Kustermans, Locally compact quantum groups in the universal setting. Int. J. Math. 12 (2001) 289–338. [5] J. Kustermans, and S. Vaes, Locally compact quantum groups. Ann. Scient. Éc. Norm. Sup. 4e série, t. 33 (2000) 837–934.

Fatemeh Khosravi, Department of Pure Mathematics, Ferdowsi university of Mashhad, Mashhad, Iran e-mail: [email protected]

Paweł Kasprzak, Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Poland e-mail: [email protected]

103 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

WEAK AMENABILITY OF GENERALIZED MATRIX BANACH ALGEBRAS

Hosein Lakzian

Abstract Let A and B be two Banach algebras, M be a Banach (A, B)-module and N be a Banach (B, A)-module. In this paper, we ﬁrst introduce the concept of generalized matrix Banach algebras of order 2 denoted by [ ] AM G = . Then we present some results clarifying the relation between n-weak amenability of G NB and that of A and B. Our results improve some results in the literature.

2010 Mathematics subject classiﬁcation: Primary 46H20, Secondary 46H25, 15A78. Keywords and phrases: Generalized matrix Banach algebra, n-weakly amenable, bilinear mapping, module morphism.

1. Introduction Consider two Banach algebras A and B, two (A, B)-module M (i.e. a left A-module and a Right B-module with compatible actions) and (B, A)-module N and two bounded bilinear mappings Φ : M×N → A and Ψ : N×M → B which are bimodule morphisms on each of their coordinates and satisfying the following equalities. m(nm′) = (mn)m′ and n(mn′) = (nm)n′ (n, n′ ∈ N, m, m′ ∈ M); Where mn := Φ(m, n) and nm = Ψ(n, m). Then the set [ ] {[ ] } AM a m G = = a ∈ A, b ∈ B, m ∈ M, n ∈ N , NB n b form a Banach algebra under matrix-like addition and matrix-like multiplication and the norm given by a m = ||a|| + ||m|| + ||b|| + ||n||, n b G which is called generalized matrix Banach algebra of order 2. Generalized matrix algebra was ﬁrst introduced by Sands in [4]. The main examples of Generalized matrix Banach algebras are triangular Banach algebras, trivial generalized matrix algebras, full matrix algebras and inﬂated algebras over a unital algebra. See [3] for more examples.

104 H. Lakzian

Note that the dual X∗ of X together with the actions ( f a)(x) = f (ax) and (b f )(x) = f (xb) is a Banach (B, A)-module where a ∈ A, b ∈ B, x ∈ X and f ∈ X∗ and the second dual X∗∗ of X under the actions (aF)( f ) = F( f a) and (bF)( f ) = F(b f ) is a Banach (A, B)-module, where a ∈ A, b ∈ B, f ∈ X∗ and F ∈ X∗∗. Similarly (2n − 1)-th dual X(2n−1) of X is a Banach (B, A)-module and (2n)-th dual X(2n) of X is a Banach (A, B)-module. Let X be a Banach A-module. A derivation D is a bounded linear operator D : A → X such that D(ab) = D(a)b + aD(b), for all a, b ∈ A. A derivation D is called inner if there exists x ∈ X such that D(a) = δx(a) = ax − xa, for all a ∈ A. The ﬃ 1 , = Z1(A,X) cohomology group of A with coe cients in X is denoted by H (A X) N1(A,X) , where Z1(A, X) is the linear space of bounded derivations from A into X, and N1(A, X) is the linear subspace of bounded inner derivations from A into X. First the notion of weak amenability for a commutative Banach algebra A was deﬁned by Bade et al [1]. In 1998, the concept of n-weak amenability was introduced by Dales et al [2]. They called A, n-weakly amenable if H1(A, A(n)) = 0, and A is said to be permanently weakly amenable if it is n−weakly amenable for all n ∈ N. To clarify the relation between n-weak amenability of G and that of A and B, we need to characterize the derivations from G into G(n). The discussion naturally splits into two cases of odd and even integers.

2. Derivations and (2n+1)-weak amenability Deﬁnition 2.1. Suppose that M is a Banach (A, B)-bimodule and N is a Banach (B, A)- bimodule. For every a(2n+1) ∈ A(2n+1) and b(2n+1) ∈ B(2n+1) we deﬁne

(2n+1) τa(2n+1),b(2n+1) : N → M n 7→ na(2n+1) − b(2n+1)n

(2n+1) νb(2n+1),a(2n+1) : M 7→ N m 7→ mb(2n+1) − a(2n+1)m and (2n+1) (2n+1) Z(2n+1)(A, M, B, N) = {( f, g); f : N → M and g : M → N are A and B − module homomorphism and g(m)n + m f (n) = 0 and f (n)m + ng(m) = 0 for each n ∈ N and m ∈ M}.

Note that for each a(2n+1) ∈ Z(A(2n+1)) and b(2n+1) ∈ Z(B(2n+1)),

(τa(2n+1),b(2n+1) , νb(2n+1),a(2n+1) ) ∈ Z(2n+1)(A, M, B, N), where Z(A(2n+1)) = {a(2n+1) ∈ A(2n+1); aa(2n+1) = a(2n+1)a, a ∈ A}. So we deﬁne

W(2n+1)(A, M, B, N) = {( f, g) ∈ Z(2n+1)(A, M, B, N); f = τa(2n+1),b(2n+1) , (2n+1) (2n+1) (2n+1) (2n+1) g = νb(2n+1),a(2n+1) , a ∈ Z(A ), b ∈ Z(B )},

105 Weak amenability of generalized matrix Banach algebras

In the sequel let A possess a bounded approximate identity (eα)α∈Λ and M and N be left and right approximately unital Banach A−modules, respectively. We will investigate the relation between (2n + 1)−weak amenability of G with (2n + 1)−weak amenability of A and B. Theorem 2.2. If A and B are (2n + 1)-weakly amenable, then H1(G, G(2n+1)) = , , , Z(2n+1)(A M B N) . W(2n+1)(A,M,B,N)

Corollary 2.3. Let A and B are (2n + 1)−weakly amenable and Z(2n+1)(A, M, B, N) = W(2n+1)(A, M, B, N). Then G is (2n + 1)−weakly amenable. Also we have the following lemma. Lemma 2.4. Let M be a closed ideal of A, with an approximate identity. Then Z(2n+1)(A, M, A, M) = W(2n+1)(A, M, A, M). From Theorem 2.2 and Lemma 2.4 we have the following: [ ] AA Corollary 2.5. Let G = . Then H1(A, A(2n+1)) = 0 if and only if AA H1(G, G(2n+1)) = 0.

Proposition 2.6. Let G be (2n + 1)−weakly amenable. Then Z(2n+1)(A, M, B, N) = W(2n+1)(A, M, B, N).

3. Derivations and (2n)-weak amenability Deﬁnition 3.1. For each a(2n) ∈ A(2n), b(2n) ∈ B(2n) deﬁne (2n) τb(2n),a(2n) : M → M m 7→ mb(2n) − a(2n)m

(2n) νa(2n),b(2n) : N → N n 7→ na(2n) − b(2n)n (2n) (2n+1) Z(2n)(A, M, B, N) = {( f, g); f : M → M and g : N → N are A and B − module homomorphismand f (m)n + mg(n) = 0 and g(n)m + n f (m) = 0 for each n ∈ N and m ∈ M}, and

W(2n)(A, M, B, N) = {( f, g) ∈ Z(2n)(A, M, B, N); f = τa(2n),b(2n) , g = νb(2n),a(2n) , a(2n) ∈ Z(A(2n)), b(2n) ∈ Z(B(2n))}, where Z(A(2n)) = {a(2n) ∈ A(2n); a(2n)a = aa(2n), a ∈ A}. There are a similar results as before for the even cases which we listed some of them in the following. , , , Theorem 3.2. If A and B are (2n)−weakly amenable then H1(G, G(2n)) = Z(2n)(A M B N) . W(2n)(A,M,B,N)

106 H. Lakzian

Proposition 3.3. Let G be (2n)−weakly amenable. Then

Z(2n)(A, M, B, N) = W(2n)(A, M, B, N). Lemma 3.4. Let M be an ideal of A, with an approximate identity. Then

Z(2n)(A, M, A, M) = W(2n)(A, M, A, M). [ ] AA Corollary 3.5. If A has a bounded approximate identity, then is (2n)−weakly AA amenable if and only if A is (2n)−weakly amenable. From Theorem 2.1 in [2], Corollaries 2.5 and 3.5 we have the following: [ ] AA Corollary 3.6. If A is an approximately unital C∗-algebra, then is perma- AA nently weakly amenable.

References [1] Bade, W.G., Curtis, P.C. and Dales, H.G., Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. 55 (1987), 359–377. [2] H.G. Dales, F. Ghahramani and N. Grønbæk, Derivations into iterated duals of Banach algebras, Studia Math. 128 (1) (1998), 19–54. [3] Yanbo Li,Feng Wei, Semi-centralizing maps of generalized matrix algebras, Linear Algebra and its Applications 436 (2012) 1122–1153. [4] A.D. Sands,Radicals and Morita contexts, J. Algebra 24 (1973) 335–345.

Hosein Lakzian, Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran e-mail: h−[email protected]

107 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

GENERIC CONTINUITY OF KC-FUNCTIONS

Alireza Kamel Mirmostafaee

Abstract We will show that every KC-function f : X×Y → Z is strongly quasi-continuous provided that X is Baire, Y a W-space and Z is regular. It follows that if Y is a Moore W-space and G is a Baire left topological group, then every KC-action π : G × Y → Y is jointly continuous. Our results enable us to obtain conditions which imply that a semitopological group is automatically a paratopological.

2010 Mathematics subject classiﬁcation: Primary 43A20, 43A22 Secondary 54H11. Keywords and phrases: Joint-continuity, semi-topological group, paratopological group.

1. Introduction There are many papers which deal with the problem of determining the points of continuity for a two variable function (see for example [2–4]). In particular, when the range of functions are not necessarily metric, any solution of the problem can be combined with group actions to emerge some strong results on joint continuity of group actions because the group actions spread the points of continuity around. One the most interesting result in this directs was obtained by Ellis [1] who proved that every separately continuous action π : G × X → X is jointly continuous provided that G is a locally compact semitopological group and X is a locally compact space. In this paper, we use a topological game argument to show that every KC-function f : X × Y → Z is strongly quasi-continuous, provided that X is a Baire space, Y is a W-space and Z is regular. In particular, when Z is a Moore space, it follows that for each y ∈ Y the set of joint continuity of f is a dense Gδ subset of X × {y}. We apply our results to prove that every KC-action π : G × Y → Y is jointly continuous when Y is a Moore W-space and G is a Baire left topological group. In particular case, when G = Y, it follows that a Baire semitopological group G, is paratopological provided that G is a Moore W-space.

2. Results In this section, we will introduce several topological games which will be used in the sequel. Each topological game is described by two types of rules; the playing

108 A. K. Mirmostafaee rules, that determine how to play the game, and the winning rule which determines the winner. The winning rule differs from game to game and, actually, identifies the game. Let (X, τ) be a topological space. The game BM(X) between two players α and β is done as follows. In step n, the player β picks a nonempty open set Un ⊂ Vn−1, where α α Un−1 is the previous move of -plaper, and answers by∩ selecting a nonempty open ⊂ α , ∞ , ∅ set Vn Un. The player wins the play (Ui Vi)i≥1 if ( n=1 Vn) . Otherwise the player β is said to have won the play. We say that the player α has a winning strategy for the game BM(X) if there exists a strategy s, such that α wins all plays provided that he/she acts according to the strategy s. In this case, we say that X is an α-favorable space, otherwise X is said to be an α-unfavorable space for this game. Similarly, winning strategy for the player β and β-favorablity are defined. In 1976, Gruenhage defined a generalization of first countable spaces by means of the following topological game. Let Y be a topological space and y0 ∈ Y. The topological game G(Y, y0) is played by two players O and P as follows. In step n, O-player selects an open set Hn+1 with y0 ∈ Hn+1 and then P answers by choosing a point yn+1 ∈ Hn+1. We say that O wins the game g = (Hn, yn)n≥1 if yn → y0. We call y ∈ Y a W-point (respectively w-point) in Y if O has (respectively P fails to have) a winning strategy in the game G(Y, y). A space Y in which each point of Y is a W-point (respectively w-point) is called a W-space (respectively w-space). Let X, Y and Z be topological spaces, a function φ : X → Z is called quasi- continuous at a point x ∈ X if for any neighborhood sets V of x and any neighborhood W of φ(x) there exists a nonempty open set G ⊂ V such that φ(G) ⊂ W. The function φ : X → Y is called quasi-continuous if it is quasi-continuous at each point of X. By a Kempisty continuous function (KC-function for short), we mean a function f : X × Y → Z which is quasi-continuous in the first variable and continuous in the second variable. A mapping f : X×Y → Z is called strongly quasi-continuous at (x, y) ∈ X×Y if for each neighborhood W of f (x, y) in Z and for each product of open sets U × V ⊂ X × Y containing (x, y), there is a nonvoid open set U1 ⊂ U and a neighborhood V1 ⊂ V of y such that f (U1 × V1) ⊂ W. Theorem 2.1. Let Y be a topological space and Z be a regular space. If either (1) X is a Baire space and the player O has a winning strategy in G(Y, y0) or (2) X is an α-favorable space and the player P does not have a winning strategy in G(Y, y0). Then every KC-function f : X × Y → Z is strongly quasi-continuous on X × {y0}. Sketch of the proof. If the result is not true, there is an open set W containing z0 = f (x0, y0) and there is some product of open sets U × V ⊂ X × Y containing (x0, y0) ′ ′ such that for each open set U ⊂ U and each neighborhood H ⊂ H of y0, there is some (x′, y′) ∈ U′ × H′ such that f (x′, y′) < W. Since Z is regular, there is an open subset G with f (x0, y0) ∈ G and G ⊂ W. ′ By quasi-continuity of f (·, y0), there is a non-empty open subset U ⊂ U such that

109 Continuity of KC-functions

′ f (U ×{y0}) ⊂ G. We define simultaneously a strategy s for β in BM(X) and a strategy t for P in G(Y, y0) by induction as follows. ≥ = , n BM = , n Let for n 1, the partial plays pn (Ui Vi)i=1 in (X) and gn (Hi yi)i=1 in G(Y, y0) are specified. Since by our assumption f (Vn × Hn) is not contained in G, there is some (xn, yn) ∈ Vn × Hn such that f (xn, yn) < G. By quasi-continuity of x 7→ f (x, yn), there is a non-empty open subset Un+1 of Vn such that f (Un+1 × {yn}) ∩ G = ∅. Define s(U1, V1,..., Un, Vn) = Un+1 and t(H1, y1,..., Hn) = yn. = , = , If (a) or (b) holds, there are a s-play p ∩(Un Vn) and t-play g (Hn yn) which α O ∗ ∈ are won by and respectively. Let x n≥1 Un. There is an open neighborhood ∗ H of y0 such that f (x , y) ∈ G for all y ∈ H. Since O wins the play g = (Hn, yn), ∗ there is some n0 ∈ N such that yn ∈ H for all n ≥ n0. Hence f (x , y0) ∈ G. However, our construction shows that f (x, yn) < G for all x ∈ Un. This contradiction proves the Theorem. □

The following result follows immediately from Theorem 2.1. Corollary 2.2. Let Y be a topological space and Z be a regular space. If either (1) X is a Baire space and Y is a W-space or (2) X is an α-favorable space and Y is a w-space. Then every KC-function f : X × Y → Z is strongly quasi-continuous. ∈ U Let Z be a topological space z Z and be a collection∪ of subsets of Z, then the star of z with respect to U is deﬁned by st(z, U) = {U ∈ U : z ∈ U}.A sequence {Gn} of open covers of Z is said to be a development of Z if for each z ∈ Z, the set {st(z, Gn): n ∈ N} is a base at z.A developable space is a space which has a development. A Moore space is a regular developable space. Corollary 2.3. Let Z be a Moore space. If either (1) X is a Baire space and Y is a W-space or (2) X is an α-favorable space and Y is a w-space. × → ∈ Then for every KC-function f : X Y Z and y0 Y, there is a dense Gδ subset Dy0 × { } of X such that f is jointly continuous at each point of Dy0 y0 . A compact space Y is called Corson compact if for some κ, Y embeds in κ {x ∈ R : xα = 0 for all but countably many α ∈ κ}. Corollary 2.4. Let X be a Baire space, Y be a Corson compact and Z be a regular space. Then every KC-function f : X × Y → Z is strongly quasi-continuous. In particular, if Z is a Moore space, then f is jointly continuous on a dense subset of X × Y.

Proof. Since every Corson compact is a W-space, the result follows from Theorem 2.1 and Corollary 2.3. □

Let G be a group equipped with a topology. The group G is called left topological if for each g ∈ G, the left translation h ∈ G → gh ∈ G is continuous. By trivial change

110 A. K. Mirmostafaee in the above deﬁnition, a right topological group can be deﬁned. If G is both left and right topological, then G is called semitopological. A semitopological group is called paratopological if the product mapping is jointly continuous. If in addition the inverse function x 7→ x−1 is continuous, then G is said to be a topological group. Let G be a left topological group and Y be a topological space. We say that G acts on X if there exists a function π : G × Y → Y such that π(gh, y) = π(g, π(h, y)) (g, h ∈ G, y ∈ Y). (1) Theorem 2.5. Let Y be a Moore space and G be a left topological group. If either (1) G is a Baire space and Y is a W-space or (2) G is an α-favorable space and Y is a w-space. Then every KC-action π : G × Y → Y jointly continuous. roof , ∈ × P . Let (g0 y0) G Y. By Corollary 2.3, there is a dense Gδ subset Dy0 of G π × { } { } { } such that is jointly continuous at each point of Dy0 y0 . Let gα and yα converge ∈ π to g and y0 respectively and take some arbitrary point h Dy0 . Since is continuous at (h, y0) and −1 lim hg gα = h, lim yα = y , α α 0 −1 we see that limα π(hg gα, yα) = π(h, y0). Therefore by using (1), we have −1 −1 lim π(gα, yα) = lim π(gh , π(hg gα, yα)) α α −1 = π(gh , π(h, y0)) = π(g, y0). This proves our result. □ The following result which follows from Theorem 2.5, generalizes [4, Theorem 4]. Corollary 2.6. Let G be a semitopological group which is also a Moore space. If G is a Baire W-space, then it is a paratopological group.

References [1] R. Ellis, Locally compact transformation groups, Duck Math. J. 24 (1957) 119–125. [2] A. K. Mirmostafaee, Norm continuity of quasi-continuous mappings and product spaces, Topol- ogy Appl. 157 (2010) 530–535. [3] I. Namioka, Separate continuity and joint continuity Paciﬁc J. Math. 51 (1974) 515–531. [4] Z. Piotrowski, Separate and joint continuity in Baire groups, Tatra Mt. Math. Publ. 14 (1998) 109-116.

Alireza Kamel Mirmostafaee, Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran e-mail: [email protected]

111 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

USING A REFINABLE FUNCTION FOR THE CONSTRUCTION OF MULTIRESOLUTION ANALYSIS IN L2(G)

Narges Mohammadian

Abstract For any locally compact abelian (LCA) and second countable group G, we aim to construct a multires- 2 2 olution analysis (MRA) in L (G) by Riesz family of shifts of a reﬁnable function φ ∈ L◦(G) based on a p uniform lattice L in G that at ﬁrst, we investigate certain Banach spaces L◦ (G), 1 ⩽ p ⩽ ∞.

2010 Mathematics subject classiﬁcation: Primary47A55, 39B52, Secondary 34K20, 39B82.. Keywords and phrases: Reisz family, multiresolution analysis, reﬁnable function, LCA group.

1. Introduction p An MRA refers to the family {V j} j∈Z of subspaces of L (G), 1 ⩽ p ⩽ ∞, which is generated by the lattice translates of the dilates of a function φ. In such conditions, there is a function φ in V0 that lattice translates of φ form an unconditional basis for V0. Such a function is called scaling function. The idea of MRA was introduced by Meyer and Mallat, which provides a natural framework for construction of wavelet bases. ZJia and Micchelli in [1] proved that the Riesz family of integer translates of a certain basis refinable function are sufficient to lead to a multiresolution analysis of Lp(Rs) for 1 ⩽ p < ∞. Later Zhou[4] developed this theory to the case p = ∞. In 1994 Dahlke generalized the definition of MRA to LCA groups, and he displayed that under specified conditions, the generalized B-splines generated an MRA. Kamyabi Gol and Raisi Tousi illustrated in [2] the conditions under which a function generates an MRA based on the spectral functions in the case of LCA groups. In this paper, compared to [2] under a weaker assumption (Riesz family vs. orthonor- mality), but an additional assumption (refinability of φ), we discuss the construction of a multiresolution approximation in L2(G), by Riesz family of shifts of a certain refinable function φ.

2. Preliminaries and related background

Let G be an LCA group with the identity 1G and the dual group Gˆ. For a closed subgroup H of G, let H⊥:={ξ ∈ Gˆ; ξ(H) = {1}}, denotes as the the annihilator of

112 N. Mohammadian

H in Gˆ. A discrete subgroup L of G is called a uniform lattice if it is co-compact. Now a fundamental domain for a uniform lattice L in G, is a measurable set S L in G, such that every x ∈ G can be uniquely written as x = ks, for k ∈ L and 1 p p p s ∈ S L.Consider the dilation operator D : L (G) −→ L (G) by D f (x) = δα f (α(x)), 1 ⩽ p < ∞, (in fact, δα is a proper positive constant depending on α such that the operator D becomes an isometrically isomorphism). Now, we introduce the notion of 2 multiresolution approximation in L (G), following [3]. A sequence {V j} j∈Z of closed subspaces of L2(G) forms a multiresolution approximation of L2(G) if it satisﬁes the following conditions:

(i) V j ⊆ V j+1, ∀ j ∈ Z. j j−1 (ii) f ∈ V j =⇒ D TkD f ∈ V j, for all j ∈ Z, k ∈ L. − 1 2 (iii) f ∈ V j ⇐⇒ δα D f ∈ V j+1. 2 (iv) There is an isomorphism from l (L) onto V0 which commutes with shift opera- ∩tors. (v) ∈Z V = {0}. ∪ j j = 2 (vi) j∈Z V j L (G), We recall that for a locally compact group G, a topological automorphism α : G → G n is said to be contractive if limn→∞ α (x) = 1G for all x ∈ G. p Now, we introduce Banach spaces L◦ (G), 1 ⩽ p ⩽ ∞. For a function φ on G and uniform lattice L in G, let ∑ φ◦(x):= |φ(k−1 x)|, k∈L then φ◦ is a L-periodic function. Write ◦ |φ| = ∥φ ∥ p , p : L (S L) and let, p L◦ (G) = {φ : G −→ C; |φ|p < ∞} (1 ⩽ p ⩽ ∞). p L◦ (G) equipped with the norm |.|p, is a Banach space, and obviously ∥φ∥p ⩽ |φ|p, for all 1 ⩽ p ⩽ ∞. 1 1 p p Note that L◦(G) = L (G). Also, if φ ∈ L (G) is compactly supported, then φ ∈ L◦ (G), for all 1 ⩽ p ⩽ ∞. ∑ φ ∗′ φ −1. φ ∈ Now, semidiscrete convolution a is deﬁned by k∈L (k )a(k) for all p ∞ ′ L◦ (G), 1 ⩽ p ⩽ ∞, and a sequence a ∈ l (L). We also denote by φ∗ the mapping a → φ ∗′ a, a ∈ l∞(L). We recall that the shifts of φ, under the lattice L in G is said to be a Riesz family of p L (G), if there exist constants Ap, Bp > 0 such that ′ Ap∥a∥p ⩽ ∥φ ∗ a∥p ⩽ Bp∥a∥p (1 ⩽ p ⩽ ∞), for all a ∈ lp(L). p ′ Let S p(φ) be the image of l (L) of the mapping φ∗ . In this case the set of shifts of φ under the lattice L in G is a Riesz basis of S p(G).

113 Multiresolution Analysis

3. Multiresolution Analysis 2 In this section for a refinable function φ ∈ L◦(G), we consider V0 = S 2(φ) and j 2 V j = D V0, where D is dilation operator. We construct an MRA of L (G) by a Riesz family of shifts of φ under the lattice L in G. p A function φ ∈ L◦ (G) is said to be refinable, if it satisfies the following refinement equation: ∑ φ = b(k)DTkφ(.) k∈L ∑ 1 p −1 = δα b(k)φ(k α(.)), (1) k∈L for some b ∈ l1(L), that is called the mask of the refinement equation. 2 j Theorem 3.1. Let φ ∈ L (G), V0 = S 2(G) and V j = D V0. If φ is refinable and shifts of φ are Riesz family under the lattice L in G, then (V j) j∈Z forms a multiresolution approximation of L2(G). φ ∈ 2 = = j φ Theorem 3.2. Let∩ L◦(G), V0 S 2(G) and V j D V0. If the set of shifts of is a = { } Riesz family, then j∈Z V j 0 Remark 3.3. Theorem 3.2 is valid for every function φ ∈ Lp(G), 1 ⩽ p < ∞. But in the case p = ∞, Theorem 3.2 may fail to hold. For example let φ be the characteristic j function of interval [0, 1) ⊂ R and let V0 = S ∞(φ), V j = D V0. The set of integer translates of φ is a Riesz basis of V0, but 1 ∈ V j for all j ∈ Z. To prove property (vi) we need the following propositions. The following proposi- φ ∈ 1 φ η = η ∈ ⊥ \{ } tion shows that for a refinable function L (G), ˆ( ) 0 for all L 1Gˆ . Proposition 3.4. If φ ∈ L1(G) is refinable and α : G → G is a topological automorphism such that αˆ −1, is contractive and αˆ(L⊥) ⊆ L⊥, then φˆ(η) = 0 for all η ∈ ⊥ \{ } L 1Gˆ . Moreover, ∑ φ −1. = φ . (k ) ˆ(1Gˆ ) k∈L p Proposition 3.5. Let φ ∈ L◦ (G), 1 ⩽ p ⩽ ∞, and the shifts of φ be a Riesz family of p ξ ∈ b |φ ξη | > , L (G) under the lattice L in G; then, for all G, supη∈L⊥ ˆ( ) 0 φ , Propositions 3.4 and 3.5 guarantee ˆ(1Gˆ ) 0. After normalization, we may assume φ = ˆ(1Gˆ ) 1; thus, we can state property (vi) as follows: φ ∈ 2 φ Theorem 3.6. If ∪ L◦(G) is refinable, such that shifts of under the lattice L are 2 Riesz family. Then j∈Z V j, is dense in L (G). Example 3.7. Let G be the following LCA group,

G = {x = (xn)n∈Z, xn ∈ Z2 = {0, 1}, ∃N ∈ Z s.t. ∀n > N ⇒ xn = 0},

114 N. Mohammadian with the operation given by 1 + 2 = 1 + 2 . (x x )n xn xn mod 2 ∑ , ∞ → | | | | = j We identify G with [0 ) as a measure space by x x where x j∈Z x j2 . This induces the Haar measure of [0, ∞) on G. We will be interested in the following subgroups,

L = {x ∈ G, x j = 0 f or j < 0}, G D = = {x ∈ G, x = 0 f or j ⩾ 0}. L j The subgroup D is known as the Cantor group. We have that L is countable, closed, 2 and discrete, and that D is compact. Consider the Hilbert space H = L (G, µG). The dilation ρ : H → H and translation T : H → H are defined respectively by (ρ f )(x)) j = f (x j−1) and Tk f (x) = f (x − k) for f ∈ H, x ∈ G, k ∈ L. Let the scaling function be ϕ(x) = χD(x), the characteristic function of D. We have −1 (ρ φ)(x) = φ(x) + φ(x + 1), so χD is satisfied in refinable equation and shifts of φ j are an orthonormal basis of H. suppose V0 = S 2(φ) and V j = D V0, therefore by Theorem 3.1, V js in which j ∈ Z, construct a multiresolution approximation of H.

References [1] R. Q. Jia, C. A. Micchelli,Using the reﬁnement equation for the construction of prewavelets II: Powers of two, in Curves, Surfaces, P. J. Laurent, A. Le Méhauté, and L. L. Schumaker, eds., Academic Press, New York, 1991, 209-246. [2] R. A. Kamyabi Gol, R. Raisi Tousi, Some equivalent multiresolution conditions on locally compact abelian groups, Proc. Indian Acad. Sci. (Math. Sci.), 120(3)(2010), 317-331. [3] S. G. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(R), Trans. Amer. Math. Soc., 315(1989), 69-87. [4] D. X. Zhou, Stability of reﬁnable functions, multiresolution analysis and Haar bases, SIAM J. Math. Anal., 27(1996), 891-904.

Narges Mohammadian, Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran e-mail: [email protected]

115 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

BANACH MODULES AND COMODULES OF BANACH COALGEBRAS

Somayeh Mohammadzadeh

Abstract Suppose A is a Banach coalgebra. A Banach space X is considered as a Banach comodule over A and ∗ then as a Banach A -module with module operations φ1 and φ2. When X = A, it will be shown that the ∗ φ∗ φt∗t convolution product of Banach algebra A , equals to both operations 1 and 2 . Then some relations among these operations and some dual of them are investigated.

2010 Mathematics subject classiﬁcation: Primary 46H25, Secondary 46H20. Keywords and phrases: Banach coalgebra, Banach module, Banach algebra, comodule.

1. Introduction A Banach coalgebra A over a ﬁeld K is a Banach space A equipped with a bounded linear map ∆ : A −→ A⊗ˆ A called the coproduct that is coassociative in the sense that

(∆ ⊗ idA) ◦ ∆ = (idA ⊗ ∆) ◦ ∆; where ⊗ˆ is the projective tensor product and id is the identity map. A Banach coalgebra (A, ∆) is called counital if there is a bounded linear map ε : A −→ K called the counit if (ε ⊗ idA) ◦ ∆ = idA = (idA ⊗ ε) ◦ ∆. , ∆ ∈ If (A ) is a Banach coalgebra∑ and a A, we use the Sweedler notation for ∆(a) ∈ A⊗ˆ A with ∆(a) = a(1) ⊗ a(2). In the following, (A, ∆, ε) will be assumed to be a counital Banach coalgebra, unless otherwise is stated. The dual space A∗ of A is a Banach algebra via the convolution operation ∑ ∗ ( f ∗ g)(a) = f (a(1)) ⊗ g(a(2))( f, g ∈ A , a ∈ A). For more about these concepts one can refer to [2, 5]. Suppose that f : X × Y −→ Z is a bounded bilinear mapping on normed spaces X, Y and Z. The adjoint of f is the bounded bilinear map f ∗ : Z∗ × X −→ Y∗ deﬁned by ⟨ f ∗(z∗, x), y⟩ = ⟨z∗, f (x, y)⟩ (x ∈ X, y ∈ Y, z∗ ∈ Z∗).

116 S. Mohammadzadeh

Using this method, the higher rank adjoints of f can be defined by setting f ∗∗ = ( f ∗)∗. The mapping f t will be considered as the bounded bilinear map from Y × X into Z defined by f t(y, x) = f (x, y). The mapping f has two extensions f ∗∗∗ and f t∗∗∗t on X∗∗ × Y∗∗. f is called (Arens) regular if these two extensions coincide. If the product of a Banach algebra A is regular as a bounded bilinear map from A × A into A, Then A is called Arens regular. The first topological center of f is defined by

∗∗ ∗∗ ∗∗∗ ∗∗ ∗∗ t∗∗∗t ∗∗ ∗∗ ∗∗ ∗∗ Zℓ( f ) = {x ∈ X ; f (x , y ) = f (x , y ) for every y ∈ Y }.

The second topological center, Zr( f ) is deﬁned similarly. For more about topological centers and the Arens regularity of bounded bilinear mappings we refer to [1, 4].

2. Main results A Banach space X is said to be a right Banach comodule over A if there exists a bounded linear map ρ : X −→ X⊗ˆ A such that

(idX ⊗ ε) ◦ ρ = idX and

(ρ ⊗ idA) ◦ ρ = (idX ⊗ ∆) ◦ ρ. Similarly, a left Banach comodule X is deﬁned by the existence of a bounded linear map λ : X −→ A⊗ˆ X for which

(ε ⊗ idX) ◦ λ = idX and

(idA ⊗ λ) ◦ λ = (∆ ⊗ idX) ◦ λ. Proposition 2.1. If X is a (right / left) Banach comodule over A, then X is a (left / right) ∗ ∗ ∗ Banach A -module with module operations φ1 : A × X −→ X and φ2 : X × A −→ X deﬁned by

φ1( f, x) = (idX ⊗ f ) ◦ ρ(x) and

φ2(x, f ) = ( f ⊗ idX) ◦ λ(x), for all x ∈ X and f ∈ A∗. In conditions of the above proposition, X∗ will be a (right / left) Banach A∗-module φ∗ φt∗t with module operations 1 and 2 . Also the Banach coalgebra A is itself a Banach comodule over A with respect to the comodule operation ∆ : A −→ A⊗ˆ A. In this case, ∗ A will be a A -module; we denote its operations with φ1 and φ2 again. ∗ ∗ φ∗ φt∗t Proposition 2.2. Consider A as an A -module with module operations 1 and 2 . Then these operations are both equal to the convolution product of A∗; in fact, φ∗ = ∗ = φt∗t 1 2 .

Now we apply the following theorem of [3] for operations φ1 and φ2.

117 Banach modules and comodules

Theorem 2.3. Let X, Y and Z be normed spaces and let f : X × Y −→ Z be a bounded bilinear mapping. Then ∗∗∗ ∗∗ ∗∗ ∗∗ ∗ ∗∗ t∗ (i) f (x , y ) ∈ Z¯, for every x ∈ Zr( f ) and y ∈ Zr( f ); ∗∗∗ ∗∗ ∗∗ t∗ (ii) f (x, y ) ∈ Z¯, for every x ∈ X and y ∈ Zr( f ); ∗∗∗ ∗∗ ∗∗ ∗ (iii) f (x , y) ∈ Z¯, for every x ∈ Zr( f ) and y ∈ Y. Corollary 2.4. If (A∗, ∗) is an Arens regular Banach algebra, then (i) A is an A∗∗∗-module; (ii) A∗ is an ideal of A∗∗∗.

References [1] H. G. Dales, Banach algebras and automatic continuity, London Math. Soc., Monographs 24, Clarendon Press, Oxford, (2000). [2] A. Lyubinin, Nonarchimedean coalgebras and coadmissible modules, p-Adic Numbers, Ultramet- ric Analysis and Applications, (2014) 105-134. [3] S. Mohammadzadeh and S. Barootkoob, Arens regularity and strong irregularity of certain bilinear mappings, Submitted. [4] S. Mohammadzadeh and H. R. E. Vishki, Arens regularity of module actions and the second adjoint of a derivation, Bull. Austral. Math. Soc., 77 (2008), 465-476. [5] T. Timmermann, An invitation to quantum groups and duality, European Mathematical Society, Germany, (2008).

Somayeh Mohammadzadeh, Department of Mathematics, Faculty of science University of Bojnord, Bojnord, Iran e-mail: [email protected]

118 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

FRAME FOR OPERATORS IN FINITE DIMENTIONAL HILBERT SPACE

Vahid Reza Morshedi∗ and Mohammad Janfada

Abstract In this paper, we study frames for operators (K-frames) in ﬁnite dimentional Hilbert spaces and express the dual of K-frames. Some properties of K-dual frames are investigated. Furthermore, the notion of their oblique K-duals and some properties are presented.

2010 Mathematics subject classiﬁcation: Primary 42C15, Secondary 42C30. Keywords and phrases: K-frame; K-dual; oblique K-duals..

1. Introduction Let K ∈ B(H), the space of all bounded linear operators on a Hilbert space H.A sequence {φ j} j∈J is said to be a K-frame for H if there exist constants A, B > 0 such that ∑ ∗ 2 2 2 A∥K x∥ ≤ |⟨x, φ j⟩| ≤ B∥x∥ , (x ∈ H). (1) j∈J

We call A, B the lower and the upper K-frame bounds for {φ j} j∈J, respectively. If K = IH , then {φ j} j∈J is the ordinary frame. If only the right inequalitiy holds, then {φ j} j∈J is called a Bessel sequence. Suppose that Φ := {φ j} j∈J is a K-frame for H. H → ℓ2 J = {⟨ , φ ⟩} The operator TΦ : ( ) deﬁned by TΦ(x) x j j∈J is called the∑ analysis ∗ ℓ2 J → H ∗ { } = φ operator. TΦ is bounded and TΦ : ( ) is given by TΦ( c j j∈J) j∈J c j j. ∗ H → H TΦ is called the pre-frame∑ or synthesis operator. The operator S Φ : deﬁned = ∗ = ⟨ , φ ⟩φ Φ by S Φ(x) TΦTΦ(x) j∈J x j j is called the frame operator of . Note that, frame operator of a K-frame is not invertible on H in general, but it is invertible on the subspace R(K) ⊂ H, that R(K) is the range of K. Given a positive integer N. Throughout this paper, we suppose that H N is a real or complex N-dimensional Hilbert space. By ⟨·, ·⟩ and ∥.∥ we denote the inner product on N H and its corresponding norm, respectively. Denote by PW the orthogonal projection of H onto a closed subspace W ⊆ H.

∗ speaker

119 Vahid Reza Morshedi,Mohammad Janfada

2. Finite K-frames

In this section, we present K-frame theory in ﬁnite-dimensional Hilbert spaces. Let K ∈ B(H N) and Φ = {φ }M be a family of vectors in H N. If A∥K∗ x∥2 = ∑ j j=1 ∑ M |⟨ , φ ⟩|2 Φ ∥ ∗ ∥2 = M |⟨ , φ ⟩|2 j=1 x j , then is called an A-tight K-frame and if K x j=1 x j , then Φ is called a tight K-frame. If ∥φ j∥ = 1 for all j = 1, 2, ..., M, this is an unit norm K-frame. For an arbitrary K-frame, we obtain the optimal lower and upper K-frame bounds by eigenvalues of its frame operator.

, ∈ H N Φ = {φ }M Proposition 2.1. Let 0 K B( ). Let j j=1 be a K-frame for R(K) with K-frame operator S Φ with eigenvalues λ1 ≥ λ2 ≥ ... ≥ λN > 0. Then λ1 is the optimal λ λ , N upper K-frame bound and if N 0 then ∥K∥2 is the optimal lower K-frame bound. Now, we introduce a constructive method to extend a given frame to a tight K- frame.

∈ H N Φ = {φ }M H N Theorem 2.2. Let K B( ). Let j j=1 be a frame for . Assume {λ }N λ ≥ λ ≥ that the frame operator S Φ has the eigenvalues j j=1, ordered as 1 2 ... ≥ λ > 0. Let {e }N be a corresponding eigenbasis. Then the collection N √ j j=1 { φ }M ∪ { λ − λ }N λ H N K j j=1 1 j Ke j j=2 is a 1-tight K-frame for . In the following proposition, we express two inequality of A-tight K-frames.

Φ = {φ }M H N Proposition 2.3. (i) If j j=1 is an A-tight K-frame for , then

2 2 max ∥φ j∥ ≤ A∥K∥ . j=1,2,...,M

Φ = {φ }M H N (ii) If j j=1 is an unit norm A-tight K-frame for , then

A∥K∥2N ≥ M.

In the last part of this section, we study conditions under which a linear combination of two K-frames is K-frame too.

∈ H N Φ = {φ }M Ψ = {ψ }M Deﬁnition 2.4. Let K B( ) and j j=1 and j j=1 be K-frames for N H . Φ and Ψ are called strongly disjoint if R(TΦ) ⊥ R(TΨ), where TΦ and TΨ are the analysis operators of the sequences Φ and Ψ, respectively.

∈ H N Φ = {φ }M Ψ = {ψ }M Theorem 2.5. Suppose that K B( ) and j j=1 and j j=1 are strongly disjoint tight K-frames for H N. Also, assume that A, B ∈ B(H N) are operators such ∗ ∗ ∗ ∗ N that AKK A + BKK B = IN×N, then {AΦ + BΨ} is a K-frame for H . In particular, ∗ 1 N = × {αΦ + βΨ} H if KK 2(|α|2+|β|2) IN N, then is a K-frame for .

120 Frame for operators in ﬁnite dimentional Hilbert space

3. Dual of K-frame In this section, we introduce the concept of K-dual of K-frames in H N and its properties are discussed. Also, the oblique K-dual is investigated. Φ = {φ }M H N Ψ = {ψ }M Definition 3.1. If j j=1 is a K-frame for , a sequence j j=1 is called a K-dual frame for Φ if ∑M N Kx = ⟨x, ψ j⟩φ j, (x ∈ H ). (2) j=1 Φ = {φ }M Ψ = {ψ }M The systems j j=1 and j j=1 are referred to as a K-dual frame pair. Proposition 3.2. Let Φ = {φ }M be a tight K-frame for H N. Then Tr(K) = ∑ j j=1 M ⟨φ , ψ ⟩, Ψ = {ψ }M Φ = {φ }M j=1 j j where j j=1 is a K-dual of j j=1. ν = {ν }M In the following theorem, we characterize the scalar sequences j j=1 for {φ }M {ψ }M ν = ⟨φ , ψ ⟩ which there exists a K-dual pair of frames j j=1 and j j=1 such that j j j for all j = 1, 2, ..., M. ∈ H N ν = {ν }M ⊂ C > = Theorem 3.3. Let K B( ) and j j=1 with M dim(R(K)) rank(K) {φ }M {ψ }M H N be given. Suppose that there exist K-dual frame pairs j j=1 and j j=1 for such ν = ⟨φ , ψ ⟩ = , , ..., ∗ {θ }M that j j j for all j 1 2 M. Then there exists a tight K -frame j j=1 and a corresponding dual frame Γ = {γ }M for H N such that ν = ⟨θ , γ ⟩ for all ∑ j j=1 j j j = , , ..., = M ν j 1 2 M. Furthermore Tr(K) j=1 j. In the following result we characterize K-duals of a K-frame. Φ = {φ }M H N Ψ = {ψ }M Proposition 3.4. Let j j=1 be a K-frame for . Then j j=1 is a K-dual for Φ if and only if R(TΦ) ⊥ R(TΘ), where TΘ is the analysis operator of the sequence Θ = {θ }M = {ψ − ∗ −1 φ }M j j=1 j K S Φ PS Φ(R(K)) j j=1. Oblique dual frames in finite dimentional Hilbert space were studied in [5]. In the last part of this section, we study this notion for K-frames. U W H N Φ = {φ }M Definition 3.5. Let and be two subspaces of and suppose that j j=1 Ψ = {ψ }M H N W = {φ = , , ..., } U = {ψ and j j=1 are in and span j : j 1 2 M , span j : j = 1, 2, ..., M}. The sequence Ψ = {ψ }M is an oblique K-dual frame of the K-frame ∑ j j=1 Φ = {φ }M W = M ⟨ , ψ ⟩φ , ∈ W. j j=1 on if Kx j=1 x j j for all x In the following two propositions a characterization of the oblique K-dual frames pair. W H N Φ = {φ }M Proposition 3.6. Suppose that is a subspace of and sequences j j=1, Ψ = {ψ }L Γ = {γ }L H N Φ ∪ Γ = W j j=1 and j j=1 in satisfy that span( ) . Then the following statements are equivalent: (i) Φ ∪ Ψ is an oblique K-dual frame∑ of Φ ∪ Γ on W. ∈ W − = L ⟨ , ψ ⟩γ (ii) For any x , (K S Φ)x j=1 x j j.

121 Vahid Reza Morshedi,Mohammad Janfada

Ψ = {ψ }M Φ = {φ }M W Proposition 3.7. If j j=1 is an oblique K-dual frame of j j=1 on and Φ is K-minimal, then the oblique K-dual frame of Φ on W is unique in the sense that Γ = {γ }M Φ ψ = γ , = , ..., if j j=1 is another oblique K-dual frame of , then j j j 1 M, where Ψ, Γ are restricted in W. Here, we state that if Φ is a K-frame for R(K), then we can make an oblique K- M d {φ } ∪{ } , { } dual frame of algebraic multiplicity of j j=1 e j j j0 where e j j=1 is an orthonormal {λ }d eigenbasis of the frame operator S Φ with associated eigenvalues j j=1. ∈ H N Φ = {φ }M W = Theorem 3.8. Let K B( ) and j j=1 be a K-frame for R(K) with W = { }d dim d. Also, let e j j=1 be an orthonormal eigenbasis of the frame operator d Φ {λ } , λ S with associated eigenvalues j j=1. Then for any eigenvalue 0 j0 , the 1 λ −λ 3 1 ∗ M ( j0 j) ∗ ∗ { √ φ } ∪ { √ + γ } , sequence λ K j j=1 λ K e j K j j: j j0 , is an oblique K-dual frame of j0 j0 2 (λ −λ ) 3 √1 M j√0 j d N { φ j} ∪ { e j} j: j, j on W, where {γ j} ⊂ H satisﬁes λ j=1 λ 0 j0, j=1 j0 j0

∑d ∗ ⟨x, K γ j⟩e j = 0, (x ∈ W).

j0, j=1

References [1] O. Christensen, A. M. Powell, and X. C.Xiao, A note on ﬁnite dual frame pairs, Proc. Amer. Math. Soc., (2012), 3921-3930. [2] O. Christensen, Frames and Bases: An Introductory Course, Birkhauser,¨ Boston, (2008). [3] D. Han and D. Larson, frames, Bases and group representations, Mem. Amer. Math. Soc., (2000). [4] X. Xiao, Y. Zhu and L. Gavru˘ ta¸ , Some properties of K-frames in Hilbert spaces, Results. Math.,(2013) 1243-1255. [5] X. C. Xiao, Y. C. Zhu and X. M. Zeng, Oblique dual frames in ﬁnite-dimensional Hilbert spaces, Int. J. Wavelets Multiresolut. Inf. Process, (2013).

Vahid Reza Morshedi, Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran e-mail: va−[email protected]

Mohammad Janfada, Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran e-mail: [email protected]

122 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

A NOTE ON GENERALIZED (σ, τ)-DERIVATIONS IN BANACH ALGEBRAS

Abolfazl Nazi Motlagh

Abstract Let A be a Banach algebra , σ and τ are linear mappings (or homomorphisms) on Banach algebra A. A linear map d : A → A is called (σ, τ)-derivation if d(ab) = d(a)σ(b) + τ(a)d(b) a, b ∈ A . A linear mapping δ : A → A is called a (σ, τ)-derivation if there exists a (σ, τ)- derivation d : A → A such that δ(ab) = δ(a)σ(b) + τ(a)d(b)(a, b ∈ A). In this talk, we investigate automatic continuity of these derivations on Banach algebras.

2010 Mathematics subject classiﬁcation: Primary 46H25: Secondary 46L57.. Keywords and phrases: Banach algebra, generalized derivation, generalized inner derivation.

1. Introduction Recently, a number of analysts [2–5] have studied various generalized notions of derivations in the context of Banach algebras. There are some applications in the other fields of research. Such mappings have been extensively studied in pure algebra. Throughout the paper, A is always a Banach algebra over complex field C, let σ and τ be two linear mappings (or homomorphisms) on A.A generalized concept of derivation is as follows Definition 1.1. A linear mapping d : A −→ A is a (σ, τ)-derivation if d(ab) = d(a)σ(b) + τ(a)d(b) for all a, b ∈ A, and is a (σ, τ)-inner derivation if there exists x ∈ A such that d(a) = xσ(a) − τ(a)x for all a ∈ A (see [2] and [5] and references therein). Example 1.2. (i) Every ordinary derivation of an algebra A into an A-bimodule is an idA-derivation, where idA is the identity mapping on the algebra A. (ii) Every point derivation d : A → C at the character θ on A is a θ-derivation.

123 A. N. Motlagh

Deﬁnition 1.3. A linear mapping δ : A → A is called generalized (σ, τ)-derivation if there exists a (σ, τ)-derivation d : A → A such that δ(ab) = δ(a)σ(b) + τ(a)d(b)(a, b ∈ A) (see [3] and references therein). Example 1.4. Basic examples are (i) (σ, τ)-derivations, (ii) (σ, τ)-semi inner derivations (i.e., maps of the form x ↣ aσ(x) + τ(x)a for some a, b ∈ A) and (iii) the σ-module maps (i.e., linear maps satisfying φ(ab) = φ(a)σ(b) for all a, b ∈ A). In particular, if 1 ∈ A, then δ(a) = δ(1)σ(a) + d(a) for all a ∈ A that d is a (σ, id)-derivation.

2. Innerness of generalized (σ, τ)-derivations on Banach algebras In this section, we introduce generalized (σ, τ)-inner derivation and we get some results about generalized (σ, τ)-derivation. Deﬁnition 2.1. A linear mapping δ on Banach algebra A is called generalized (σ, τ)- inner derivation, if there exists σ-module map ψ and x ∈ A such that

δ(a) = ψ(a) − τ(a)x (a ∈ A).

Moslehian and Niazi in[5] called a map dx : A −→ A a (σ, τ)-inner derivation if dx(a) = xσ(a) − τ(a)x for some x ∈ A. If we take ψ(a) = xσ(a), then this deﬁnition covers the notation of Moslehian and Niazi. Theorem 2.2. Let δ be a bounded generalized (σ, τ)-derivation on Banach algebra A. Then δ is a generalized (σ, τ)-derivation if and only if there exists an (σ, τ)- inner derivation dx on A such that dx(a) = xσ(a) − τ(a)x and δ(ab) = δ(a)σ(b) + τ(a)d(b)(a, b ∈ A).

Proof. Suppose that δ is a generalized (σ, τ)-inner derivation, i.e. there exist σ- module map ψ and x ∈ A such that

δ(b) = ψ(b) − τ(b)x (b ∈ A).

Now for all a, b ∈ A we have

δ(ab) = ψ(ab) − τ(ab)x = ψ σ − τ τ ( (a) (b) ()a) (b)x = ψ(a) − τ(a)x σ(b) + τ(a)xσ(b) − τ(a)τ(b)x ( ) ( ) = ψ(a) = τ(a)x σ(b) + τ(a) xσ(b) − τ(b)x

= δ(a)σ(b) + τ(a)dx(b).

124 A note on generalized (σ, τ)-derivations in Banach algebras conversely, let δ be a generalized (σ, τ)-inner derivation and there exist a inner derivation dx such that δ(ab) = δ(a)σ(b) + τ(a)dx(b). Deﬁne ψ : A −→ A by ψ(a) = δ(a) + τ(a)x. Then ψ is a bounded linear map. Now we have ψ(ab) = δ(ab) + τ(ab)x = δ σ + τ + τ τ (a) (b) (a)(dx(b) (a) (b))x = δ(a)σ(b) + τ(a) xσ(b) − τ(b)x + τ(a)τ(b)x = δ σ + τ σ ((a) (b) ()a)x (b) = δ(a) + τ(a)x σ(b) = τ(a)σ(b) □

Corollary 2.3. Let every (σ, τ)-derivation on A be (σ, τ)-inner derivation. Then every generalized (σ, τ)-derivation on A is generalized (σ, τ)-inner derivation.

3. Automatic continuity of generalized (σ, τ)-derivations on Banach algebras The following proposition characterizes generalized (σ, τ)-derivations by σ-module mappings. Proposition 3.1. A linear mapping δ : A → A is a generalized (σ, τ)-derivation if and only if there exist a (σ, τ)-derivation d : A → A and a σ-module map φ : A → A such that δ = d + φ. Proof. At ﬁrst suppose δ be a generalized (σ, τ)-derivation on A, then there exists a (σ, τ)-derivation d on A such that δ(ab) = δ(a)σ(b) + τ(a)d(b)(a, b ∈ A). Therefore φ = δ − d is a σ-module map, because for all a, b ∈ A we have φ = δ − (ab) ((ab) d(ab) ) ( ) = δ(a)σ(b) + τ(a)d(b) − d(a)σ(b) + τ(a)d(b) = (δ(a) − d(a))σ(b) = φ(a)σ(b). Then φ is a σ-module map and δ = d + φ. conversely, let d be a (σ, τ)-derivation on A, φ be a σ-module map on A and put δ = d + φ. Then δ clearly is a linear map and δ(ab) = d(ab) + φ(ab) = σ + τ + φ σ (d(a) (b) )(a)d(b) (a) (b) = d(a) + φ(a) σ(b) + τ(a)d(b) = δ(a)σ(b) + τ(a)d(b) for all a, b ∈ A. Hence δ is a generalized (σ, τ)-derivation. □

125 A. N. Motlagh

In the following result we investigate continuity of σ-module maps. Lemma 3.2. Let A have a bounded left approximate identity. σ-module map φ on A is a bounded mapping if σ is a bounded mapping. Theorem 3.3. Let A have a bounded left approximate identity, σ be bounded linear map on A and δ = d + φ be a generalized (σ, τ)-derivation. Then δ is bounded if and only if d is bounded. Proposition 3.4. Let σ be a continuous linear mapping on C∗-algebra A. Then every ∗ − (σ, σ)-derivation on A is automatically continuous. Deﬁnition 3.5. Let A be a ∗-algebra, σ and τ be two linear mappings on A. A linear mapping δ : A −→ A is called generalized ∗ − (σ, τ)-derivations if there exists a ∗ − (σ, τ)-derivation d : A −→ A such that δ(ab) = δ(a)σ(b) + τ(a)d(b), (a, b ∈ A) Theorem 3.6. Let σ be a continuous linear mapping on C∗-algebra A. Then every generalized ∗ − (σ, σ)-derivation on A is automatically continuous. Proposition 3.7. If σ and τ are continuous ∗-linear mappings on C∗-algebra A, then every ∗ − (σ, τ)-derivation on A is automatically continuous. Theorem 3.8. If σ and τ are continuous ∗-linear mappings on C∗-algebra A, then every generalized ∗ − (σ, τ)-derivation on A is automatically continuous.

References

[1] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer - Verlag, New York, (1973). [2] C. Park and D. Y. Shin, Generalized (θ, ϕ)-derivations on Banach algebras, Korean J. Math., 22 (2014), 139-150. [3] M. S. Moslehian, Hyers-Ulam-Rassias stability of generalized derivations, Intern. J. Math. Math. Sci, (2006). [4] M. Mirzavaziri and M. S. Moslehian, Automatic continuity of σ-derivations on C∗-algebras, it Proc .Amer. Math. Soc., (2006), 3319-3327. [5] M. S. Moslehian and A. N. Motlagh, Some notes on (σ, τ)-amenability of Banach algebras, Studia Univ. "BABES-BOLYAI", Mathematica, Volume LIII, (2008).

Abolfazl Nazi Motlagh, Department of Mathematics, University of Bojnord, Bojnord, Iran e-mail: [email protected] and [email protected]

126 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

CONTRACTIBILITY OF NON-ARCHIMEDEAN BANACH ALGEBRAS

Abolfazl Niazi Motlagh

Abstract

In this talk we investigate contractibility of non-Archimedean Banach algebras.

2010 Mathematics subject classiﬁcation: Primary 39B52; Secondary 39B82, 13N15, 46S10. Keywords and phrases: Banach algebra, derivation, approximately contractibility, non-Archimedean Banach algebra..

1. Introduction Let K be a field. A non-archimedean absolute value on K is a function |.| : K −→ [0, ∞) such that for any a, b ∈ K we have (i) |a| ≥ 0 and equality holds if and only if a = 0, (ii) |ab| = |a||b|, (iii) |a + b| ≤ max{|a|, |b|}. Condition (iii) is called the strict triangle inequality. By (ii), we have |1| = |−1| = 1. Thus, by induction, is concluded from (iii) that |n| ≤ 1 for each integer n.In all, we always assume that |.| is non-trivial, i.e., that there is an a0 ∈ K such that |a0| < 0, 1. Let X be a linear space over a scalar field K with a non-archimedean non-trivial valuation |.|. A function ∥.∥ : X −→ R is a non-archimedean norm (valuation) if it satisfies the following conditions: (I) ∥x∥ = 0 if and only if x = 0; (II) ∥rx∥ = |r|∥x∥ for all r ∈ K and x ∈ X; (III) the strong triangle inequality (ultrametric); namely, ∥x+y∥ ≤ max{∥x∥, ∥y∥(x, y ∈ X). Then (X, ∥.∥) is called a non-archimedean space. It is concluded from (III) that

∥xm − x∥ ≤ max{∥x j+1 − x j∥ : l ≤ j ≤ m − 1}(m > l).

Therefore, a sequence {xm} is Cauchy in X if and only if {xm+1 − xm} converges to zero in a non-archimedean space. By a complete non-archimedean space we mean one in which every Cauchy sequence is convergent. A non-archimedean Banach algebra is a complete non-archimedean algebra A which satisﬁes ∥ab∥ ≤ ∥a∥∥b∥ for all a, b ∈ A.

127 A. N. Motlagh

For more detailed deﬁnitions and history of stability of functions on (normed, Banach, non-archimedean)spaces, we refer to [1].

2. Approximate derivations Throughout this section, A is a non-archimedean Banach algebra on non-archimedean filed K that the characteristic of K is not 2 and X is a non-Archimedean Banach A- bimodule. We say a mapping f : A −→ X is approximately ∆-derivation if it satisfies in a functional equality(inequality)∆ such that there are derivation D : A −→ X and real valued function φ : A −→ R such that ∥ f (a) − D(a)∥ ≤ φ(a). Proposition 2.1. Suppose that k is a fixed integer greater than 2 and |k| < |2|. Let A be an unital non-archimedean Banach algebra,X is a non-Archimedean Banach A-bimodule and f : A −→ X be a mapping such that

(a + b + cd ) ∥ f (a) + f (b) + c f (d) + f (c)d∥ ≤ k f (1) k for all a, b, c, d, ∈ A. Then f is a derivation. Proof. By taking a = b = c = d = 0 in (1)we have ∥2 f (0)∥ ≤ ∥k f (0)∥. So by |k| < |2| we get (|2| − |k|)∥ f (0)∥ ≤ 0 and therefore f (0) = 0. Now we show that f is an odd function. Set b = −a and c = d = 0 in (1),therefore we have ∥ f (a) + f (−a)∥ ≤ |k|∥ f (0)∥ and so f (−a) = − f (a) for all a ∈ A. In this step we will show that f (a2) = a f (a) + f (a)a for all a ∈ A and therefore we can conclude f (1) = 0. For this purpose enough to take a := a2, b := 0, c := −a and d := a in(1); thus, we get

(a2 + 0 − aa) ∥ f (a2) + f (0) − a f (a) + f (−a)a∥ ≤ k f = 0. k Now, letting c := 1 and d := −a − b in(1), we have

(a + b − (a + b)) ∥ f (a) + f (b) − f (a + b)∥ ≤ k f = 0. k As a result, we have f (a + b) = f (a) + f (b). In the last step set a := ab, b := 0, c : −a and d := b(1), so we can see that

(ab + 0 + a(−b)) ∥ f (ab) + f (0) − a f (b) − f (a)b∥ ≤ k f = 0. k Therefore f (ab) = a f (b) + f (a)b and this completes the proof. □ Theorem 2.2. Suppose that k is a ﬁxed integer greater than 2 and |k| < |2|. Let r < 1, θ be nonnegative real numbers and f : A −→ X be an odd mapping such that f (1) = 0 and

(a + b + cd ) (∆)∥ f (a) + f (b) + c f (d) + f (c)d∥ ≤ k f k + θ(∥a∥r + ∥b∥r + ∥cd∥r) (2)

128 CONTRACTIBILITY OF NON-ARCHIMEDEAN BANACH ALGEBRAS for all a, b, c, d ∈ A. Then there exists a unique derivation D : A −→ X such that 2 + |2|r ∥ f (a) − D(a)∥ ≤ θ∥a∥r (a ∈ A). (3) |2|r Theorem 2.3. Suppose that k is a ﬁxed integer greater than 2 and |k| < |2|. Let r > 1, θ be nonnegative real numbers and f : A −→ X be an odd mapping such that f (1) = 0 and

(a + b + cd ) ∥ f (a) + f (b) + c f (d) + f (c)d∥ ≤ k f k + θ(∥a∥r + ∥b∥r + ∥cd∥r) (4) for all a, b, c, d ∈ A. Then there exists a unique derivation D : A −→ X such that 2 + |2|r ∥ f (a) − D(a)∥ ≤ θ∥a∥r (a ∈ A). (5) |2|r | | < | | < 1 , θ Theorem 2.4. Suppose that k is a ﬁxed integer greater than 2 and k 2 . Let r 3 be nonnegative real numbers and f : A −→ X be an odd mapping such that f (1) = 0 and

(a + b + cd ) ∥ f (a) + f (b) + c f (d) + f (c)d∥ ≤ k f k + θ∥a∥r.∥b∥r.∥cd∥r) (6) for all a, b, c, d ∈ A. Then there exists a unique derivation D : A −→ X such that θ|2|r ∥ f (a) − D(a)∥ ≤ ∥a∥3r (a ∈ A). (7) |2|3r | | < | | > 1 , θ Theorem 2.5. Suppose that k is a ﬁxed integer greater than 2 and k 2 . Let r 3 be nonnegative real numbers and f : A −→ X be an odd mapping such that f (1) = 0 and

(a + b + cd ) ∥ f (a) + f (b) + c f (d) + f (c)d∥ ≤ k f k + θ∥a∥r.∥b∥r.∥cd∥r) (8) for all a, b, c, d ∈ A. Then there exists a unique derivation D : A −→ A such that θ|2|r ∥ f (a) − D(a)∥ ≤ ∥a∥3r (a ∈ A). (9) |2|

3. Approximate Contractible non-Archimedean Banach algebras If every bounded derivation is inner, then A is said to be contractible. The non- Archimedean Banach algebra A is called approximately contractible if for every approximate derivation there exists real valued function φ : A −→ R and an x ∈ X such that ∥xa − ax − f (a)∥ ≤ φ(a).

129 A. N. Motlagh

Theorem 3.1. A non-Archimedean Banach algebra A is approximately contractible if and only if A is contractible. Proof. Let A be a contractible and f : A −→ X is a approximate derivation. By Theorem2.2( there) exists a bounded derivation D : A −→ X defined by D(a):= n a ∈ A limn→∞ 2 f 2n , a which satisfies 2 + |2|r ∥ f (a) − D(a)∥ ≤ θ∥a∥r (a ∈ A). (10) |2|r Since A is contractible, there is some x ∈ Xsuch that D(a) = xa − ax. ∥ − − ∥ = ∥ − ∥ ≤ 2+|2|r θ∥ ∥r A Hence xa ax f (a) D(a) f (a) |2|r a Therefore is approximately contractible. Conversely, let A be approximately contractible and D : A −→ X be a bounded derivation. Then D is trivially an approximate derivation. Due to the approximate contractibility of A, there exists real valued function φ : A −→ R and an x ∈ X such that ∥xa − ax − D(a)∥ ≤ φ(a). Replacing a by 2na in the later inequality we can conclude ∥xa − ax − D(a)∥ ≤ 2−nφ(a). Hence xa − ax = D(a). It follows that A is contractible. □ One can similarly define notation approximate amenability and establish the following theorem. Theorem 3.2. A Non-Archimedean Banach algebra A is approximately amenable if and only if A is amenable.

References [1] M. S. Moslehian and TH. M. Rassiase, stability of functional equations in non-Archimedean space, Appl. Anal. Discrete Math 1 (2007), 325-334. [2] S. M. Ulam,Problems in Modern Mathematics, Science Editions, John Wiley & Sons, New York, (1964). [3] Y. J. Cho, C. Park and R. Saadati,Functional inequalities in non-Archimedean Banach spaces, Applied Mathematics Letters, (2010) 1238-1242.

Abolfazl Niazi Motlagh, Department of Mathematics, University of Bojnord, P. O. Box 1339, Bojnord, Iran e-mail: [email protected] and [email protected]

130 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

A NOTE ON THE STAHL’S THEOREM

Hamed Najafi

Abstract Let H be a Hilbert space and B(H) denotes the algebra of all bounded linear operators on H and A be a bounden operator in H. Let A, B ∈ B(H) be positive operators and Φ be a positive linear functional on B(H). We show that, if f is a non-negative operator decreasing function, then the function t → Φ ( f (A + tB)) can be written as a Laplace transform of a positive measure.

2010 Mathematics subject classiﬁcation: Primary 15A15, , Secondary 15A16 . Keywords and phrases: BMV conjecture, laplace transform, operator monotone functions..

1. Introduction Let H be a Hilbert space and B(H) denotes the algebra of all bounded linear operators on H and A be a bounden operator in H. An operator A is called positive if ⟨Ax, x⟩ ≥ 0 holds for every x ∈ H and then we writ A ≥ 0. For self-adjoint operators A, B ∈ B(H) we say A ≥ B if A − B ≥ 0. For a continuous function f and a self adjoint operator A with spectra in domain of f , the operator f (A) is defined by standard functional calculus. In particular, if H∑is a finite dimensional Hilbert space and A has the = n µ , spectral decomposition A i=1 iPi where Pi are the projections corresponding to the eigenspaces of eigenvalue µi, then ∑n f (A) = f (µi)Pi. i=1 A function f on the interval I is called operator increasing if A ≥ B implies that f (A) ≥ f (B), for each self adjoint operators A, B ∈ B(H) with spectra in I. Also, f is called operator decreasing function if − f is operator increasing function. The Bessis-Moussa-Villani conjecture states( that) for a self adjoint Matrix A and positive Matrix B, the function f (t) = Tr expA−tB can be represented as the Laplace transform ∫ ∞ f (t) = exp−tx dµ(x), (1) 0 of a positive measure µ on [0, ∞) [1]. This conjecture has attracted a lot of attention in mathematics and physics. Despite a lot of afford to prove the conjecture, it remained

131 H. Najafi open until 2012. Eventually, Stahl [5] proved this conjecture. In [3], Hansen got a similar results.

Theorem 1.1. [3] If f is a non-negative operator decreasing function on [0, ∞), then for positive matrices A, B the map t → Tr f (A + tB) can be written as the Laplace transform of a positive measure.

In proof of this theorem, Hansen used the theory of Frechet diﬀerentials and the Bernsteins theorem highlighting the measure µ in (1) exists if and only if f is completely monotone or (−1)n f n(t) ≥ 0, for each n = 0, 1, 2,... and t > 0. In this note, we extended the results of Hansen and prove that for an arbitrary Hilbert space H and any positive linear functional Φ on B(H). Also, in a special case, we obtained a partial extension of a famous equivalent statement for Bessis-Moussa-Villani conjecture which state for each p ≤ 0 and positive p semi-deﬁnite matrices A and B, the function hp(t) = Tr (A + tB) is completely monotone [4]. Indeed, we show that for a positive linear functional Φ on B(H), the p function ϕp(t) = Φ ((A + tB) ) is completely monotone for each −1 ≤ p ≤ 0 and positive operators A, B ∈ B(H).

2. Main Results

The following theorem states the main results.

Theorem 2.1. Let A, B ∈ B(H) be positive operators and f be an operator decreasing function on (0, ∞). For any positive linear functional Φ on B(H) the function ϕ(t) = Φ ( f (A + tB)) is operator decreasing. In particular, if f is non-negative then ϕ is completely monotone.

By replacing f by − f we obtain the following corollary.

Corollary 2.2. Let A, B ∈ B(H) be positive operators and f be an operator increasing function on (0, ∞). Then, for any positive linear functional Φ on B(H) the function ϕ(t) = Φ ( f (A + tB)) is an operator increasing function.

Example 2.3. The functions ln (t + 1) is a non-negative operator increasing function on [0, ∞). Hence, for positive operators A, B and positive linear functional Φ, the function t → Φ ( f (A + tB)) is a non-negative operator increasing function on [0, ∞).

The function f (t) = tp is operator increasing for 0 ≤ p ≤ 1 and operator decreasing for −1 ≤ p ≤ 0. Therefore, As an example we the following corollary is given.

Corollary 2.4. Let A, B be positive operators in B(H) and −1 ≤ p ≤ 1. For a positive linear functional Φ on B(H), the function ϕ(t) = Φ ((A + tB)p) is operator increasing if 0 ≤ p ≤ 1 and operator decreasing if −1 ≤ p ≤ 0.

132 A Note On The Stahl’s Theorem

References [1] D. Bessis, P. Moussa and M. Villani, Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics, J. Math. Phys., 16 (1975) 2318-2385. [2] R. Bhatia, Matrix Analysis, Springer, New York, 1997. [3] F. Hansen, Trace functions as Laplace transforms, J. Math. Phys., 47 (2006) , 1-11. [4] E. Lieb and R. Steiringer, Equivalent forms of the Bessis-Moussa-Villani conjecture, J. Stat. Phys., 115 (2004) 185-190. [5] R., H., Stahl, Proof of the BMV conjecture, Acta Math, 211 (2013), 255-290.

Hamed Najafi, Department of Pure Mathematics Ferdowsi University of Mashhad Mashhad , Iran e-mail: hamednajaﬁ[email protected]

133 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

ON SOME SOLVABLE EXTENSIONS OF THE HEISENBERG GROUP II

Mehri Nasehi

Abstract In this paper we consider two families of four or five-dimensional solvable Lie groups which are extensions of the Heisenberg group. We first calculate the energy of an arbitrary left-invariant vector field X on these spaces. Then we obtain the exact form of all harmonic maps on these spaces and prove that if these spaces admit a non-trivial harmonic map, then any harmonic map is a critical point for the energy functional restricted to vector fields of the same length. Later on, we investigate the existence of left-invariant Yamabe solitons and explicitly describe all geodesic vectors on these spaces.

2010 Mathematics subject classiﬁcation: Primary 22E25, Secondary 53C30. Keywords and phrases: Solvable Lie groups, Heisenberg group, Harmonic maps, Yamabe solitons, geodesic vectors.

1. Introduction Heisenberg groups play an important role in geometric analysis, mathematical physics and quantum mechanics. In the past many paper were devoted to the study of the three-dimensional Heisenberg group H3. Moreover in [2] some solvable extensions of the Heisenberg group H3 are investigated and all homogeneous Riemannian structures on these spaces are obtained. Recently in [5] we have investigated some problems related to left-invariant harmonic vector fields, left-invariant Ricci solitons, Einstein- like metrics and contactizations on these spaces. Our aim in this paper is to investigate some other problems related to harmonic maps, geodesic vectors, left-invariant Yam- abe solitons and the energy of an arbitrary left-invariant vector field X on these spaces. By this study we extend the study of left-invariant Ricci solitons in [2] on these spaces and prove that if these spaces admit a non-trivial harmonic map, then any harmonic map is a critical point for the energy functional restricted to vector fields of the same length.

2. Harmonic maps and their properties on some solvable extensions of the Heisenberg group Here we remind some facts about some solvable extension of the Heisenberg group. The one-dimensional extensions of the Heisenberg group: These spaces are four- dimensional and denoted by A4(λ, µ), where λ and µ are positive real numbers. By

134 M. Nasehi

[2], the non-zero Lie brackets are given by [e1, e2] = λe3, [e4, e1] = µe1, [e4, e2] = µe2, [e4, e3] = 2µe3. Also the left-invariant Riemannian metric gλ,µ and the non-zero components of the Levi-Civita connection are given in [2]. The two-dimensional extensions of the Heisenberg group: These spaces are five- dimensional and denoted by A5(λ, µ, ν), where λ, µ, ν ∈ R and λ, µ > 0. By [2], the non-zero brackets are given by [e1, e2] = λe3, [e4, e1] = µe1, [e4, e2] = µe2, [e4, e3] = 2µe3, [e4, e5] = νe5. Also the left-invariant Riemannian metric gλ,µ,ν and the non- zero components of the Levi-Civita connection are given in [2]. Here to obtain the energy of an arbitrary left-invariant vector field X on some solvable extensions of the Heisenberg group we remind some facts and for more details we refer to [1]. Let (M, g) be an oriented n-dimensional Riemannian manifold and (TM, gs) be its tangent bundle. Then the energy of∫ the smooth vector field , → , s = n , + 1 ∥ ∇ ∥2 , s X :(M g) (TM g ) is defined by E(X) 2 vol(M g) 2 M X dv where g is the Sasakian metric and M is compact. Here we suppose that Ω is a relatively compact domain of these spaces and calculate the energy of X|Ω. Proposition 2.1. Let (G, g) be one of the two families of four or five-dimensional solvable metric Lie groups (A4(λ, µ), gλ,µ) and (A5(λ, µ, ν), gλ,µ,ν) and X be an arbitrary left-invariant vector field on G. Then the energy of X|Ω on A4(λ, µ) and A5(λ, µ, ν) are = { + µ2∥ ∥2 + λ2 2 + 2 + 2 + µ2 2 + µ2 2} Ω respectively given by EΩ(X) 2 X 2 (k1 k2 k3) 3 k3 5 k4 vol = { 5 + µ2∥ ∥2 + λ2 2 + 2 + 2 + µ2 2 + µ2 + ν2 2 + ν2 − µ2 2} Ω, and EΩ(X) 2 X 2 (k1 k2 k3) 3 k3 (5 )k4 ( )k5 vol where EΩ(X) denotes the energy of X|Ω and ∥X∥ is the norm of X. The critical points for the energy functional are harmonic maps. These vector fields are characterized by Euler-Lagrange equations. In fact a vector field X defines a non- zero harmonic∑ map from (M, g) to (TM, gs) if and only if∑X satisfies the conditions ∗ ∇ ∇X = (∇ ∇ X − ∇∇ X) = 0 and tr[R(∇.X, X).] = R(∇ X, X)e = 0, where ei ei ei ei ei i i i {e1, ··· , en} is an orthonormal local frame field on (M, g). Thus we can obtain the exact form of all harmonic maps on some solvable extensions of the Heisenberg group as follows.

Theorem 2.2. (a) The metric Lie group (A4(λ, µ), gλ,µ) never admits any harmonic λ, µ, ν , map. (b) The∑ metric Lie group (A5( ) gλ,µ,ν) with an arbitrary left-invariant vector = 5 = field X i=1 kiei admits a non-trivial harmonic harmonic map X k5e5 if and only if ν = 0. Let χP(M) = {W ∈ χ(M): ∥W∥2 = ρ2}, where ρ , 0 is a real constant. Then we denote by E|χP(M) the energy functional restricted to vector fields of the same length. ∗ By the Euler-Lagrange equations, X is a critical point for E|χP(M) if and only if ∇ ∇X is collinear to X. Thus it is natural to ask whether any of the vector fields which are given in Theorem 2.2 is a critical point for the energy functional restricted to vector fields of the same length. To answer this question we prove the following result.

Theorem 2.3. (i) Let X = k1e1 + ··· + k4e4 be an arbitrary left-invariant vector ﬁeld on the metric Lie group (A4(λ, µ), gλ,µ). X is a critical point for the energy functional

135 On some solvable extensions of the Heisenberg group II restricted to vector fields of the same length if and only if X has one of the forms X = k1e1 + k2e2, X = k3e3 or X = k4e4. (ii) Let X = k1e1 + ··· + k5e5 be an arbitrary left-invariant vector field on the metric Lie group (A5(λ, µ, ν), gλ,µ,ν). X is a critical point for the energy functional restricted to vector fields of the same length if and only if X has one of the forms X = k1e1 + k2e2, X = k3e3, X = k4e4 or X = k5e5. Thus by theorem 2.2 and 2.3 we obtain the following result. Corollary 2.4. Let (G, g) be one of the two families of four or five-dimensional solvable metric Lie groups (A4(λ, µ), gλ,µ) and (A5(λ, µ, ν), gλ,µ,ν). If G admits a non-trivial harmonic map, then any harmonic map is a critical point for the energy functional restricted to vector fields of the same length.

3. Left-invariant Yamabe solitons on some solvable extensions of the Heisenberg group In this section to obtain left-invariant Yamabe solitons on two families of four or five-dimensional Lie groups we remind that a Riemannian manifold (M, g) is a Yamabe soliton if it admits a vector field X such that LXg = (τ − λ)g, where LX denotes the Lie derivative in the direction of X, τ is the scaler curvature and λ is a real number. A Yamabe soliton is non-trivial when τ , λ. For more details see [4]. Theorem 3.1. Let (G, g) be one of the two families of four or five-dimensional solvable metric Lie groups (A4(λ, µ), gλ,µ) and (A5(λ, µ, ν), gλ,µ,ν). Then it is not a non-trivial left-invariant Yamabe soliton.

Proof. Assume that X = k1e1 + k2e2 + k3e3 + k4e4 is an arbitrary left-invariant vector ﬁeld on A4(λ, µ). Then the non-zero Lie derivative components are LXg(e1, e1) = LXg(e2, e2) = LXg(e3, e3) = −2k4µ, LXg(e1, e3) = k2λ, LXg(e1, e4) = k1µ, LXg(e3, e4) = 2k3µ, LXg(e2, e3) = −k1λ and LXg(e2, e4) = k2µ. Notice that λ, µ > 0, then by using the τ = − µ2 − λ2 = τ − λ , scaler curvature 22 2 in [5] and the Yamabe soliton formula LXg ( )g we obtain the result. For (A5(λ, µ, ν), gλ,µ,ν) we have a similar argument. □

4. Geodesic vectors on some solvable extensions of the Heisenberg group = G , Let (M H g) be a homogeneous Riemannian manifold with the reductive decomposition G = M ⊕ H. Then a homogeneous geodesic through the origin ◦ ∈ M is a geodesic γ(t) which is an orbit of a one-parameter subgroup of G, that is γ(t) = exp(tZ)(◦) where t ∈ R and Z is a non-zero vector of G. By [3] a non-zero vector Z ∈ G is a geodesic vector if and only if ⟨[Z, Y]M, ZM⟩ = 0, for all Y ∈ M.

Theorem 4.1. (a) X is a geodesic vector of the metric Lie group (A4(λ, µ), gλ,µ) if and = , = + , = + + only if X has one of the forms (ai) X k4e4 (aii) X k4e4 k5V1 (aiii1 ) X k1e1 k2e2 (k k k +k k2+k k2) (k2k2+2k k k k +k2k2+k2k2) + + 1 6 3 8 1 8 2 − 6 3 6 3 8 1 8 1 8 2 + + (k6k3 k8k1) + , = k3e3 (k k ) e4 2 V1 k6V2 k V3 k8V4 (aiii2 ) X 6 2 (k6k2) 2 k (k2+k2) ∑ −k2(k2+k2) + + + , = 8 4 3 + 4 + + 8 4 3 − k8k4 + k1e1 k2e2 k3e3 k4e4 (aiii3 ) X (k k ) e1 i=3 kiei k5V1 2 V2 k V3 5 3 (k5k3) 3

136 M. Nasehi

k k2 , = + + + , = + − 6 4 + + k6k4 k8V4 (aiii ) X k5V1 k6V2 k7V3 k8V4 (aiii ) X k1e1 k4e4 2 V1 k6V2 V3 4 5 k k1 = + , 1 and (aiii6 ) X k4e4 k5V1 where with respect to the reductive decomposition G = A4(λ, µ) + H, the set {V1, ··· , V4} is a base for H. (b) X is a geodesic vector of the metric Lie group (A5(λ, µ, ν), gλ,µ,ν) if and only if X has = + + , ν = , = + , ν , , = one of the forms (bi)X k4e4 k5e5 k6V1 0 (bii)X k4e4 k6V1 ∑ 0 (biii)X + , ν = , = , ν , , = + , = 5 + k4e4 k5e5 0 (biv)X k4e4 0 (bv)X k4e4 k5e5 (bvi1 ) X i=1 kiei k6V1 , = + , , = + + + with k5 0, (bvi2 ) X k3e3 k5e5 with k5 0 (bvii1 ) X k1e1 k2e2 k3e3 k k k +k k2+k k2 k2k2+2k k k k +k2k2+k2k2 + 1 7 3 9 1 9 2 − 7 3 7 3 9 1 9 1 9 2 + + k7k3 k9k1 + , = + + k k e4 2 V1 k7V2 k k9V4 (bvi2 ) X k1e1 k2e2 7 2 k7k2 2 k (k2+k2) k2(k2+k2) + = 9 4 3 + + + − 9 4 3 − k9k4 + , = k3e3 k4e4(bvi3 ) X k k e1 k3e3 k4e4 k6V1 2 k V3 k9V4 (bvi4 ) X 6 3 k6k3 3 k k2 + + + , = + − 7 4 + + k7k4 , = k6V1 k7V2 k8V3 k9V4 (bvi5 ) X k1e1 k4e4 2 V1 k7V2 k V3 and (bvi6 ) X k1 1 k4e4 + k6V1, where with respect to the reductive decomposition G = A5(λ, µ, ν) + H, the set {V1, ··· , V4} is a base for H. Proof. To prove the case (a) we notice that, if λ , 2µ, then we have the homogeneous Riemannian structure (2.5) in [2]. By Taking into account this structure we have = −λ , −λ = −λ G˜a,b = either a 2 or a 2 . If a 2 , then the reductive decomposition is {0} + A4(λ, µ) = span{e1, e2, e3, e4} [2] and X is a geodesic vector of this type if and only if its components (k1, ··· , k4) with respect to the basis {e1, ··· , e4} satisfy λ − µ + = , λ + − +µ = , µ = , µ 2 +µ 2 + 2µ = . k2 k3 k4( k1 bk2) 0 k1 k3 k4( bk1 k2) 0 2 k4k3 0 k1 k2 2k3 0 Thus µ > 0 gives us the case (ai). The remaining cases can be proved by a similar way.

References [1] M. Aghasi and M. Nasehi, On the geometrical properties of solvable Lie groups, Adv. Geom. 15 (2015) 507-517. [2] W. Batat, P. M. Gadea and J. A. Oubina, Homogeneous Riemannain structures on some solvable extensions of the Heisenberg group, Acta Math. Hungar. (2012), 24 pages. [3] G, Calvaruso, O. Kowalski, R. Marinosci, Homogeneous geodesics in solvable Lie groups, Acta Math. Hungar. 101, 4 (2003), 313-322. [4] M. Nasehi and M. Aghasi, On the geometrical properties of hypercomplex four-dimensional Lie groups, to appear in Georgian Math. J [5] M. Nasehi, On some solvable extensions of the Heisenberg group , The 4th Seminar on Harmonic Analysis and Applications (2016) 144-148.

Mehri Nasehi, Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran e-mail: [email protected]

137 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

ON THE STABILITY OF THE CUBIC FUNCTIONAL EQUATION ON N-ABELIAN GROUPS

Mahdi Nazarianpoor ∗ and Ghadir Sadeghi

Abstract

Most the literature on the stability of the cubic functional equation focus on the case where the relevant domain is a normed space. In this paper, we investigate the stability of the cubic functional equation on n-Abelian groups.

2010 Mathematics subject classiﬁcation: Primary 46L53, Secondary 46L10. Keywords and phrases: Hyers-Ulam stability, cubic functional equation, n-Abelian group.

1. Introduction

In 1940, S. M. Ulam [3] proposed the following question concerning the stability of group homomorphisms: Let G1 be a group and (G2, d) a metric group. Given ε > 0, does there exist a δ > 0 such that if a function h : G1 → G2 satisﬁes the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G1, then there exists a homomorphism H : G1 → G2 such that d(h(x), H(x)) < ε for all x ∈ G1? In the next year D. H. Hyers [1] answers the problem of Ulam under the assumption that the groups are Banach spaces: Let X be a normed space and Y a Banach space. Suppose that for some ε > 0, the mapping f : X → Y satisﬁes || f (x + y) − f (x) − f (y)|| ≤ ε for all x, y ∈ X. Then there exists a unique additive mapping T : X → Y such that || f (x) − T(x)|| ≤ ε for all x ∈ X. Jun and Kim [2] introduced the following cubic functional equation

f (2x + y) + f (2x − y) = 2 f (x + y) + 2 f (x − y) + 12 f (x) (1) for functions from a vector space into a Banach space and they established the general solution and the generalized Hyers-Ulam stability for the functional equation (1).

∗ speaker

138 M. Nazarianpoor and G. SADEGHI

2. Preliminaries In this section we consider the stability of the cubic functional equation f (x2y) + f (x2y−1) − 2 f (xy) − 2 f (xy−1) − 12 f (x) = 0 (2) for the pair (G, X) when G is an arbitrary group and X is a real Banach space. Every solution of functional equation (2) is said to be a cubic mapping. We prove that if G is an n-Abelian group with n ∈ N, then the cubic functional equation is stable. The Jun and Kim result is a particular case of this result. In this sequel we will write the arbitrary group G in multiplicative notation. Throughout the section X denotes a Banach space.

Deﬁnition 2.1. The cubic functional equation (2) is said to be stable for the pair (G, X) (write (G, X) is CS for short) if for every function f : G → X such that

∥ f (x2y) + f (x2y−1) − 2 f (xy) − 2 f (xy−1) − 12 f (x)∥ ≤ δ (3) for all x, y ∈ G and for some δ ≥ 0, there is solution T of functional equation (2) and a constant ϵ ≥ 0 dependent only on δ satisfying

∥ f (x) − T(x)∥ ≤ ϵ. (4) Deﬁnition 2.2. We say that a mapping f : G → X is a quasi-cubic mapping if there exists a nonnegative number δ such that

∥ f (x2y) + f (x2y−1) − 2 f (xy) − 2 f (xy−1) − 12 f (x)∥ ≤ δ for all x, y ∈ G. It is clear that the set of all quasi-cubic mappings from G into X is a real linear space relative to the ordinary operations. We denote it by KC(G, X). The subspace of KC(G, X) consisting of all cubic functions will be denoted by C(G, X). The mapping f : G → X is said to be a pseudo-cubic mapping if it is a quasi-cubic mapping satisfying f (xn) = n3 f (x) for any x ∈ G and any n ∈ N. We denote the space of all pseudo-cubic mappings from G into X by PC(G, X).

3. Main results Lemma 3.1. Suppose that f ∈ KC(G, X). Then for any k, m ∈ N with m ≥ 2, there is a δm > 0 such that for each x ∈ G, the following inequality

1 k ∥ f (xm ) − f (x)∥ ≤ 2b (5) m3k m = 1 δ holds, where bm m3 m.

139 cubic functional equation on N-Abelian groups

Lemma 3.2. Let f ∈ KC(G, X). For any x ∈ G and any m ∈ N, the sequense 1 mk ( m3k f (x ))k is a Cauchy sequence. Deﬁnition 3.3. For any f ∈ KC(G, X), the function fˆ is deﬁned as

1 k fˆ(x) = lim f (x2 ). k→∞ 8k Theorem 3.4. KC(G, X) = PC(G, X) ⊕ B(G, X). Theorem 3.5. The cubic functional equation (2) is stable for the pair (G, X) if and only if PC(G, X) = C(G, X). Theorem 3.6. Let X, Y be a Banach space over reals. Then the cubic functional equation (2) is stable for the pair (G, X) if and only if it is stable for the pair (G, Y). Theorem 3.7. Let n ∈ N and G be an n-Abelian group. Then the cubic functional equation (2) is stable on group G. It is well known that every Abelian group is an n-Abelian group for any n ∈ N. Thus we get the following result . Corollary 3.8. The cubic functional equation (2) is stable on any Abelian group.

References [1] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27 (1941), 222-224. [2] K. W. Jun, M. M Kim, , The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl, 274(2), (2002), 267-278. [3] S. M. Ulam, A collection of Mathematical problems, interscience Publ, New York, (1960).

Mahdi Nazarianpoor , Department of Mathematics and Computer Sciences, University of Hakim Sabzevari, sabzevar, Iran e-mail: [email protected]

Ghadir Sadeghi, Department of Mathematics and Computer Sciences, University of Hakim Sabzevari, Sabzevar, Iran e-mail: [email protected]

140 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

TERNARY N-WEAK AMENABILITY OF JB-TRIPLES

Mohsen Niazi, Mohammad Reza Miri and Hamid Reza Ebrahimi Vishki

Abstract One of the noteworthy amenability problems, is to exploring any relationship between the amenability of an algebra structure and its bidual space. In this paper we consider this issue in the context of triple systems and prove that a JB-triple is ternary n-quasi-weakly amenable whenever its bidual space is ternary n-weakly amenable.

2010 Mathematics subject classiﬁcation: Primary 17C65, Secondary 17A40, 46H70, 46H25. Keywords and phrases: JB-triple, Banach ternary module, triple derivation, ternary n-weak amenability.

1. Introduction A Jordan triple is a complex vector space E, with a triple product π : E × E × E → E, π(a, b, c) = [a, b, c], (a, b, c ∈ E) which is bilinear and symmetric in the outer variables and conjugate linear in the middle one satisfying the so-called Jordan identity: [a, b, [c, d, e]] = [[a, b, c], d, e] − [c, [b, a, d], e] + [c, d, [a, b, e]], (a, b, c, d, e ∈ E). When E is a Banach space and π is continuous, we say that E is a Jordan Banach triple system (JB-triple system for short). For example a Banach ∗-algebra is a JB-triple , , = 1 ∗ + ∗ with respect to the triple product [a b c] 2 (ab c cb a). A JB∗-triple is a JB-triple system E satisfying the following axioms: (1) For any a in E the mapping x 7→ [a, a, x] is a hermitian operator on E with non-negative spectrum; (2) ∥[a, a, a]∥ = ∥a∥3 for all a in A. A well-known construction due to Dineen shows that the bidual of a JB∗-triple is a JB∗-triple. Instead of this method, which only applicable for JB∗-triples and relies on the so-called local reﬂexivity principle, by mimicking Arens’ method and using the Aron-Berner extensions (as described in [3]), we construct a concrete extension π∗∗¯∗∗¯ of the triple product π for a JB-triple system E. An easy veriﬁcation reveals that π∗∗¯∗∗¯ is bilinear in the outer variables and conjugate linear in the middle one. Further, it can be

141 M. Niazi, M. R. Miri and H. R. E. Vishki readily checked that (E∗∗, π∗∗¯∗∗¯ ) is a JB-triple system if and only if π∗∗¯∗∗¯ is symmetric in the outer variables. When this property holds, the JB-triple system E is called regular. It is easy to check that E is regular in the case where π∗∗¯∗∗¯ is separately weak∗ continuous, and it is worth to note that every JB∗-triple enjoys this property. Similarly we can extend the triple product π to a triple product π[n] on the (2n)-dual E(2n) of a JB-triple system E, where π[n] is defined inductively by the following formulae: π[1] = π∗∗¯∗∗¯ , π[n+1] = π[n]∗∗¯∗∗¯ , (n ∈ N). It is again worth to mention that every JB∗-triple is permanently regular, that is (E(2n), π[n]) is a JB∗-triple for each n ∈ N. In the following we present a module structure for Jordan triples. According to the conjugate linearity of the triple product of a Jordan triple in the middle variable, it is reasonable to have two types of module structures on Jordan triples (cf. [2, Subsection 2.2] and [4, Definition 3.1]). Definition 1.1. Let E be a Jordan triple and X be a complex vector space. We consider the following mappings and axioms:

π1 : X × E × E → X, π1(x, a, b) = [x, a, b]1;

π2 : E × X × E → X, π2(a, x, b) = [a, x, b]2;

π3 : E × E × X → X, π3(a, b, x) = [a, b, x]3.

(1) π1 is linear in the first and second variables and conjugate linear in the third variable. π2 is conjugate linear in each variable. π3 is conjugate linear in the first variable and linear in the second and third variables. ′ (1) Each of the mappings π1, π2 and π3 is linear in the first and third variables and conjugate linear in the second variable.

(2) [x, b, a]1 = [a, b, x]3, and [a, x, b]2 = [b, x, a]2 for every a, b ∈ E and x ∈ X.

(3) Let [·, ·, ·] denote any of the mappings [·, ·, ·]1, [·, ·, ·]2, [·, ·, ·]3 or the triple product of E. Then the (Jordan) identity [a, b, [c, d, e]] = [[a, b, c], d, e] − [c, [b, a, d], e] + [c, d, [a, b, e]], holds for every a, b, c, d, e, where one of them is in X and the other ones are in E.

When the mappings π1, π2 and π3 satisfy the axioms (1), (2) and (3), X is called a ternary E-module of type (I) and when they satisfy the axioms (1)′, (2) and (3), X is called a ternary E-module of type (II).

When E is a JB-triple system, X is a Banach space and the module actions π1, π2 and π3 are continuous we say that X is a Banach ternary E-module. Proposition 3.2 in [4] shows that the dual space of a Banach ternary E-module of type (I) (respectively, (II)) is a Banach ternary E-module of type (II) (respectively, (I)).

142 Ternary n-Weak Amenability of JB-triples

As every JB-triple system E is a Banach ternary E-module of type (II), under its own triple product as module actions, the just quoted proposition shows that E∗ is a Banach ternary E-module of type (I) and E∗∗ is a Banach ternary E-module of type (II), . . . etc. This procedure shows that the iterated dual space E(n) is a Banach ternary E-module of type (I) whenever the integer n is odd and is a Banach ternary E-module of type (II) whenever the integer n is even. Definition 1.2. A ternary derivation from a JB-triple system E into a Banach ternary E-module of type (I) (respectively, (II)) X is a continuous, conjugate linear (respectively, linear) mapping D : E → X, satisfying D([a, b, c]) = [D(a), b, c] + [a, D(b), c] + [a, b, D(c)], for every a, b, c in E. The set of all these ternary derivations is denoted by Dt(E, X). Let E be a JB-triple system and X be a Banach ternary E-module. Applying axiom (3) of Definition 1.1, we see that the mapping δ(b, x)(a) = [b, x, a] − [x, b, a], (a ∈ E) where b ∈ E and x ∈ X, is a ternary derivation. A finite sum of these derivations is called a ternary inner derivation. The set of all ternary inner derivations from E to X is denoted by Innt(E, X). ∗ The JB-triple system E will be called ternary weakly amenable if Dt(E, E ) = ∗ Innt(E, E ). We can extend this concept for any iterated dual space of E. For (n) any integer n ∈ N, we say that E is ternary n-weakly amenable if Dt(E, E ) = (n) ∗∗ (n+2) ∗∗ (n+2) Innt(E, E ). Since δ(b , e ): E → E , (n ∈ N), is a ternary derivation for every b∗∗ ∈ E∗∗ and e(n+2) ∈ E(n+2), it is not hard to see that the mapping ∗ ◦δ ∗∗, (n+2) → (n) (n−1) → (n+1) Jn−1 (b e ): E E is also a ternary derivation, where Jn−1 : E E denote the canonical embedding. We call a finite sum of this type of derivations a ternary quasi-inner derivation. For any n ∈ N, a Jordan triple E is said to be ternary n-quasi-weakly amenable if every continuous ternary derivation D : E → E(n) is a ternary quasi-inner derivation.

2. Main results The following result is a ternary version of [1, Theorem 2]. Theorem 2.1. Let (E, π) be a regular JB-triple system. If every ternary derivation ∗ ∗∗ ∗∗ ∗ ∗∗ D : E → E satisﬁes D (E ) ⊆ Zl(π ), then ternary weak amenability of E implies ternary quasi-weak amenability of E. In the following result we provide some alternative conditions under which a similar conclusion of Theorem 2.1 can be obtained. Theorem 2.2. Let (E, π) be a regular JB-triple system. If π∗∗¯∗∗¯ (E∗∗, E∗∗, E) ⊆ E and π∗∗¯∗∗¯ (E, E, E∗∗) ⊆ E, then ternary weak amenability of E∗∗ implies ternary quasi-weak amenability of E.

143 M. Niazi, M. R. Miri and H. R. E. Vishki

Corollary 2.3. Let E be a regular JB-triple system such that E is a triple ideal of E∗∗. Then ternary weak amenability of E∗∗ implies ternary quasi-weak amenability of E. An analogue of Theorem 2.1 can be obtained for the iterated dual spaces of E. Surprisingly the restrictive condition of Theorem 2.1 can be dropped for n ≥ 2. This result is a ternary version of [1, Theorem 1]. Theorem 2.4. Let E be a permanently regular JB-triple system. Then for every integer n ≥ 2, ternary n-weak amenability of E∗∗ implies ternary n-quasi-weak amenability of E.

References [1] S. Barootkoob, H.R. Ebrahimi Vishki, Lifting derivations and n-weak amenability of the second dual of a Banach algebra, Bull. Austral. Math. Soc. 83 (2011) 122–129. [2] T. Ho, A.M. Peralta and B. Russo, Ternary weakly amenable C∗-algebras and JB∗-triples, Quart. J. Math. 64 (2013), 1109–1139. [3] A.A. Khosravi and H.R. Ebrahimi Vishki, Aron-Berner extensions of trilinear maps with application to the biduals of a JB∗-triple, Preprint. [4] M. Niazi, M.R. Miri and H.R. Ebrahimi Vishki, Ternary Weak Amenability of Bidual of a JB∗−Triple, Banach J. Math. Anal. to appear 2017.

Mohsen Niazi, Department of Mathematics, University of Birjand, Birjand, Iran e-mail: [email protected]

Mohammad Reza Miri, Department of Mathematics, University of Birjand, Birjand, Iran e-mail: [email protected]

Hamid Reza Ebrahimi Vishki, Department of Pure Mathematics, Centre of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, Mashhad, Iran e-mail: [email protected]

144 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

CREATION AND ANNIHILATION OPERATORS ON DE SITTER MANIFOLD

Ardeshir Rabeie

Abstract In this paper by using the Kirillov orbit method, we show that the phase space for a scalar massive particle ∗ 1 1 on 1+1-de Sitter space is a cotangent bundle T (S ) which is isomorphic to the complex circle S C. The eigenstates in associated Hilbert space which are the coherent states, are obtained from the heat kernel of complex circle. By these states and the Berezin integral quantization, we obtain the creation and annihilation operators on de Sitter.

2010 Mathematics subject classiﬁcation: Primary 34B45, Secondary 83D05.

Keywords and phrases: Coherent states, orbit, quantization.

1. Introduction

Study of the simple harmonic oscillator in quantum mechanics is one of the most important application of the harmonic analysis in physics. The phase space of this motion is isomorphic to C. For each point of this space we associate a vector in Hilbert space which have the three properties of Coherent states(CS). By using of this vector and the Berezin integral quantization, we can obtain the creation and annihilation operators for simple harmonic oscillator. Another application of harmonic analysis is in general relativity. The general relativity is known by Einstein’s equations in Minkowskian space (space-time):

1 Rµν − gµν − Λ gµν = −8πGTµν. (1) 2

145 Rabeie

In this equation, gravitational force is considered as curvature of space-time. The de

Sitter metric (gµν) is one of the solutions of this equation with null energy-momentum

(Tµν = 0) and positive cosmological constant (Λ > 0). This metric is visualized by a hyperboloid embedded in a ﬁve-dimensional Minkowski space which is known as 1+3- de Sitter space-time. The quantum calculation for a movement particle on this space is very diﬃcult and therefore we limit our discussion on 1+1-de Sitter space. In this work, we obtain the creation and annihilation operators for simple harmonics oscillator and a movement massive particle on 1+1-de Sitter space by Berezin quantization method [3]: ∫

O f = f (χ)|χ >< χ| µ(dχ) ,, X where µ(dχ) and |χ > are respectively, the relevant measure on phase space(X) and CS. Also, f (χ) is a classical observable (a function on phase space) .

2. Creation and annihilation operators for simple harmonics oscillator

The phase space of one particle which doing simple harmonics oscillation is isomorphic to the complex space C. Correspond to each point of this space (z = √1 (q + ip) ∈ C), we can introduce a vector in Hilbert space. This vector is called 2 the standard coherent state and given by [4]:

∑∞ n − |z|2 z |z >= |q, p >= e 2 √ |n > . (2) n=0 n!

These states satisfy the resolution of the unity :

∫ ∑∞ d2z |z >< z| = |n >< n| = Id, (3) C π n=0

d2z where π is lebesgue measure and {|n >} are orthonormal basis in Fock representation. Usually, having the resolution of the unity is equivalence to doing the Berezin’s quantization on harmonics oscillator. Therefore, from eq. (1) we can obtain the

146 Creation and annihilation operators on de Sitter manifold creation and annihilation operators as follows: ∫ ∑∞ d2z √ Oz = z |z >< z| = n + 1|n + 1 >< n|, (4) C π n=0 ∫ ∑∞ d2z √ Oz = z |z >< z| = n|n − 1 >< n| . (5) C π n=0

3. Creation and annihilation operators on 1+1-de Sitter

The co-adjoint orbit is mathematically a phase space for a Hamiltonian system whose group G is the symmetric group. Moreover, for all simple or semi-simple Lie groups, one can identify the Lie algebra and its dual. Therefore, for de-Sitter group which is a simple group, the adjoint orbits of movement massive particle on 1+1-de Sitter is the phase space. On the other hand, the symmetric covering group for 1+1-de Sitter space is group SU(1, 1) that is given by [1]:        i θ   ψ ψ   φ φ   e 2 0   cosh sinh   cosh isinh  , ∋ =    2 2   2 2  . SU(1 1) g  θ      (6)  −i   ψ ψ   φ φ  0 e 2 sinh cosh −isinh cosh | {z } | 2 {z 2 } | 2{z 2 } “space translation " “time translation " “Lorentz transformation " Group SU(1, 1) is simple and one can show that the associated phase space is identiﬁed with cotangent space T ∗(S 1). This cotangent bundle is isomorphic to the complex 1 circle “S C "(see [2] for m = r = ℏ = 1): { } −→ −→ sinh(p)−→ S 1 = a = cosh(p) x + i p ∈ C2 = {a = cos(β + ip) , a = sin(β + ip)} , C p 1 2 (7) β, ∗ 1 2 = 2 = , 2 = 2 + 2 where the ( p) play the role of pair varieties of T (S ) and a x 1 p p1 p2.

1 In other words, we must construct our CS on S C. For this purpose, we use the heat 1 −→ −→ kernel on complex circle ( i.e. ρτ( a , x ) ) which was presented by Hall-Mitchell [2]. |Ψτ > This kernel is related to CS ( a ) as follows : ∑∞ − θ˜− π 2 1 −→ −→ τ 1 ( 2 n) 1 1 ρ ( a , x ) =< x|Ψ >= √ e 2τ , ⃗a ∈ S , ⃗x ∈ S , τ a πτ C 2 n=−∞

147 Rabeie where τ is a positive real parameter and θ˜ is a complex angle with 0 ≤ Re θ˜ ≤ π. By ∑ ∑ +∞ = +∞ b θ˜ = −→.−→ = the Poisson summation “ n=−∞ f (n) n=−∞ f (n) " and “ cos x a cos(ip) " we obtain : ∑ −→ | x > = e−inβ|n >, n ∑ 2 τ 1 − τn np −inβ |Ψ > = √ e 2 e e |n > , (8) a N n ∑ ∑ N = ϑ p | iτ = −τn2 2np < ∞, | >< | = . where 3( i π ) n e e n n n Id N − p2 µ dβ, dp = √ e τ dβdp Also, by using the equation (1) and the measure “ ( ) 2π πτ ", we can present the creation and annihilation on 1+1-de Sitter: ∫ ∫ ∑ (iβ−p) τ τ −τ(n+ 1 ) O iβ−p = e |Ψ >< Ψ | µ(dβ, dp) = e 2 | n + 1 >< n | (9), e −→ −→ −→ a a ∈ 1 . = x S x p 0 n ∫ ∫ ∑ (−iβ−p) τ τ −τ(n− 1 ) O −iβ−p = e |Ψ >< Ψ | µ(dβ, dp) = e 2 | n − 1 >< n(10)| . e −→ −→ −→ a a ∈ 1 . = x S x p 0 n

4. Conclusion

By using the coherent states and the Berezin quantization method, we showed that the creation and annihilation operators for a massive particle on 1+1-de Sitter space −iβ−p are respectively, the operators Oz and Oz where z = e (like simple harmonics oscillator).

References

[1] A. Rabeie, E. Huguet and J. Renaud, Wick ordering for coherent state quantization in 1 + 1 de Sitter space, Physics Letters A 370, (2007), 123-125. [2] B. Hall and J. J. Mitchell, Coherent states on spheres, J.Math.Phys., (2002), 1211-1236. [3] F. A. Berezin, Quantization, Math.USSR Izvestija, (1974), 1109-1165. [4] J. P. Gazeau, Coherent states in quantum physics, WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim, (2009).

Ardeshir Rabeie, Department of Physics

148 Creation and annihilation operators on de Sitter manifold

Razi University of Kermanshah Kermanshah, Iran e-mail: [email protected]

149 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

WAVELET METHOD FOR TIME-FRACTIONAL CONVECTION-DIFFUSION EQUATIONS

Parisa Rahimkhani∗ and Yadollah Ordokhani

Abstract In this paper, a new method based on the Bernoulli wavelets expansion together with operational matrix of fractional integration is proposed to solve time-fractional partial diﬀerential equations. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically. The suggested method reduces this type of equation to the solution of a algebraic system. They have been used in modeling turbulent ﬂow chaotic dynamics of classical conservation systems.

2010 Mathematics subject classiﬁcation: Primary 35R11, Secondary 34K28, 65M70 . Keywords and phrases: Bernoulli wavelet, time-fractional partial diﬀerential equations, Collocation method, Caputo derivative .

1. Introduction Recently, fractional diﬀerential operators are indisputably found to play a fundamental role in the modeling of a considerable number of phenomena. Because of the nonlocal property of fractional derivative, they can utilize for modeling of memory-dependent phenomena and complex media such as porous media and anomalous diﬀusion.

2. Basic definitions 2.1. The fractional derivative and integral Definition 2.1. The Riemann-Liouville fractional integral operator of order ν ≥ 0 is defined as [4]  ∫  1 t f (s) ν  −ν ds, ν > 0, t > 0, I f (t) =  Γ(ν) 0 (t−s)1 f (t), ν = 0.

Deﬁnition 2.2. Caputo’s fractional derivative of order ν is deﬁned as [4] ∫ ν 1 t f (n)(s) D f t = ν+ − ds, n − < ν ≤ n, n ∈ N. ( ) Γ(n−ν) 0 (t−s) 1 n 1 ∗ speaker

150 P. Rahimkhani and Y. Ordokhani

2.2. Bernoulli wavelet The Bernoulli wavelet ψnm(t) = ψ(k, nˆ, m, t) involve four ar- guments, nˆ = n − 1, n = 1, 2,..., 2k−1, k is assumed any positive integer, m is the degree of the Bernoulli polynomials and t is the normalized time. They are deﬁned on the interval [0, 1) by [3] { k−1 − n n+ 2 β˜ k 1 − , ˆ ≤ < ˆ 1 , 2 m(2 t nˆ) k−1 t k−1 ψ , (t) = 2 2 n m 0, otherwise, where   1, m = 0, β˜ (t) =  √ 1 β , > , m  − m(t) m 0 (−1)m 1(m!)2 α (2m)! 2m and m = 0, 1,..., M−1. Here βm(t) are the well-known Bernoulli polynomials of order m [3].

3. Operational matrix of the fractional integration The Bernoulli wavelets can be expanded into an M terms Bernoulli polynomials as

Ψ2k−1 M×1(t) = Φ2k−1 M×M BM×1(t), where Φ is the transformation matrix of the Bernoulli wavelet to the Bernoulli polynomials [3]. The fractional integral of Bernoulli polynomials vector can be written as IνB(t) = F(ν)B(t), where F(ν) is given in [3]. Therefore, the Bernoulli wavelets operational matrix of the fractional integration P(ν)(IνΨ(t) = P(ν)Ψ(t)), is P(ν) = ΦF(ν)Φ−1.

4. Problem statement and approximation method We consider the following time-fractional convection-diﬀusion equation ∂νu(x, t) ∂u(x, t) ∂2u(x, t) + a(x) + b(x) = f (x, t), 0 ≤ x, t < 1, 0 < ν ≤ 1, (1) ∂tν ∂x ∂x2 with initial and boundary conditions

u(x, 0) = f0(x), u(0, t) = g0(t), u(1, t) = g1(t). (2) For solving this problem, we approximate ∂2+νu(x, t) ≃ ΨT (x)UΨ(t), (3) ∂x2∂tν

151 Wavelet method for time-fractional ... where U = [ui, j]2k−1 M×2k−1 M is an unknown matrix which should be found. By fractional integration of order ν of Eq. (3) with respect to t, we get

∂2u(x, t) ∂2u(x, t) ∂2 f (x) ≃ ΨT x UP(ν)Ψ t + = ΨT x UP(ν)Ψ t + 0 . 2 ( ) ( ) 2 ( ) ( ) 2 (4) ∂x ∂x t=0 ∂x By fractional integration of order 2 of Eq. (3) with respect to x, and using Eq. (2) we have ∂νu(x, t) ∂νg (t) ∂νg (t) ≃ (P(2)Ψ(x))T UΨ(t) − x(P(2)Ψ(1))T UΨ(t) + (1 − x) 0 + x 1 . (5) ∂tν ∂tν ∂tν By fractional integration of order 2 of Eq. (4) with respect to x and considering Eq. (2), we get u(x, t) ≃ (P(2)Ψ(x))T UP(ν)Ψ(t) − x(P(2)Ψ(1))T UP(ν)Ψ(t) + µ(x, t), (6) where µ , = + − − ′ + − + − + + ′ . (x t) g0(t) f0(x) f0(0) x f0(0) x(g1(t) g0(t)) x( f0(1) f0(0) f0(0)) By fractional diﬀerentiation of order 1 of Eq. (6) with respect to x, we obtain ∂u(x, t) ∂µ(x, t) ≃ (P(1)Ψ(x))T UP(ν)Ψ(t) − (P(2)Ψ(1))T UP(ν)Ψ(t) + . (7) ∂x ∂x Replacing Eqs. (4)-(7) in Eq. (1) and collocating this equation at the 2k−1 M zeros of shifted Legendre polynomials P2k−1 M(x) and P2k−1 M(t).

5. Illustrative test problems Example 5.1. Consider the following time-fractional diﬀusion equation ([2]) ∂νu(x, t) ∂2u(x, t) 1 − = 2( t2−ν − 1), 0 ≤ x, t < 1, 0 < ν ≤ 1, (8) ∂tν ∂x2 Γ(3 − ν) where u(0, t) = t2, u(1, t) = 1 + t2, u(x, 0) = x2. The exact solution to this problem is u(x, t) = x2 + t2. By solving this problem for k = 2, M = 1 we obtain the exact solution. Example 5.2. Consider the initial boundary values problem of fractional partial diﬀerential equation of order ν ([1], [5]) ∂νu(x, t) ∂u(x, t) ∂2u(x, t) + x + = 2tν + 2x2 + 2, 0 ≤ x, t < 1, 0 < ν ≤ 1, (9) ∂tν ∂x ∂x2 , = Γ(ν+1) 2ν, , = + Γ(ν+1) 2ν, , = 2. where u(0 t) 2 Γ(2ν+1) t u(1 t) 1 2 Γ(2ν+1) t u(x 0) x , = 2 + Γ(ν+1) 2ν. The exact solution to this problem is u(x t) x 2 Γ(2ν+1) t By solving this problem for k = 2, M = 1 we obtain the exact solution.

152 P. Rahimkhani and Y. Ordokhani

Table 1. Comparison of absolute error for ν = 0.5, t = 0.25 for Example 1. x Re f.[2] Our method J = 1, m = 3 J = 2, m = 2 k = 2, M = 1 0.2 4.4 × 10−3 8.8 × 10−2 0 0.4 5.1 × 10−2 9.8 × 10−2 0 0.6 7.1 × 10−2 3.4 × 10−1 0 0.8 2.8 × 10−2 4.3 × 10−1 0

Table 2. Comparison of absolute error for ν = 0.5, t = 0.5 for Example 2. x Re f.[1] Re f.[5] Our method m = 64 m = 25 k = 2, M = 1 0.3 1.9 × 10−3 2.3 × 10−5 0 0.5 1.0 × 10−6 2.8 × 10−5 0 0.7 1.7 × 10−3 2.0 × 10−5 0 0.9 1.7 × 10−2 4.7 × 10−6 0

References [1] Y. Chen, Y. Wu, Y. Cui, Z. Wang and D. Jin, Wavelet method for a class of fractional convection-diffusion equation with variable coefficients, Journal of Computational Science, 1 (2010) 146−149. [2] S. Irandoust-pakchin, M. Dehghan, S. Abdi-mazraeh and M. Lakestani, Numerical solution for a class of fractional convection-diffusion equations using the flatlet oblique multiwavelets, Journal of Vibration and Control, 20 (2014) 913−924. [3] P. Rahimkhani, Y. Ordokhani and E. Babolian, An efficient approximate method for solving delay fractional optimal control problems, Nonlinear Dyn. 86 (2016) 1649−1661. [4] P. Rahimkhani, Y. Ordokhani and E. Babolian, Fractional-order Bernoulli wavelets and their applications, Appl. Math. Model.40 (2016) 8087−8107. [5] A. Saadatmandi, M. Dehghan and M.R. Azizi, The Sinc-Legendre collocation method for aclass of fractional convection-diffusion equations with variable coefficients, Communications in Nonlinear Science and Numerical Simulation, 17 (2012) 4125−4136.

Parisa Rahimkhani, Department of Mathematics, University of Mathematical Sciences, Tehran, Iran e-mail: [email protected]

Yadollah Ordokhani, Department of Mathematics, University of Mathematical Sciences, Tehran, Iran e-mail: [email protected]

153 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

DRAZIN AND MOORE-PENROSE SPECTRUM

Zahra Rahmati Nasrabad∗ and Asadollah Niknam

Abstract In this paper, we are going to deﬁne and study the Drazin and Moore-Penrose spectrum and obtain some results for Drazin and Moore-Penrose inverses.

2010 Mathematics subject classiﬁcation: Primary 47A55, Secondary 39B52. Keywords and phrases: Drazin spectrum, Moore-Penrose spectrum, Drazin inverse, Moore-Penrose inverse.

1. Introduction An element of a complex Banach algebra A is called regular (or relatively regular) if there is x ∈ A such that axa = a. If a is relatively regular, then it has generalized inverse, which is an element b ∈ A satisfying the equations aba = a and bab = b .A relation between a relatively regular element and its generalized inverse is reﬂexive in the sense that if b is a generalized inverse of a , then a is a generalized inverse of b. We say that an element a ∈ A is Drazin invertible if there is x ∈ A such that xa = ax, ax2 = x, ak = ak+1 x (1) for some nonnegetive integer K. For any Drazin invertible a ∈ A such that x is unique; we write x = aD, and call it the Drazin inverse of a. We write AD for the set of all Drazin invertible elements of A. An element a ∈ A is Moore-Penrose invertible if there exists a x ∈ A such that xax = x, axa = a, (ax)∗ = ax, (xa)∗ = xa. (2) There is at most one element x satisfying above. If a is Moore-Penrose invertible, the unique solution of (1.2) is called the Moore-Penrose inverse of a and is denoted by a†. The set of all Moore-Penrose invertible elements of A is denoted by A†. Let A be a unital C∗- algebra. The Drazin spectrum of an element a ∈ A is the set spDR(a) = {λ ∈ C; λe − a is not Drazin invertible}. Let A be a unital C∗- algebra. The Moore-Penrose spectrum of an element a ∈ A is the set spMP(a) = {λ ∈ C; λe − a is not Moore-Penrose invertible}.

∗ speaker

154 Zahra rahmati nasrabad,Asadollah Niknam

2. Main results Now we state some properties of Drazin and Moore-Penrose spectrums. To achieve our goal, we need to express the following theorems. Theorem 2.1. Consider C∗-algebra A, and a, x ∈ A. Then (i) the equation a = axa and (ax)∗ = ax are equivalent to a = x∗a∗a. (ii) the equation a = axa and (xa)∗ = xa are equivalent to a = aa∗ x∗. (iii) the equation x = xax and (ax)∗ = ax are equivalent to x = xx∗a∗. (iv)the equation x = xax and (xa)∗ = xa are equivalent to x = a∗ x∗ x. Theorem 2.2. Consider C∗-algebra A, and a ∈ A. Then the following statements are equivalent: (i) x ∈ A is the Moore-Penrose inverse of a. (ii)a∗ = xaa∗ and x = xx∗a∗. (iii) a = aa∗ x∗ and x∗ = axx∗. (iv) a∗ = a∗ax and x = a∗ x∗ x. (v) a = x∗a∗a and x∗ = x∗ xa. Theorem 2.3. Let a ∈ A be normal. Then the following are true. (i) a ∈ A† ⇐⇒ a ∈ AD. (ii) if a ∈ A† , then a† is normal and commute with a. Theorem 2.4. An element a of a C∗ -algebra A is Moore-Penrose invertible if and only if a∗a (respectively aa∗) is Drazin invertible. If a ∈ A† , then a† = (a∗a)Da∗ = a∗(aa∗)D. Theorem 2.5. An element of a C∗ -algebra is Moore-Penrose invertible if and only if it is regular. Theorem 2.6. Let a, b ∈ A† , Then

b† − a† = −b†(b − a)a† + (e − b†b)(b∗ − a∗)(a†)∗a† + b†(b†)∗(b∗ − a∗)(e − aa†). (3) 1 Theorem 2.7. If a, b ∈ A† are such that ∥b − a∥ < ∥a†∥−1 and ∥bb† − aa†∥ < 1 then 2 ∥b†∥ ≤ 4∥a†∥. (4)

† Theorem 2.8. Let an, a be nonzero elements of A such that an −→ a in A . Then the following conditions are equivalent † † † † 1. an −→ a anan −→ aa

† † † 2. anan −→ a asup∥an∥ < ∞.

Theorem 2.9. Let a(t) be a C∗ -algebra valued function deﬁned on an interval J such † that 0 , a(t) ∈ A for all t ∈ J and that a(t) is deﬀerentiable at t0 . Then the function

155 Drazin and Moore-Penrose spectrum

† a (t) is diﬀerentiable at t0 if and only if one of the condition of preceding theorem is † ′ † ′ satisﬁed. The derivative (a ) = (a ) (t0) is given by (a†)′ = −a†a′a† + (e − a†a)(a′)∗(a†)∗a† + a†(a†)∗(a′)∗(e − aa†), (5)

∗ † ′ ∗ † ′ where a, a , a , a stand for a(t0), a (t0), a (t0), a (t0), respectively. Theorem 2.10. Let a(t) be a C∗ -algebra valued function deﬁned on an interval J † such that 0 , a(t) ∈ A for all t ∈ J and that a(t) is deﬀerentiable at t0. The following conditions are equivalent. † 1. a (t)is continuous at t0 † 2. a(t)a (t)is continuous at t0 † 3. a (t)a(t)is continuous at t0 there is δ > 0 such that sup ∥a†(t)∥ < ∞. In the following theorems we state two properties of Drazin spectrum. Theorem 2.11. Let A be unital Banach algebra and consider a ∈ A . Then the following statements are equivalent. (i) σ(a) is at most countable. (ii) σDR(a) is at most countable. Theorem 2.12. Let A be a unital Banach algebra and consider a, b ∈ A. Then

σDR(ab) = σDR(ba).

Now by above theorems we stablish a new properties of Moore-Penrose spectrum

Theorem 2.13. spMP is non-empty. Is it true that if we have a Banach algebra A that every element of A has Moore- Penrose inverse then A ≃ C?

References [1] R. A. Lubansky and Koliha-Drazin invertible from a regularity, Math. Proc. Roy. Ir. Acad. 107A (2007), 137-141. [2] E. Boasso, Drazin spectra of Banach space operators and Banach algebra elements, J. Math., 148 (2001). [3] J. J. Koliha, A generalized Drazin inverse, Glasgo Mathematical Journal (1996), 367-81. [4] V. Rakocevitcˇ , On the continuity of the Moore-Penrose inverse in C∗-algebras, Mathematica Montisnigiri2 (1993), 89-92. [5] V. Rakocovic and Y. Wei, A weighted Drazin inverse and applications, Linear Algebra Appl., (2002), 25-39.

Zahra Rahmati Nasrabad, Department of Pure Mathematics, Ferdowsi University of Mashhad

156 Zahra rahmati nasrabad,Asadollah Niknam

Mashhad, Iran e-mail: [email protected]

Asadollah Niknam, Department of Pure Mathematics, Ferdowsi University of Mashhad Mashhad, Iran e-mail: [email protected]

157 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

CYCLIC AMENABILITY OF CERTAIN PRODUCT OF BANACH MODULES

Mohammad Ramezanpour

Abstract Let A and X be Banach algebras and let X be an algebraic Banach A-module. Then the ℓ1-direct sum A × X equipped with the multiplication

(a, x)(b, y) = (ab, ay + xb + xy)(a, b ∈ A, x, y ∈ X) is a Banach algebra, which we denote it by A ▷◁ X. Module extension algebras, Lau product and also the direct sum of Banach algebras are the main examples satisfying this framework. I this note we investigate the cyclic amenability of A ▷◁ X. We then apply the results to the special cases to improve some older results.

2010 Mathematics subject classiﬁcation: Primary 46H20, Secondary 46H25. Keywords and phrases: Banach algebra, derivation, Cyclic amenability.

1. Introduction Let A and X be Banach algebras and let X be a Banach A-module. We say that X is an algebraic Banach A-module if for every x, y ∈ X and a ∈ A,

a(xy) = (ax)y, (xy)a = x(ya), (xa)y = x(ay), ∥ax∥ ≤ ∥a∥∥x∥ and ∥xa∥ ≤ ∥a∥∥x∥.

Then a direct veriﬁcation shows that the direct sum A × X under the multiplication

(a, x)(b, y) = (ab, ay + xb + xy)(a, b ∈ A, x, y ∈ X) and the norm ∥(a, x)∥ = ∥a∥ + ∥x∥ is a Banach algebra which is denoted by A ▷◁ X. Module extension algebras, Lau product and also the direct sum of Banach algebras are the main examples satisfying this framework. Let A be a Banach algebra. A derivation D : A → A∗ is called cyclic derivation if it satisfying the property D(a)(c) + D(c)(a) = 0 for all a, c ∈ A. Clearly inner derivations are cyclic. A Banach algebra is called cyclic amenable if every continuous cyclic derivations D : A → A∗ is inner. This notion was presented by Gronbaek [2]. He investigated the hereditary properties of this concept, found some relations between cyclic amenability of a Banach algebra and the trace extension property of its ideals.

158 M. Ramezanpour

In this note we give general nessesary and suﬃcent conditions for A ▷◁ X to be cyclic amenable. In particular, we improve and extend some recent results on cyclic amenability of module extension algebras, Lau product and also the direct sum of Banach algebras.

2. Main Results Let A be a Banach algebra, and X be a Banach A-bimodule. Then the dual space X∗ of X becomes a dual Banach A-bimodule with the module actions deﬁned by ( f a)(x) = f (ax) and (a f )(x) = f (xa), for all a ∈ A, x ∈ X and f ∈ X∗. A derivation from A into X is a linear mapping D : A → X satisfying D(ac) = D(a)c + aD(c)(a, c ∈ A).

If x ∈ X then dx : A → X deﬁned by dx(a) = ax − xa is a derivation which is called an inner derivation. A derivation D : A → A∗ is said to be cyclic if D(a)(c) + D(c)(a) = 0 for all a, c ∈ A. If every continuous cyclic derivation D : A → A∗ is inner then A is called cyclic amenable. Let A and X be Banach algebras and let X be a Banach A-module. For Banach algebra A ▷◁ X, one can directly checked that the dual (A ▷◁ X)∗ as a Banach (A ▷◁ X)- module enjoys the following module operations: ( f, g)(a, x) = ( f a + g · x, gx + ga), (a, x)( f, g) = (a f + x · g, xg + ag), for all a ∈ A, f ∈ A∗, x ∈ X and g ∈ X∗; where g · x ∈ A∗ is given by (g · x)(a) = g(xa). To clarify the relation between cyclic amenability of A ▷◁ X and that of A and X, we need the next result which characterize the continuous cyclic derivations on A ▷◁ X. Lemma 2.1. D : A ▷◁ X → (A ▷◁ X)∗ is a continuous cyclic derivation if and only if , = − ∗ , + D(a x) (DA(a) DX(x) DX(a) TX(x)) such that, ∗ ∗ (1) DA : A → A and TX : X → X are continuous cyclic derivations. ∗ (2) DX : A → X is a continuous derivation. such that

TX(ax) = DX(a)x + aTX(x), TX(xa) = xDX(a) + TX(x)a.

Moreover, D = d( f,g) is inner derivation if and only if DA = d f , DB = dg and TB = dg are inner derivations. Here we gives general necessary and suﬃcient conditions for A ▷◁ X to be cyclic amenable. Theorem 2.2. [3] The Banach algebra A ▷◁ X is cyclic amenable if and only if

159 Cyclic amenability of certain product of Banach modules

(i) A is cyclic amenable. (ii) If T : X → X∗ is a continuous cyclic derivation and there is a continuous derivation D : A → X∗ such that T(ax) = D(a)x + aT(x), T(xa) = xD(a) + T(x)a. Then T is inner derivation. (iii) If D : A → X∗ is a continuous derivation such that xD(a) = D(a)x = 0. Then ∗ there is g ∈ X such that D = dg and xg = gx. As an immediate consequence we have the following. Theorem 2.3. Suppose that X2 is dense in X. Then A and X are cyclic amenable if and only if A ▷◁ X is cyclic amenable. Let X be a Banach A-module and deﬁne xy = 0 for all x, y ∈ X. Then X is a algebraic Banach A-module and A ▷◁ X is the module extension algebra A ⋉ X, as introduced in [5]. Applying Theorem 2.2, for A ⋉ X we arrive at the following result. Corollary 2.4. A module extension Banach algebra A ⋉ X is cyclic amenable if and only if (i) A is cyclic amenable. (ii) The only A-module morphism T : X → X∗ such that T(y)(x) + T(x)(y) = 0 for all x, y ∈ X, is zero. (iii) H1(A, X∗) = {0}; that is, every continuous derivation from A into X∗ is inner. Let A and B be Banach algebras and let θ be a nonzero multiplicative linear functional on A. Then B endowed with the module operations ab = ba = θ(a)b is an algebraic Banach A-module and A ▷◁ B is the θ-Lau product Banach algebra A θ × B. This product was introduced by Lau for certain class of Banach algebras and followed by Sangani Monfared [4] for the general case. As another consequence of Theorem 2.2, we use it for the θ-Lau product Banach algebra A θ × B. Then we get the following characterization for cyclic amenability of A θ × B which extends the related result in [1].

Corollary 2.5. The θ-Lau product Banach algebra A θ× B is cyclic amenable if and only if (i) A and B are cyclic amenable. (ii) Either ⟨B2⟩ = B or every continuous point derivation at θ is zero. Let A and B be Banach algebras. Then B endowed with the module operations ab = ba = 0 is an algebraic Banach A-module and A ▷◁ B is the direct product Banach algebra A ⊕ B. Applying Theorems 2.2 for A ⊕ B, we get the next result. Corollary 2.6. The direct product A ⊕ B is cyclic amenable if and only if (i) Both A and B are cyclic amenable. (ii) A2 is dense in A or B2 is dense in B.

160 M. Ramezanpour

Acknowledgments The Financial Support of the Research Council of Damghan University with the Grant Number 93/Math/123/225 is also Acknowledged.

References [1] E. Ghaderi, R. Nasr-Isfahani and M. Nemati, Some notions of amenability for certain products of Banach algebras, Colloq. Math. 130 (2) 2013 147–157. [2] N. Gronbæk¨ , Weak and cyclic amenability for noncommutative Banach algebras, Proc. Edinburgh Math. Soc. 35 (2) (1992) 315–328. [3] M. Ramezanpour, Some cohomological properties of certain product of Banach modules, submitted. [4] M. Sangani Monfared, On certain products of Banach algebras with applications to harmonic analysis, Studia Math. 178 (3) (2007) 277–294. [5] Y. Zhang, Weak amenability of module extensions of Banach algebras, Trans. Amer. Math. Soc. 354 (10) (2002) 4131–4151.

Mohammad Ramezanpour, School of Mathematics and Computer Science, Damghan University, Damghan 41167, Iran. e-mail: [email protected]

161 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

ON THE CONTINUOUS WAVE PACKET FRAMES

Atefe Razghandi∗ and Ali Akbar Arefijamaal

Abstract b Consider the wave packet group GΘ = H ⋉Θ (K × K) where H and K are locally compact groups, K is also abelian and Θ : H → Aut(K × Kb) is a continuous homomorphism. In this article, we extend the notation 2 of Zak transform on L (GΘ) and introduce a frame condition to generate wave packet frames on GΘ.

2010 Mathematics subject classiﬁcation: Primary 43A32, Secondary 43A25. Keywords and phrases: Wave packet frame, Zak transform, Locally compact abelian group.

1. Introduction The Zak transform has been used in applications in physics and signal theory [5]. An approach to deﬁne the Zak transform on semidirect product groups of the form Gτ = H ⋉τ K introduced in [3]. Many locally compact groups are non-abelian although they can be cosidered as semidirect product of locally compact groups. Dual of non- abelian locally compact groups is based on the dual group. But the dual of non abelian locally compact groups is considerably more intricate and consists of all classes of equivalence of its irreducible representations [2]. In this paper, by using a version of duality for semidirect product group, we extend some classical results from abelian case to non-abelian semidirect product groups which are compatible and useful in application. Throughout this article, we assume Gτ = H⋉τ K is the semi-direct product group of locally compact group H and locally compact abelian group K. The mapping h 7→ τh is a homomorphism of H into the group of automorphisms of K such that the mapping (h, k) 7−→ τh(k) from H × K onto K is continuous. The group law is given by ′ ′ ′ ′ (h, k).(h , k ) = (hh , kτh(k )), ((h, k) ∈ Gτ).

Then Gτ is a (not necessarily abelian) locally compact group. Moreover, the left Haar , = δ measure of Gτ is dmGτ (h k) (h)dmH(h)dmK(k), where mH and mK are the left Haar measures of H and K, respectively and the positive continuous homomorphism δ on H is given by (15.29 of [4])

dmK(k) = δ(h)dmK(τh(k)). (1) ∗ speaker

162 A. Razghandi and A. Arefijamaal

It is worthwhile to mention that the above τ-dual action on K induces such τ-dual on b b K. More precisely, we can deﬁne homomorphism bτ : H → Aut(K) via h 7→ bτh, given by

bτh(ω):= ωh = ω ◦ τh−1 (2) b for all ω ∈ K, where ωh(k) = ω(τh−1 (k)) for all k ∈ K. Also, the continuity of the homomorphism bτ : H → Aut(Kb) given in (2) guaranteed by Theorem 26.9 of [4]. b Hence, the semi-direct product Gbτ = H ×bτ K is a locally compact group with the left Haar measure , ω = δ −1 ω . dmGbτ (h ) (h) dmH(h)dmKb( ) (3) b Furthermore, the continuous action (h, k) 7→ τh(k) induces a mapping Θ : H → (K×K) given by h 7→ Θh where Θh(k, ω) = (τh(k), ωh). (4)

In [3], it is shown that Θ is a well-deﬁned homomorphism and (h, k, ω) 7→ Θh(k, ω) is b continuous. Thus Θ induces the semi direct product group GΘ = H ⋉Θ (K ×K) which is called the wave packet group. It is a locally compact group with the left Haar measure , , ω = ω dmΘ(h k ) dmH(h)dmK(k)dmKb( ) and the modular function ∆ , , ω = ∆ , GΘ (h k ) H(h) b for all (h, k, ω) ∈ GΘ, for more details see Theorem 3.2 of [3]. For (k, ω) ∈ K × Kand 2 b 2 b 2 b f ∈ L (K × K) the translation operator T(k,ω) : L (K × K) → L (K × K) is deﬁned by

−1 T(k,ω) f (x, ξ) = f (xk , ξω). H { }∞ ⊆ H H Let be a separable Hilbert space. A sequence fi i=1 is called a frame for if there are constants A, B > 0 satisfying ∑∞ 2 2 2 A ∥ f ∥ ≤ | ⟨ f, fi⟩ | ≤ B ∥ f ∥ . k=0 { }∞ If fi i=1 is a frame, the frame operator is deﬁned by ∑∞ S : H → H, S f = ⟨ f, fi⟩ fi. k=0 The series converging unconditionally and S is a bounded, invertible, and self-adjoint operator. This leads to the frame decomposition ∑∞ ⟨ ⟩ −1 −1 f = S S f = f, S fi fi. i=1

163 On the continuous wave packet frames

2. Main result The Zak transform was first introduced and used in 1950 by Gelfand for a problem in differential equations. Weil defined this transform on arbitrary locally compact abelian group with respect to arbitrary closed subgroup. Subsequently the Zak transform was redicovered in quantum mechanic by Zak. Finally the continuous Zak transform for locally compact groups is established in [1]. b Definition 2.1. The Zak transform of f ∈ Cc(Gθ) is defined on K × K by ∫ , ω, = , , γ ω γ γ . Zc f (h x) f (h y ) (y) (x)dmk(y)dmKb( )

c 2 It is shown that Zc : Cc(Gθ) → Cc(Gθ) is an isometry in L -norms and so that it can 2 2 be uniquely extended into the Zak transform Zc : L (GΘ) → L (GΘb).

2 2 b If f = f1 ⊗ g1 that f1 ∈ L (H) and g1 ∈ L (K × K) then 2 2 b 2 b f1 ⊗ g1 ∈ L (H) ⊗ L (K × K) = L (H × K × K). Also ∫ ∫ 2 2 ∥Z f ∥ = | f h g y, γ ω y γ x dm y dmb γ | dm h dmb ω dm x c 2 H×K×Kb 1( ) 1( ) ( ) ( ) K ( ) K ( ) H( ) K ( ) K ( ) ∫ ∫ ∫ = | |2 | , γ ω γ γ |2 ω f1(h) dmH(h) ×b ×b g1(y ) (y) (x)dmK (y)dmKb( ) dmKb( )dmK (x) H K K K K 2 = ∥ f ∥2 . gb . 1 2 1 2

Therefore Zc f = f1 ⊗ gb1 and

∥Zc f ∥2 = ∥ f1∥2∥gb1∥2 = ∥ ∥ ∥ ∥ f1 2 g1 2 = ∥ ⊗ ∥ f1 g1 2 = ∥ f ∥2. Let H be a locally compact group and K an locally compact abelian group with the dual group Kb . For h ∈ H and f ∈ L2(K × Kb) deﬁne the dilation of f by h via

Dh f (k, ω) = f (τh−1 (k),bτh−1 (ω)).

b 2 b Theorem 2.2. Let GΘ = H ⋉θ (K × K) and ψ ∈ L (K × K). Then F (ψ) = {T(k,ω)Dhψ;(h, k, ω) ∈ Gθ}∫ is a continuous frame with bounds A, B if and only if 2 ≤ γψ ≤ γψ = | ψ ξ, | . A B a.e. where H (ZcDh )( y) dmH(h)

References [1] A. Areﬁjamaal, The continuous Zak transform and generalized Gabor frame, Mediterr. J. Math. 10(2013) 353-365. [2] G. B. Folland, , A course in abstract harmonic analysis, CRC press, Boca Raton,1995.

164 A. Razghandi and A. Arefijamaal

[3] A. Ghaani Farashahi, Abstract harmonic analysis of wave packet transform on locally compact abelian groups, Banach. J. Math. Anal. to appear. [4] E. Hewitt, K. A. Ross, Abstract Harmonic Analysis, Springer-Verlag, 1969. [5] A. J. E. M. Janssen, , Bargmann transform, Zak transform, and coherent ststes, J. Math. phys. 23 (1982) 720-731.

Atefe Razghandi, Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran e-mail: [email protected]

Ali Akbar Arefijamaal, Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran e-mail: areﬁ[email protected]

165 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

NEW PRODUCT ON BANACH SPACE LP(G)

Fatemeh Roohi Afrapoli∗ and Fatemeh Esmaeelzadeh

Abstract Let G be a locally compact ablian (LCA) group and ϕ : G −→ G is a topological isomorphism. For 1 ≤ p < ∞, Lp(G) is denoted the usual Lebesgue space with respect to the measure on G. We deﬁne a new product on Banach space Lp(G) and investigate its properties. As a result the new deﬁned product on L2(G) the same as inner product on it.

2010 Mathematics subject classiﬁcation: Primary 43A05, Secondary 43A15. Keywords and phrases: Uniform lattic, ϕ, p-product, Weil’s formula.

1. Introduction and Preliminaries Let G be a locally compact abelian (LCA) group. It is well known that such a group possesses a Haar measure that is unique up to a multiplication by constants. Suppose that H is a close subgroup of G, G/H as a quotient group. Let dx, dh and dx˙ be the Haar measures on G, H and G/H, respectively, such that ∫ ∫ ∫ f (x)dx = f (xh)dhdx˙, f ∈ L1(G). G G/H H This formula is known as Weil’s formula.

In this note, we deﬁne a new product on Banach space Lp(G), then obtain a type norm on Lp(G) and we show that some inequalities with respect the new product.

2. main result Throughout this paper let G be a second countable locally compact abelian group. In this case, we always have a uniform lattice for G. Consider L is a uniform lattice in G, and ϕ : G −→ G is a topological isomorphism, then G/ϕ(L) is a (LCA) group with the Haar measure dx˙. For f, g ∈ Lp(G), 1 ≤ p < ∞, it is well-known that f gp−1 ∈ L1(G). Now by using Weil’s formula we have

∗ speaker

166 F. Author, S. Author and T. Author

∫ ∑ ∫ ∑ | f gp−1(xϕ(k−1))|dx˙ = | f gp−1(xϕ(k−1))|dx˙ G/ϕ(L) G/ϕ(L) k∈L ∫ ϕ(k)∈ϕ(L) = | f gp−1(x) | dx ∫G ∫ ≤ ( | f (x) |p dx)1/p( | gp−1(x) |q dx)1/q G G p−1 ≤ ∥ f ∥p∥g ∥q, = ϕ , /ϕ where x˙ x (L) dx˙ is the Haar∑ measure on G (L) and q is the conjugate exponent ∈ , p−1 ϕ −1 to p. Thus for almost all x G k∈L f g (x (k )) converges. Now we define a new product for f, g ∈ Lp(G) where 1 < p < ∞. Definition 2.1. Let f, g ∈ Lp(G) ( 1 < p < ∞). The ϕ, p-product of f, g is defined by ∑ p−1 −1 [ f, g]ϕ,p(x) = f g (xϕ(k )), k∈L for all x ∈ G. We define ϕ, p-norm of f as ∥ ∥ = | |, | | 1/p . f ϕ,p(x) [ f f ]ϕ,p (x) Also for p = ∞ we define: ∑ −1 ∥ f ∥ϕ,∞(x) = supx˙∈G/ϕ(L) | f (xϕ(k )) | . ϕ(k)∈ϕ(L)

Note that [ f, g]ϕ,p is ϕ(L)-periodic, i.e. for every l ∈ L and x ∈ G, [ f, g]ϕ,p is constant on ϕ(L)-cosets. So one may consider the ϕ, p-product of f, g ∈ Lp(G) as a p p 1 mapping [., .]ϕ,p : L (G) × L (G) −→ L (G/ϕ(L)) deﬁned by ∑ p−1 −1 [ f, g]ϕ,p(x ˙) = f g (xϕ(k )), k∈L for all x˙ ∈ G/ϕ(L).

Example 2.2. In the above definition suppose that G = R, L = Z and fix a ∈ R+. ϕ R −→ R, ϕ = Then : given by (x) ax is a topological isomorphism∑ and the mapping ., . 2 R × 2 R −→ 1 , , = − [ ]ϕ,2 : L ( ) L ( ) L ([0 1]), defined by [ f g]ϕ,2 n∈Z f g(x na) is inner product of f and g introduced by Casazza and Lammers in [1]. If ϕ is the identity function on R , p = 2 then the ϕ, p-product is exactly the product defined by Ron and Shen [5]. It is obvious that for Lp(G) (1 < p < ∞), i.e. for f, g ∈ Lp(G), c ∈ C, we have:

(1) ∥ f ∥ϕ,p = 0 if and only if f = 0 a.e.

167 New product on Banach space Lp(G)

(2) ∥c f ∥ϕ,p = c∥ f ∥ϕ,p. 1 (3) ∥ f + g ∥ϕ,p≤∥ f ∥ϕ,p + ∥ g ∥ϕ,p . (Triangle Inequality) . Proposition 2.3. Let f, g, h ∈ Lp(G) where 1 ≤ p < ∞ and q is the conjugate exponent to p. Then p−1 ′ | [| f |, | g |]ϕ,p |≤∥ f ∥ϕ,p∥ g ∥ϕ,q (Holder sinequality) (1) Proof. Put gp−1 = k. It is clear k ∈ Lq(G). We have ∑ ∥ ∥q = | |, | | = | || q−1 | ϕ −1 . k ϕ,q [ k k ]ϕ,q k k (x (k )) k∈L

If ∥ f ∥ϕ,p= 0 or ∥ k ∥ϕ,q= 0(since f = 0 or k = 0 a.e.) or if ∥ f ∥ϕ,p= ∞ or ∥ k ∥ϕ,q= ∞, the result is trivial. Moreover, we observe that if

| [| f |, | k |]ϕ,p |≤∥ f ∥ϕ,p∥ k ∥ϕ,q holds for a particular f and k, then it also holds for all scalar multiples of f and k, for if f and k are replaced by a f and bk, both sides of (1) change by a factor of | ab |. It therefore suﬃces to prove that (1) holds when ∥ f ∥ϕ,p=∥ k ∥ϕ,q= I, where I is an identity operator. To this end, by using Lemma 6.1 in [2] with a =| f p(xϕ(l−1)) |, b =| kq(xϕ(l−1)) | and λ = 1/p to obtain | f (xϕ(l−1)) | | k(xϕ(l−1)) |≤ 1/p | f p(xϕ(l−1)) | +1/q | kq(xϕ(l−1)) |,

| f | | k | (xϕ(l−1)) ≤ 1/p | f f p−1(xϕ(l−1)) | +1/q | kkq−1(xϕ(l−1)) |, ∑ ∑ | f | | k | (xϕ(l−1)) ≤ 1/p( | f | | f p−1 | (xϕ(l−1)) l∈L l∑∈L + 1/q( | k | | kq−1 | (xϕ(l−1)), l∈L and so ∑ ∑ | | f | | k | (xϕ(l−1)) | ≤ 1/p( | f | | f p−1 | (xϕ(l−1))) l∈L l∑∈L + 1/q( | k | | kq−1 | (xϕ(l−1))). l∈L Thus ∑ −1 | | f | | k | (xϕ(l )) ≤ 1/p[| f |, | f |]ϕ,p + 1/q[| k |, | k |]ϕ,q | l∈L = / ∥ ∥p + / ∥ ∥q 1 p f ϕ,p 1 q k ϕ,q = (1/p + 1/q)I = I

=∥ f ∥ϕ,p∥ k ∥ϕ,q .

1 Minkoweski Inequality

168 F. Author, S. Author and T. Author

Therefore ∑ −1 | | f || k | (xϕ(l )) |≤ ∥ f ∥ϕ,p∥ k ∥ϕ,q . l∈L Now put k = gp−1. So we have p−1 | [| f |, | g |]ϕ,p | ≤ ∥ f ∥ϕ,p∥ g ∥ϕ,q . □

Proposition 2.4. Let f, g ∈ Lp(G) for 1 ≤ p < ∞ and c ∈ C. Then the following properties hold: ∫ ∫ p−1 f, g ϕ, xdx = f g x dx. (i) G/ϕ(L)[ ] p ˙ ˙ G ( ) 1/p−1 (ii) [c f, g]ϕ,p = c[ f, g]ϕ,p = [ f, c g]ϕ,p (iii) ∫[ f + h, g]ϕ,p = [ f, g]ϕ,p + [h, g]ϕ,p ∥ f ∥p x dx = ∥ f ∥p (iv) G/ϕ(L) ϕ,p( ˙) ˙ Lp(G) If f, g ∈ L2(G), we get ∫

[ f, g]ϕ,p(x ˙)dx˙ =< f, g >L2(G) . G/ϕ(L) (for more details see [5]).

References [1] P. G. Casazza and M. C. Lammers,Bracet products for Weyi-Heisenberg frames, Advances in Gobor analysis. Appl. Numer. Harmon. Anal., Birkhauser Boston, (2003), 71-98. [2] G. B. Folland, Real analysis, John Viley, New York| (1984). [3] G. B. Folland, A course in abstract harmonic analysis, CRC Press (1995). [4] R. Q. Jia and C. A. Micchelli, Using the renement equation for the construction of pre-wavelets II: powers of two, Curves and Surfaces (P. J. Laurent, A. Le Mt’ehautt’e, and L. L. Schumaker, eds.), Academic Press, New York, (1991). [5] sc R. A. Kamyabi Gol and Raisi R. Tosi, Bracket products on locally compact Abelian groups, J. Sci. Islam. Repub. Iran, (2008).

Fatemeh Roohi Afrapoli, Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran e-mail: [email protected]

Fatemeh Esmaeelzadeh, Department of Mathematics, Bojnourd Branch, Islamic Azad University, Bojnourd, Iran e-mail: [email protected]

169 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

MARTINGALE INEQUALITIES IN NONCOMMUTATIVE PROBABILITY SPACES

Ghadir Sadeghi

Abstract We establish some inequalities for martingales in the framework of noncommutative probability spaces.

2010 Mathematics subject classiﬁcation: Primary 46L53, Secondary 47A30. Keywords and phrases: (Noncommutative) probability space; trace; noncommutative martingale.

1. Introduction A von Neumann algebra M on a Hilbert space with unit element 1 equipped with a normal faithful tracial state τ : M → C is called a noncommutative probability space. We denote by ≤ the usual order on self-adjoint part Msa of M. For each self-adjoint operator x ∈ M, there exists a unique spectral measure E as a σ-additive mapping with respect to the strong operator topology from the Borel σ-algebra B(R) of R into the set of all orthogonal projections such that∫ for every Borel function f : σ(x)∫ → C the operator f (x) is deﬁned by f (x) = f (λ)dE(λ), in particular, χ = λ = | | ∈ M B(x) B dE( ) E(B). Of course, the modules x of x can be deﬁned by |x| = (x∗ x)1/2 by utilizing the usual functional calculus. The inequality −t x Prob(x ≥ t):= τ(χ[t,∞)(x)) ≤ e τ(e ) . (1) is known as exponential Chebyshev inequality in the literature. The celebrated Golden–Thompson inequality [1] states that for any self-adjoint elements y1, y2 in a noncommutative probability space M, + / / τ(ey1 y2 ) ≤ τ(ey1 2ey2 ey1 2) (2) and + τ(ey1 y2 ) ≤ τ(ey1 ey2 ). (3)

For p ≥ 1, the noncommutative Lp-space Lp(M) is deﬁned as the completion of M p 1/p with respect to the Lp-norm ∥x∥p := (τ(|x| )) . Further, for a positive element x ∈ M, it holds that ∫ ∞ p p−1 ∥x∥p = pt τ(χ[t,∞)(x))dt. (4) 0

170 F. Author, S. Author and T. Author

The commutative cases of discussed spaces are usual Lp-spaces and the Schatten p- classes Cp. For further information we refer the reader to [1, 2] and references therein. Let N be a von Neumann subalgebra of M. Then there exists a normal contraction positive mapping projecting EN : M → N satisfying the following properties: (i) EN(axb) = aEN(x)b for any x ∈ M and a, b ∈ N; (ii) τ ◦ EN = τ. Moreover, EN is the unique mapping satisfying (i) and (ii). The mapping EN is called the conditional expectation of M with respect to N. Let N ⊆ A j (1 ≤ j ≤ n) be von Neumann subalgebras of M. We say that the A j are order independent over N if for every 2 ≤ j ≤ n, the equality

E j−1(x) = EN(x) holds for all x ∈ A j, where E j−1 is the conditional expectation of M with respect to the von Neumann subalgebra generated by A1,..., A j−1; cf. [1, 3]. A ﬁltration of M is an increasing sequence (M j, E j)0≤ j≤n of von Neumann subalge- E bras∪ of M together with the conditional expectations j of M with respect to M j such ∗ ⊆ that j M j is w –dense in M. It follows from M j M j+1 that

Ei ◦ E j = E j ◦ Ei = Emin{i, j} . (5)

1 for all i, j ≥ 0. A finite sequence (x j)0≤ j≤n in L (M) is called a martingale (supermartingale, resp.) with respect to filtration (M j)0≤ j≤n if x j ∈ M j and E j(x j+1) = x j (E j(x j+1) ≤ x j, resp.) for every j ≥ 0. It follows from (5) that E j(xi) = x j for all i ≥ j, in particular x j = E j(xn) for all 0 ≤ j ≤ n, in other words, each martingale can be adopted by an element. Put dx j = x j − x j−1 ( j ≥ 0) with the convention that x−1 = 0. Then dx = (dx j)n≥0 is called the martingale difference of (x j). The reader is referred to [4] for more information.

2. Noncommutative martingale inequalities In this section we provide a noncommutative Azuma inequality under a Lipschitz condition.

Theorem 2.1. (Noncommutative Azuma inequality) Let x = (x j)0≤ j≤n be a self-adjoint martingale with respect to a ﬁltration (M j, E j)0≤ j≤n and dx j = x j − x j−1 be its associated martingale diﬀerence. Assume that −c j ≤ dx j ≤ c j for some constants c j > 0 (1 ≤ j ≤ n). Then      ∑n       −λ2   dx ≥ λ ≤  ∑  Prob  j  2 exp  n 2  j=1 2 j=1 c j for all λ > 0. The next results present some noncommutative McDiarmid type inequalities.

171 Short title of the paper for running head

Corollary 2.2. (Noncommutative McDiarmid inequality) Let (M j, E j)0≤ j≤n be a ﬁltra- ∈ sa ≤ ≤ sa × · · · × sa → sa tion of M, x j M j (1 j n) and there exist mappings g j : M1 M j M such that the sequence

g0(x1,..., xn)), g1(x1, ··· , xn), ··· , gn(x1, ··· , xn) constitute a martingale satisfying

−c j ≤ g j(x1, ··· , xn) − g j−1(x1, ··· , xn) ≤ c j for any 1 ≤ j ≤ n. Then      −t2  (|g x ,..., x − g x ,..., x | ≥ t) ≤  ∑  . Prob n( 1 n) 0( 1 n)) 2 exp  n 2  2 j=1 c j ∑ = , ··· , = j Considering c j 1 and g j(X1 Xn) i=1 Xi in the previous Corollary, we reach the following Chernoﬀ type inequality for random variables:

Corollary 2.3. Suppose that X1, ··· , Xn are independent random variables with E(X j) = 0 and |X j| ≤ 1 for all j. Then    ∑n   ≥  ≤ −t2/2n . Prob  X j t 2e j=1 for all t ≥ 0. Sometimes Lipschitz conditions seem to be too strong. So we may need some more eﬀective tools. In the sequel, we prove an extension of the Azuma inequality under some mild conditions. Our ﬁrst result is indeed a noncommutative Azuma inequality involving supermartingales.

Theorem 2.4. Let x = (x j)0≤ j≤n be a self-adjoint supermartingale with respect to a ﬁltration (M j, E j)0≤ j≤n such that for some positive constants a j, b j, σ j and M satisﬁes E − E 2 ≤ σ2 + (i) j−1((x j j−1(x j)) ) j b j x j−1, (ii) x j − E j−1(x j) ≤ a j + M for all 1 ≤ j ≤ n. Then      −λ2  Prob (xn − x0 ≥ λ) ≤ exp  (∑ ) . (6)  n σ2 + + 2 + λ/  2 j=1( j Db j a j ) (M 3) for all λ > 0, where D := max1≤ j≤n−1 M j and M j is the maximum of spectrum of x j −x0.

If we take martingales and put b j = 0 in Theorem 2.4, then we get the following Azuma inequality for martingales.

Corollary 2.5. Suppose that x = (x j)0≤ j≤n is a self-adjoint martingale with respect to a filtration (M j, E j)0≤ j≤n and dx j = x j − x j−1 is its associated martingale difference such that for some positive constants a j, σ j and M satisfies

172 F. Author, S. Author and T. Author

E 2 ≤ σ2 (i) j−1((dx j) ) j , (ii) dx j ≤ a j + M for all 1 ≤ j ≤ n. Then          ∑n   −λ2   ≥ λ ≤ ( ) Prob  dx j  2 exp  ∑   n σ2 + 2 + λ/  j=1 2 j=1( j a j M 3) for all λ > 0.

References [1] M. Junge and Q. Zeng, Noncommutative Bennett and Rosenthal inequalities, Ann. Probab. 41 (2013) 4287-4316. [2] Gh.Sadeghi and M.S. Moslehian, Inequalities for sums of random variables in noncommutative probability spaces, Rocky Mount. J. Math. 46 (2016) 309-323. [3] Gh.Sadeghi and M.S. Moslehian, Noncommutative martingale concentration inequalities, Illinois J. Math. 58 (2014) 561-575. [4] Q. Xu, Operator spaces and noncommutative Lp, Lectures in the Summer School on Banach spaces and Operator spaces, Nankai University China, 2007.

Ghadir Sadeghi, Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran e-mail: [email protected]

173 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

COHERENT FRAMES

Ahmad Safapour∗ and Zohre Yazdani Fard

Abstract

In this paper we introduce a class of continuous frames in a Hilbert space H which is indexed by some locally compact group G, equipped with its left Haar measure. These frames are obtained as the orbits of a single element of Hilbert space H under some unitary representation π of G on H.

2010 Mathematics subject classiﬁcation: Primary 42C15, Secondary 46C99. Keywords and phrases: Coherent frame. Continuous frame, locally compact group, Unitary representation.

1. Introduction

In 1946 Gabor[4] introduced a method for reconstructing signals which led eventually to the theory of wavelets. Later in 1952 Duffin and Schaeffer introduced frame theory for Hilbert spaces to study some problems in nonharmonic Fourier series. Frames reintroduced in 1986 by Daubechies, Grossmann and Meyer [3]. Nowadays frames have become an alternative to orthonormal basis for reconstructing elements of a Hilbert space. Frames have been used in characterization of function spaces and other fields such as signal and image processing [2], filter bank theory and wireless communications. The concept of generalization of frames to a family indexed by some measure space was proposed by G. Kaiser and independently by Ali, Antoine and Gazeau[1]. Kaiser used the terminology generalised frames. Also, in mathematical physics these frames are referred to as coherent states.

In this paper, we consider a continuous frame { fg}g∈G indexed by a locally compact group G, equipped with the left Haar measure µ for which all the elements fg appear by the action of G on a single element f ∈ H via a unitary represntation of G on H and study canonical dual and combinations of this frames.

∗ speaker

174 A. Safapour and Z. Yazdani Fard

2. Basic Frame Theory

A countable family of elements { fi}i in Hilbert space H is a discrete frame if there exist constants A, B > 0 such that for each f ∈ H, ∑ 2 2 A|| f || ≤ ⟨ f, fi⟩ ≤ B|| f || . (1) i

A and B are the lower and upper frame bounds. The frame { fi}i is called a tight frame if A = B and a normalized tight frame if A = B = 1. A uniform frame is a frame in which all the elements have equal norms. If the upper inequality in 1 holds, then { fi}i is a Bessel sequence. If there is another frame {gi}i ⊂ H satisfying ∑ f = ⟨ f, fi⟩gi, ∀ f ∈ H i then {gi}i is said to be a dual of { fi}i. Given a bessel sequence { fi}i, the synthesis operator T : l2 → H defined by ∑ T{ci}i = ci fi i √ is linear and bounded with ||T|| ≤ B. The adjoint of T is the analysis operator T ∗ : H → l2 defined by ∗ T f = {⟨ f, fi⟩}i. ∗ The frame operator is S = TT , which is well defined and bounded. If { fi}i is a frame, −1 then {S fi}i is the canonical dual frame and every f ∈ H can be reconstructed as ∑ −1 f = ⟨ f, fi⟩S fi. i

A continuous frame for a Hilbert space H is a family { fm}m∈M indexed by a measure space (M, µ) such that • for all f ∈ H, m → ⟨ f, fm⟩ is a measurable function on M; • there exist constants A, B > 0 such that for each f ∈ H ∫ 2 2 A|| f || ≤ ⟨ f, fm⟩µ(m) ≤ B|| f || . M The discrete frames correspond to the case where M is at most countable, equipped with the counting measure µ. The continuous frame operator S : H → H is weakly deﬁned by ∫

⟨S f, g⟩ = ⟨ f, fm⟩⟨ fm, g⟩µ(m) ∀ f, g ∈ H. M S is bounded, positive and invertible. Every f ∈ H has the representation ∫ −1 f = ⟨ f, fm⟩S fmµ(m) M

175 Coherent Frames

3. Coherent Frames A unitary representation of a group G on a Hilbert space H is a linear mapping π of G on H such that π(g) is a unitary operator for every g ∈ G. Definition 3.1. Let G be a locally compact abelian group. A coherent frame for a Hilbert space H is a continuous frame {π(g)ϕ}g∈G, where π is a unitary representation of G on H and ϕ ∈ H. Obviously, coherent frames are uniform. Before we develop the theory for coherent frames, we mention a few examples of coherent frames. = R − { } × R ffi Example 3.2. Let Ga f f 0 be a ne∫ group equipped with the measure +∞ 2 1 2 |ψˆ(γ)| ψ ∈ R ψ = γ < ∞ a2 ab. If L ( ) is admissible, i.e. , C : −∞ |γ| , then the family {ψa,b} = {π , ψ} (a,b)∈Ga f f (a b) (a,b)∈Ga f f is a cohernt tight frame with frame bound Cψ for 2 2 L (R), where π is a unitary representation of Ga f f on L (R) defined by − 1 x b 2 (π(a, b) f )(x) = (TbDa f )(x) = √ f ( ), f ∈ L (R), x ∈ R. |a|2 a Example 3.3. Let G = R2 equipped with the lebesgue measure ab. If g ∈ L2(R) − {0}, a,b then the family {g }(a,b)∈G = {π(a, b)g}(a,b)∈G is a cohernt normalized tight frame, whrere π is a unitary representation of G on L2(R) defined by 2πixb (π(a, b)g)(x) = (EbTag)(x) = g(x − a)e , x ∈ R. Now, we show that the canonical dual of a coherent frame is also a coherent frame.

Lemma 3.4. Let {π(g)ϕ}g∈G be a coherent frame and let S be its frame operator. Then S commutes with π(g) for every g ∈ G.

Proposition 3.5. Let {π(g)ϕ}g∈G be a coherent frame for H. Then the canonical dual has the form {π(g)ψ}g∈G for some ψ ∈ H. Recall that the frame operator S for a coherent frame is a positive invertible 1 operator , and therefore has a positive square root operator S 2 . The inverse of S is also a positive operator, and hence also has a squre root. The following proposition tells us that every coherent frame can be associated to a normalized tight coherent frame. − 1 Proposition 3.6. Let {π(g)ϕ}g∈G be a coherent frame for H. Then {S 2 π(g)ϕ}g∈G is a coherent normalized tight frame for H. In the rest of this paper, we show that coherent frames can be combined as follows: • the direct sum of disjoint coherent frames is a coherent frame. • the tensor product of coherent frames is a coherent frame.

Deﬁnition 3.7. Let {ϕm}m∈M and {ψm}m∈M be continuous frames for Hilbert spaces H and K, respectivly.∫ {ϕm}m∈M and {ψm}m∈M are called disjoint if for all x ∈ H and for ∈ K ⟨ , ϕ ⟩ ⟨ψ , ⟩ µ = all y we have M x m H m y K (m) 0.

176 A. Safapour and Z. Yazdani Fard

Φ = {π ϕ} Ψ = {ρ ψ} Theorem 3.8. Let (g) g∈G and (g) g∈G be( coherent) frames for Hilbert π ϕ H K Φ ⊕ Ψ = { (g) } spaces and , respectivly. The direct sum ρ ψ g∈G is a coherent (g) ( ) S Φ frame for direct sum Hilbert space H ⊕ K with frame operator S Φ⊕Ψ = if and S Ψ only if Φ and Ψ are disjoint. ′ Theorem 3.9. Let Φ = {π(g)ϕ}g∈G and Ψ = {ρ(g )ψ}g′∈G′ be coherent frames for Hilbert spaces H and K, respectivly. The tensor product Φ ⊗ Ψ = {π(g)ϕ ⊗ ′ ρ(g )ψ}g∈G,g′∈G′ is a coherent frame for Hilbert tensor product space H ⊗ K. ——————————————–

References [1] S. T . Ali, J. P. Antoine, J . P. Gazeau, Continuous frames in Hil bert spaces, Annals of Physics 222 (1993), 1-37. [2] E. J. Cande`s and D. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities, Comm. Pure and Appl. Math. 56 (2004), 216-266. [3] I. Daubechies, A. Grasmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271-1283. [4] D. Gabor, Theory of communication, J. Inst. Electr. Eng. London 93 (III), (1946), 429-457.

Ahmad Safapour, Department of Mathematical Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran e-mail: [email protected]

Zohre Yazdani Fard, Department of Mathmatical Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran e-mail: [email protected]

177 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

ARENS REGULARITY OF BANACH ALGEBRAS AND MODULE ACTIONS

Abbas Sahleh and Leila Najarpisheh∗

Abstract In this paper we give some conditions under which the Arens regularity of a Banach algebra A implies the Arens regularity of certain Banach right module action of A.

2010 Mathematics subject classiﬁcation: Primary 46H25, Secondary 47A07. Keywords and phrases: Banach algebra, Arens product, second dual, Banach module action, Hilbert C∗- module.

1. Introduction For a normed space X, we denote by X′ and X′′ the first and second duals of X, respectively. We usually identify an element of X with its canonical image in X′′. Let X, Y and Z be normed spaces and f : X × Y −→ Z be a bounded bilinear map. In [1], R. Arens showed that f has two natural, but different, extensions f ′′′ and f r′′′ r from X′′ × Y′′ to Z′′. The adjoint f ′ : Z′ × X −→ Y′ of f is defined by ⟨ f ′(z′, x), y⟩ = ⟨z′, f (x, y)⟩ for every x ∈ X, y ∈ Y, z′ ∈ Z′, which is also a bounded bilinear map. Similarly by setting f ′′ = ( f ′)′ and continuing in this way, the mapping f ′′ : Y′′ × Z′ −→ X′, f ′′′ : X′′ × Y′′ −→ Z′′ are also bounded bilinear mappings. We also denote by f r the reverse map of f , that is the bounded bilinear map f r : Y × X −→ Z defined by f r(y, x) = f (x, y),(x ∈ X, y ∈ Y), and it may be extended as above to f r′′′ r : X′′ × Y′′ −→ Z′′. The map f is called Arens regular when the equality f ′′′ = f r′′′ r holds. Two natural extensions of the multiplication map π : A × A −→ A of a Banach algebra (A, π), π′′′ and πr′′′ r, are the so-called first and second Arens products, which will be denoted by □ and ♢, respectively. The Banach algebra (A, π) is said to be Arens regular if the multiplication map π is Arens regular. Let (A, π) be a Banach algebra, and X be a Banach space. Suppose πr : X×A −→ X is a bounded bilinear map. Then the pair (X, πr) is said to be a right Banach A-module if πr is associative, i.e. πr(x, π(a, b)) = πr(πr(x, a), b), for every a, b ∈ A , x ∈ X. A left A-module (πl, X) can be defined similarly. ∗ speaker

178 A. Sahleh and L. Najarpisheh

Let A be a C∗-algebra and X be a linear space which is a left A-module with a compatible scalar multiplication. The space X is called a left pre-Hilbert A-module ⟨., .⟩ × −→ if there exists an A-valued inner product X : X X A with the following properties: ⟨ , ⟩ ≥ ⟨ , ⟩ = = (i) X x x 0 and X x x 0 if and only if x 0; ⟨λ + , ⟩ = λ ⟨ , ⟩ + ⟨ , ⟩ (ii) X x y z X x z X y z ; ⟨ . , ⟩ = ⟨ , ⟩ (iii) X a x y a X x y ; ⟨ , ⟩∗ = ⟨ , ⟩ , , ∈ , ∈ , λ ∈ C (iv) X x y X y x for all x y z X a A . A left pre-Hilbert A-module X is called a left Hilbert A-module if it is complete with 1 ∗ || || = || ⟨ , ⟩|| 2 respect to the norm x X x x . One interesting example of left Hilbert C - ∗ ⟨ , ⟩ = ∗ , ∈ modules is any C -algebra A as a left Hilbert A-module via A a b ab (a b A). Let X be a left Hilbert A-module, we deﬁne L(X) to be the set of all maps T : X −→ −→ ⟨ , ⟩ = ⟨ , ⟩ , ∈ X for which there is a map S : X X such that X T x y X x S y (x y X). It is easy to see that T must be bounded A-linear and L(X) is a C∗-algebra. For x, y ∈ X, θ θ = ⟨ , ⟩ ∈ K deﬁne the operator x,y on X by x,y(z) X z y x , z X. Denote by (X) the closed linear span of {θx,y : x, y ∈ X}, then K(X) is a closed two sided ideal in L(X). The reader is referred to [4] for more details on Hilbert C∗-modules.

2. Main results Suppose that A is a Banach algebra. It is worth to mention that, in general, there is no relation between the Arens regularity of A and the Arens regularity of right Banach A−module. For example, let A be the C∗−algebra of compact operators on a separable, infinite-dimensional Hilbert space H and let X be the trace-class operators on H. Then, a direct verification reveals that the usual A−module action on X is not Arens regular [2]. On the other hand, an arbitrary Banach algebra A can be viewed as a right Banach ′ A−module under the module action πr(a, b) = φ(a)b (for a fixed φ ∈ A with ∥φ∥ = 1), which is trivially Arens regular [3]. The following results provide an interrelation between the Arens regularity of A and that of certain module actions.

Theorem 2.1 ([5, Theorem 3.2]). Let A be a Banach algebra and let (X, πr) be a right Banach A-module. If A is Arens regular and πr(x, A) = X for some x ∈ X, then πr is Arens regular. Proposition 2.2. Let A be a Banach algebra with a bounded approximate identity and ′′ ′′ ′′ let ϕ0 be a mixed identity for A . If (A , □) is Arens regular then ϕ0□A as a Banach right (A′′, □)−module is Arens regular. Let (A, π) be an algebra. We denote by AOP the opposite algebra to (A, π), so that AOP is the same linear space as A, but the product is πr. Proposition 2.3. Let B be a unital C∗-algebra and X be a left Hilbert B−module and OP let A = L(X). Then the module action πr(x, T) = T(x)(x ∈ X, T ∈ A ) is Arens regular.

179 Arens regularity

References [1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951) 839-848. [2] J. Bunce and W. L. Paschke, Derivations on a C∗ - algebra and its double dual, J. Funct. Anal. 37 (1980) 235-247. [3] A. R. Khoddami and H. R. Ebrahimi Vishki, The higher duals of a Banach algebra induced by a bounded linear functional, Bull. Math. Anal. Appl. 3 (2011) 118-122. [4] E. C. Lance, Hilbert C∗-modules, LMS Lecture Note Series 210, Cambridge Univ. Press, 1995. [5] A. Sahleh and L. Najarpisheh, Arens regularity of bilinear maps and Banach module actions, To Appear in Bull. Iranian Math. Soc.

Abbas Sahleh, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran e-mail: [email protected]

Leila Najarpisheh, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran e-mail: [email protected]

180 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

U-CROSS GRAM MATRIXES AND THEIR ASSOCIATED RECONSTRUCTIONS

Mitra Shamsabadi∗ and Ali Akbar Arefijamaal

Abstract In this paper, we study U-cross Gram matrixes, which can be produced by frames and Riesz bases, and their properties. Then we investigate some necessary or suﬃcient conditions for invertibility of this matrixes and try to reconstruct the elements of ℓ2. It is important in application to state the inverse of U-cross Gram matrixes as the form of U-cross Gram matrixes.

2010 Mathematics subject classiﬁcation: Primary 41A58; Secondary 43A35. Keywords and phrases: U-cross Gram matrix; cross Gram matrix; Gram matrix, inverse.

1. Introduction In modern’s life, many applications in industry refer to signal processing and it’s ground is frame theory. In some areas of applications for example, sound [3] and physics [1] there are many problems that can be stated as operator theory. We know that an operator U can be stated by orthonormal basis {ei}i∈I and described by a matrix such that all it’s entries constructed by frames and their canonical dual, see [4]. Finally by a good extension an operator developed for Bessel sequences, frames and Riesz basis by Balazs, [2]. Throughout this paper H denotes a separable Hilbert space and I a countable indexing set. The identity operator on H is denoted by IH . A Riesz basis for H is a family of the form {Uei}i∈I, where {ei}i∈I is an orthonormal basis for H and U : H → H is a bounded bijective operator. A family { f } ∈ in H is a frame if there exists constants A, B > 0 such that i i I ∑ 2 2 2 A ∥ f ∥ ≤ |⟨ f, fi⟩| ≤ B ∥ f ∥ , ( f ∈ H). (1) i∈I

A and B are frame bounds. If { fi}i∈I satisﬁes in the right hand of (1), then it is called a Bessel sequence. We say that a sequence { fi}i∈I in H a frame sequence if it is a frame for span{ f } ∈ . For a Bessel sequence { f } ∈ deﬁnes the synthesis operator i i I  i i I   ∑  2   T : ℓ → H, {ci}i∈I 7→ ci fi . i∈I ∗ speaker

181 M. Shamsabadi, A. Arefijamaal

Its adjoint operator T ∗ : H → ℓ2, so called analysis operator is given by ∗ = {⟨ , ⟩} , ∈ H . T f f fi i∈I ( f ) ∑ { } H → H = ∗ = , For a frame fi i∈I, S : , which is deﬁned by S f TT f i∈I ⟨ f fi⟩ fi, for all f ∈ H, is called the frame operator.

Proposition 1.1. For a sequence { fi}i∈I in H, the following conditions are equivalent:

1. { fi}i∈I is a Riesz basis for H. 2. { fi}i∈I is complete in H and there exist constants A, B > 0 such that for every ﬁnite scalar sequence {ci}i∈I, one has

∑ ∑ 2 ∑

A |c |2 ≤ c f ≤ |c |2 . (2) i i i i i∈I i∈I i∈I

2. Main Results

Deﬁnition 2.1. Let Ψ = {ψi}i∈I be a Bessel sequence in H1 and Φ = {ϕi}i∈I a Bessel sequences in H2. For U ∈ B(H1, H2), we call GU,Φ,Ψ given by ( ) ⟨ ⟩ = ψ , ϕ , , ∈ , GU,Φ,Ψ i, j U j i (i j I) (3) H = H = U-cross Gram matrix. If 1 2 and U IH1 it is called cross Gram matrix and denote it by GΦ,Ψ. Also, if Φ = Ψ then this matrix is Gram matrix GΨ.

Let Φ = {ϕi}i∈I and Ψ = {ψi}i∈I be two Bessel sequences in H1 and H2 with upper ′ bounds B and B , respectively. For all U ∈ B(H1, H2), the following assertions hold. ℓ2 ℓ2 1. The U-cross √ Gram matrix GU,Φ,Ψ deﬁnes a bounded operator from to and ′ GU,Φ,Ψ ≤ BB ∥U∥. Furthermore, ∗ GU,Φ,Ψ = TΦUTΨ. ( )∗ 2. GU,Φ,Ψ = GU∗,Ψ,Φ.

Proposition 2.2. Let Φ, Ψ and Ξ be Bessel sequences in H. Also U1, U2 ∈ B(H). If Ψ† is every dual of Ψ, then

,Φ,Ψ † = † ,Ψ,Ξ = ,Φ,Ξ 1. GU1 GU2,Ψ ,Ξ GU1,Φ,Ψ GU2 GU1U2 . = 2. GU1,Φ,ΨGU2,Ψ,Ξ GU1S ΨU2,Φ,Ξ.

Theorem 2.3. Let Ψ = {ψ}i∈I be a Riesz bases in H and ∆ = {δi}i∈I the orthonormal 2 basis of ℓ . Then G ∗ e, G e e and G −1,Ψ,Ψ are identity matrixes. TΨ,∆,Ψ S Ψ,Ψ,Ψ S Ψ In the sequel, we ﬁnd an inverse for U-cross Gram matrix under some conditions. Theorem 2.4. If Φ and Ψ be two Riesz basis and U ∈ B(H) is an invertible operator then U-cross Gram matrix GU,Φ,Ψ has inverse and ( ) −1 = . GU,Φ,Ψ GU−1,Ψe,Φe

182 U-cross Gram matrixes and their associated reconstructions

In the following, we state some suﬃcient conditions for the invertibility of a U- cross Gram matrix.

Theorem 2.5. Let Ψ = {ψi}i∈I be a Bessel sequence and Φ = {ϕ}i∈I a Riesz basis with bounds A and B, respectively. Then GU,Φ,Ψ is invertible, if ∑ B ∥Uψi − ϕi∥ ≤ √ , i∈I BΦ where BΦ is a Bessel bound of Φ. Corollary 2.6. Suppose that Φ and Ψ are two Bessel sequences in H with Bessel bounds BΦ and BΨ, respectively. Also, GU,Φ,Ψ is invertible. 1. If V ∈ B(H) such that 1 ∥U − V∥ ≤ √ , −1 GU,Φ,Ψ BΦBΨ

then GV,Φ,Ψ is invertible. 2. If Ξ = {ξi}i∈I is a Bessel sequence in H such that ∑ 1 ∥ψi − ξi∥ ≤ √ , −1 ∥ ∥ i∈I GU,Φ,Ψ BΦ U

then GU,Φ,Ξ is invertible. 3. If Θ = {θi}i∈I is a Bessel sequence in H such that ∑ 1 ∥ϕi − θi∥ ≤ √ , −1 ∥ ∥ i∈I GU,Φ,Ψ BΨ U

then GU,Θ,Ψ is invertible.

Theorem 2.7. Let GU,Φ,Ψ be invertible. Then Ψ and Φ are Riesz sequences.

References [1] S. T. Ali, J.-P. Antoine, J.-P. Gazeau, Coherent States, Wavelets and Their Generalization, SGraduate Texts in Contemporary Physics. Springer New York, 2000 . [2] P. Balazs, Matrix-representationofoperatorsusingframes, Sampling Theory in Signal and Image Processing (STSIP) 7(1) 2008, 39-54. [3] P. Balazs, W. Kreuzer, and H. Waubke, A stochastic 2d-model for calculating vibrations in liquids and soils, Journal of Computational Acoustics, 2006. [4] O. Christensen, Frames and pseudo-inverses, J. Math. Anal. Appl, 195(2), (1995),401-414.

Mitra Shamsabadi, Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran e-mail: [email protected]

183 M. Shamsabadi, A. Arefijamaal

Ali Akbar Arefijamaal, Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran e-mail: areﬁ[email protected]

184 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

SOME PROPERTIES OF K-FUSION FRAMES AND THEIR MULTIPLIERS

Mitra Shamsabadi

Abstract In this paper, we introduce the concept of K-fusion frames and propose the duality for such frames. The relation between the local frames of K-fusion frames with their dual is studied. The elements from the range of a bounded linear operator K can be reconstructed by K-frames. Also, we establish K-fusion frame multipliers and investigate reconstruction of the range of K by them.

2010 Mathematics subject classiﬁcation: Primary 41A58; Secondary 43A35. Keywords and phrases: K-fusion frame; K-dual; multiplier.

1. Introduction The theory of frames plays an important role in wavelet theory as well as (time- frequency) analysis for functions in L2(Rd). The traditional applications of frames are signal processing, image processing, sampling theory and communication [3], moreover, recently the use of frames also in numerical analysis for the solution of operator equation by adaptive schemes is investigated [1]. Also, frame multipliers have so applications in psychoacoustical modeling and denoising [2]. For two sequences Φ := {ϕi}i∈I and Ψ := {ψi}i∈I in a Hilbert space H and a sequence m = {mi}i∈I of complex scalars, the operator Mm,Φ,Ψ : H → H given by ∑ Mm,Φ,Ψ f = mi⟨ f, ψi⟩φi, ( f ∈ H), (1) i∈I is called a multiplier. The sequence m is called symbol. If Φ and Ψ are√ Bessel se- ∞ quences for H and m ∈ ℓ , then Mm,Φ,Ψ is well deﬁned and ∥Mm,Φ,Ψ∥ ≤ BΦBΨ∥m∥∞, where BΦ and BΨ are Bessel bounds of Φ and Ψ, respectively. The invertibility of multipliers, which plays a key role in the topic, is discussed in [5]. Let {Wi}i∈I be a family of closed subspaces of H and {ωi}i∈I be a family of weights, i.e. ωi > 0, i ∈ I. The sequence {(Wi, ωi)}i∈I is called a fusion frame for H if there exist constants 0 < A ≤ B < ∞ such that ∑ ∥ ∥2 ≤ ω2∥π ∥2 ≤ ∥ ∥2, ∈ H , A f i Wi f B f ( f ) i∈I

185 M. Shamsabadi

π { } ∈ where Wi is the orthogonal projection onto Wi i∈I, for all i I. It is called a Bessel fusion sequence if we have the upper bound.⊕ A fusion frame {(Wi, ωi)}i∈I is H = said to∑ be an⊕orthonormal fusion basis if i∈I Wi. The synthesis operator → H { , ω } TW :( i∈I ∑Wi)ℓ2 for a Bessel fusion sequence (Wi ∑i) i⊕∈I is defined by { } = ω . ∗ H → TW ( fi i∈I) i∈I i fi Also, its adjoint operator TW : ( i∈I Wi)ℓ2 , which ∗ = {ω π }. is called the analysis operator, is given by TW f ∑i Wi f The bounded operator = ∗ H = ω2π S W TW TW on is called frame operator and S W i∈I i Wi . K-frames which recently introduced by Gavru¸taareˇ a generalization of frames, in the meaning that the lower frame bound only holds for that admits to reconstruct from the range of a linear and bounded operator in a Hilbert space. Definition 1.1. Let K be a bounded and linear operator on a separable Hilbert space H. A sequence F := { fi}i∈I ⊆ H is called a K-frame for H, if there exist constants A, B > 0 such that ∑ ∗ 2 2 2 A∥K f ∥ ≤ |⟨ f, fi⟩| ≤ B∥ f ∥ , ( f ∈ H). (2) i∈I Obviously, every K-frame is a Bessel sequence, hence similar to ordinary∑ frames 2 → H { } = the synthesis operator can be defined as TF : l ; TF( ci i∈I) i∈I ci fi. It is a bounded operator and its adjoint which is called the analysis operator given ∗ = {⟨ , ⟩} H → H by TF( f ) f ∑fi i∈I. Finally, the frame operator is given by S F : ; = ∗ = ⟨ , ⟩ S F f TFTF f i∈I f fi fi. Some properties of ordinary frames are not hold for K- frames, for example, the frame operator of a K-frame is not invertible and dual pair is not interchangeable in general. If K has close range then S F from R(K) onto S F(R(K)) is an invertible operator and −1∥ ∥ ≤ ∥ −1 ∥ ≤ −1∥ †∥2∥ ∥, ∈ , B f S F f A K f ( f S F(R(K))) (3) where K† is the pseudo-inverse of K.

Deﬁnition 1.2. Let { fi}i∈I be a K-frame. A Bessel sequence {gi}i∈I is called a K-dual of { fi}i∈I if ∑ K f = ⟨ f, gi⟩ fi, ( f ∈ H). (4) i∈I ∗ −1 { π } ∈ = { } ∈ The K-dual K S F S F R(K) fi i I of F fi i I which is called the canonical, is e denoted by { fi}i∈I. In the present paper, the reconstruction elements from the range of K by a K-fusion frame, where K is a closed range and bounded linear operator on H, is investigated. We also introduce K-fusion frame multipliers and discuss their invertibility.

2. Main Results

Let {Wi}i∈I be a family of closed subspaces of H and {ωi}i∈I be a family of weights, i.e. ωi > 0, i ∈ I. The sequence {(Wi, ωi)}i∈I is called a K-fusion frame for H if there

186 Some properties of K-fusion frames and their multipliers

< ≤ < ∞ exist constants 0 A B ∑ such that ∥ ∗ ∥2 ≤ ω2∥π ∥2 ≤ ∥ ∥2, ∈ H . A K f i Wi f B f ( f ) i∈I (∑ ⊕ ) → H deﬁne the synthesis operator TW : i∈I Wi by ∑ ℓ2 TW ({ fi}i∈I) = ωi fi. i∈I (∑ ⊕ ) ∗ H → Its adjoint operator TW : i∈I Wi 2 , which is called the analysis operator, ∗ ℓ is obtained by T f = {ωiπW f }i∈I, where W i   ∑ ⊕   ∑    = { } ∈ , ∥ ∥2 < ∞ .  Wi  fi i∈I : fi Wi fi  i∈I ℓ2 i∈I

Also frame operator of {W } ∈ on H, denoted by S , is given by i i I ∑ W = ∗ = ω2π . S W f TW TW i Wi f i∈I It is not diﬃcult to see that the frame operator of a K-fusion frame is not invertible on H, in general. However, S W : R(K) → S W R(K) is invertible and −1∥ ∥ ≤ ∥ −1 ∥ ≤ −1∥ †∥2∥ ∥, ∈ , B f S W f A K f ( f S W (R(K))) (5) where K† is the pseudo-inverse of K. Now, we can reconstruct R(K) by K-fusion frame elements. ∑ = −1 ∗ = ω2π −1 ∗ , ∈ H . K f S W (S W ) K f i Wi (S W ) K f ( f ) i∈I

Deﬁnition 2.1. Let W = {(Wi, ωi)}i∈I be a K-fusion frame. A Bessel fusion sequence V = {(V , υ )} ∈ is called a K-dual for W if i i i I ∑ = ω υ π −1 ∗ π , ∈ H . K f i i Wi (S W ) K Vi f ( f ) (6) i∈I

Proposition 2.2. Let W = {(Wi, ωi)}i∈I be a Bessel fusion sequence such that Wi ⊆ e ∗ −1 ∈ = { π , ω } ∈ S W (R(K)), for all i I. Then W : (K S W S W (R(K))Wi i) i I is a K-dual of W. Every K-dual of a K-fusion frame is a K∗-fusion frame.

Theorem 2.3. Let {(Wi, ωi)}i∈I be a fusion frame for H and K ∈ B(H) be a closed † range operator such that Wi ⊆ R(K ), for all i ∈ I. Then {(KWi, ωi)}i∈I is a K-fusion frame. ∞ Deﬁnition 2.4. Let W be a K-fusion frame and m := {mi}i∈I ∈ ℓ . For every Bessel fusion sequence V, the operator M , , : H → H given by ∑ m W V = ω υ π −1 ∗ π , ∈ H . Mm,W,V mi i i Wi (S W ) K Vi f ( f ) i∈I is called a K-fusion frame multiplier.

187 M. Shamsabadi

In the above deﬁnition, a K- fusion frame multiplier is a fusion frame multiplier if K = IH . For more details of fusion frame multipliers see [4]. Theorem 2.5. Let W be a K-fusion frame and V a Bessel fusion sequence. The following assertions hold. ∗ 1. Let Mm,W,V = K. Then V satisﬁes the lower K -fusion frame condition. In particular, it is a K∗-fusion frame. ∗ 2. If Mm,W,V has a K-left inverse, then V is a K -fusion frame.

Theorem 2.6. Let W = {(Wi, 1)}i∈I is a K-fusion frame and V = {(Vi, 1)}i∈I a Bessel fusion sequence. If   ∑ 1  ( )∗ 2 2  −1  AV  πW S K − πV  ≤ √ , (7) i W i † 2 i∈I BV K on R(K), where BV and AV are the upper bound and lower bound of V, respectively, then M1,W,V is invertible on R(K).

References [1] P. Balazs, W. Kreuzer, H. Waubke, A stochastic 2D-model calculating vibrations in liquids and soils, J. Comput. Acoust. 15 (3) (2007) 271-283. [2] P. Balazs, B. Laback , G. Eckel and W. A. Deutsch, Time-frequency sparsity by removing perceptually irrelevant components using a simple model of simultaneous masking, IEEE Trans. Audio Speech Lang. 18 (1) (2010) 34-49. [3] R. W. Heath and A. J. Paulraj, Linear dispersion codes for MIMO systems based on frame theory, IEEE Trans. Signal Processing 50 (2002), 2429-2441. [4] M. Shamsabadi, A. A. Arefijamaal The invertibility of fusion frame multipliers, Linear and Multilinear Algebra, (2016) , to appear. [5] D. T. Stoeva, P. Balazs, Invertiblity of multipliers, Appl. comput. Harmon. Anal., 33 (2), (2012), 292-299.

Mitra Shamsabadi, Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran e-mail: [email protected]

188 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

WAVELETS METHOD FOR SOLVING A CLASS OF FRACTIONAL PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS

Fatemeh Soltani Sarvestani∗, Mohammad Hossein Heydari and Asadollah Niknam

Abstract In this paper, a Galerkin method based on the second kind Chebyshev wavelets (SKCWs) is proposed for solving a class of fractional partial integro-diﬀerential equations with weakly singular kernels. In the proposed method, the operational matrix of fractional order integration (OMFI) for SKCWs is used to transform the problem under consideration to a linear system of algebraic equations which can be simply solved.

2010 Mathematics subject classiﬁcation: Primary 45B05; Secondary 65R20. Keywords and phrases: Second kind Chebyshev wavelets (SKCWs); Hat functions (HFs); Fractional partial integro-diﬀerential equations; Operational matrix of fractional integration..

1. Introduction The main goal of this paper is to propose an eﬃcient and accurate method based on SKCWs to solve the following fractional partial integro-diﬀerential equation: ∫ t −β ut(x, t) = f (x, t)+µ uxx(x, t)+ (t−s) uxx(x, s)ds, µ ≥ 0, 0 < β < 1, (x, t) ∈ Ω, (1) 0 with Ω = [0, 1] × [0, 1], subject to the initial condition u(x, 0) = g(x), (2) and boundary conditions

u(0, t) = h0(t), u(1, t) = h1(t). (3) In the sequel, we use the following definition of the fractional calculus, which is required for establishing our results. Definition 1.1. The Riemann-Liouville fractional integration operator of order α ≥ 0 of a function u(t) is defined in [1] by:  ∫  t  1 α−1 α  (t − s) u(s)ds, α > 0, (I u) (t) =  Γ α (4)  ( ) 0 u(t), α = 0.

∗ speaker

189 F. Soltani Sarvestani, M. H. Heydari and A. Niknam

2. The HFs and SKCWs An mˆ -set of the hat functions (HFs) is defined over the interval [0, 1] in [2]. An arbitrary∑ function f (t) defined on the interval [0, 1] may be expanded by the HFs as ≃ mˆ −1 φ = T Φ = Φ T f (t) i=0 fi i(t) F (t) (t) F, where T T F ≜ [ f0, f1,..., fmˆ −1] , Φ(t) ≜ [φ0(t), φ1(t), . . . , φmˆ −1(t)] , (5) ( ) = i , = , ,..., − and fi f mˆ −1 i 0 1 mˆ 1. Theorem 2.1. (See [2]). The fractional integration of order α in the Riemann- Liouville sense of the vector Φ(t) can be expressed as: (IαΦ)(t) ≃ Pˆ (α)Φ(t), (6) where matrix Pˆ (α) is called the OMFI of order α for the HFs. The SKCWs are defined on the interval [0, 1] by [3]:  ( ) [ ]  2 k + n−1 n  √ 2 k 1 − + , ∈ , , π 2 Um 2 t 2n 1 t 2k 2k ψnm(t) =  (7)  0, o.w, where Um(t) is the second kind Chebyshev polynomial of degree m. The set of the , = SKCWs √ is an orthogonal set on [0 1] with respect to the weight function wn(t)  ( ) [ ]  − k+1 − + 2, ∈ n−1 , n , 1 2 t 2n 1 t k k ,  2 2 . A function u(t) defined on [0 1] may 0, o.w. ∑ ∑ ≃ 2k M−1 ψ = T Ψ = be expanded by the SKCWs as u(t) n=1 m=0 cnm nm(t) C (t), where cnm k (u(t), ψnm(t)) and C and Ψ(t) are mˆ = 2 M column vectors. It can be also written ∑ wn(t) ≃ mˆ ψ = T Ψ = ψ = ψ as u(t) i=1 ci i(t) C (t), where ci cnm and i(t) nm(t), and the index i is determined by the relation i = M(n − 1) + m + 1. Thus we have: [ ] T T C ≜ [c1, c2,..., cmˆ ] , Ψ(t) ≜ ψ1(t), ψ2(t), . . . , ψmˆ (t) . (8) An arbitrary∑ function∑ of two variables u(x, t), may be expanded by the SKCWs as u(x, t) ≃ mˆ mˆ u ψ (x)ψ (t) = ΨT (x)UΨ(t), where U = [u ] and u = ( ( i=1 ) j=1 ) i j i j i j i j ψi(x), u(x, t), ψ j(t) . w (t) n wn(x)

Theorem 2.2. Let Φ(t) and Ψ(t) be the HFs and SKCWs vectors deﬁned in Eqs. (5) and (8), respectively. The vector Ψ(t) can be expanded by the HFs vector Φ(t) as Ψ(t) ≃ QΦ(t), where the mˆ × mˆ matrix Q is called the SKCWs matrix and

Qi j = ψi(( j − 1)h), i = 1, 2,..., mˆ , j = 1, 2,..., mˆ . (9) Theorem 2.3. The fractional integration of order α > 0 in the Riemann-Liouville sense of the vector Ψ(t) can be expressed as: ( ) (IαΨ)(t) ≃ QPˆ (α)Q−1 Ψ(t) ≜ P(α)Ψ(t), (10) where matrix P(α) is called the OMFI of order α for the SKCWs.

190 Wavelets for fractional partial integro-diﬀerential equations

3. Description of the proposed method ∂3u(x,t) To solve Eq. (1), we approximate ∂x2∂t by SKCWs as follows: ∂3u(x, t) ≃ Ψ(x)T UΨ(t), (11) ∂x2∂t where U = [ui j]mˆ ×mˆ should be found and Ψ(.) is SKCWs vector defined in Eq. (8). By integrating of Eq. (11) two times with respect to x, we obtain: ( ) ∂ , ( ) ut(x t) T T (2) u (x, t) ≃ u (0, t) + x | = + Ψ(x) P UΨ(t), (12) t t ∂x x 0 and by putting x = 1 in Eq. (12), and considering Eq. (2), we obtain: ∂ , ( ) ut(x t) ′ ′ T T (2) | = ≃ h (t) − h (t) − Ψ(1) P UΨ(t). (13) ∂x x 0 1 0 ′ ′ We also expand h0(t) and h1(t) by SKCWs as: ′ ≃ T Ψ , ′ ≃ T Ψ , h0(t) H0 (t) h1(t) H1 (t) (14) where H0 and H1 are coefficient vectors. Substituting Eq. (14) into Eq. (13) yields: ∂ , ( ( ) ) ut(x t) T T T T (2) eT | = ≃ H − H − Ψ(1) P U Ψ(t) ≜ U Ψ(t). (15) ∂x x 0 1 0 So, by substituting Eq. (15) into Eq. (12), we have: [ ( ) ] , ≃ Ψ T T + eT + T (2) Ψ ≜ Ψ T Λ Ψ , ut(x t) (x) EH0 XU P U (t) (x) 1 (t) (16) where X and E are coefficient vectors for x and the unit function, respectively. Now, by integrating of Eqs. (16) and (11) with respect to t, we obtain: [ ] T T T u(x, t) ≃ Ψ(x) G0E + Λ1P Ψ(t) ≜ Ψ(x) Λ2Ψ(t), (17) [ ] T T T uxx(x, t) ≃ Ψ(x) G1E + UP Ψ(t) ≜ Ψ(x) Λ3Ψ(t), (18) ′′ where G0 and G1 are coefficient vectors for g(x) and g (x), respectively. Furthermore, we expand f (x, t) as follows: f (x, t) ≃ Ψ(x)T FΨ(t), (19) where F is coefficient matrices for f (x, t). Then, by substituting Eqs. (16), (18) and (19) into Eq. (1), and using OMFI of SKCWs, we have: [ ] T (1−β) T Ψ(x) Λ1 − µΛ3 − Γ(1 − β)Λ3P Ψ(t) ≃ Ψ(x) FΨ(t). (20) It is obvious that Eq. (20) generates a set of mˆ 2 linear algebraic equations as: (1−β) Λ1 − µΛ3 − Γ(1 − β)Λ3P = F. (21) Finally, by solving the above system for U, we obtain an approximate solution for the problem using Eq. (17).

191 F. Soltani Sarvestani, M. H. Heydari and A. Niknam

Example 3.1. Consider the fractional partial integro-differential equation [4]: ∫ t − 1 ut = (t − s) 2 uxx(x, s)ds, 0 , = π subject to the initial condition u(x 0) sin( x) and the homogeneous( boundary) ( condi-) ∑∞ −1 5 3 n , = − nΓ 3 + π 2 2 π tions. The exact solution of this problem is u(x t) n=0( 1) 2 n 1 t sin( x). This problem is solved by the proposed method for k = 1 and M = 12. A comparison between the absolute errors obtained by the proposed method via the three-point explicit finite-difference method, the three-point implicit technique and the Crank- Nicolson procedure [4] with h = 0.02 at t = 1 is performed in Table 1. Table 1. The absolute errors obtained by the finite difference schemes and our method. Finite difference schemes Our method x Explicit Implicit Crank-Nicolson k = 1, M = 12 0.1 7.5 × 10−3 7.1 × 10−3 5.1 × 10−3 1.4 × 10−3 0.3 7.6 × 10−3 7.4 × 10−3 5.3 × 10−3 3.7 × 10−3 0.5 7.5 × 10−3 7.5 × 10−3 5.4 × 10−3 4.5 × 10−3 0.7 7.3 × 10−3 7.2 × 10−3 5.5 × 10−3 3.7 × 10−3 0.9 7.8 × 10−3 7.6 × 10−3 5.2 × 10−3 1.4 × 10−3

References [1] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [2] M. P. Tripathi, V. K. Baranwal, R. K. Pandey and O. P. Singh, A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions, Commun Nonlinear Sci Numer Simulat. 18 (2013) 1327–1340. [3] Li Zhu and Qibin Fan, Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet, Commun Nonlinear Sci Numer Simulat , 17 (2012), 2333–2341. [4] M. Dehghan, Solution of a partial integro-differential equation arising from viscoelasticity, International Journal of Computer Mathematics , 83 (1) (2006), 123–129.

Fatemeh Soltani Sarvestani, Department of Pure Mathematics, Faculty of Mathematics, Ferdowsi University Of Mashhad, Mashhad, Iran. e-mail: [email protected].

Mohammad Hossein Heydari, Department of Mathematics, Faculty of Science, Fasa University, Fasa, Iran. e-mail: [email protected].

Asadollah Niknam, Department of Pure Mathematics, Ferdowsi University Of Mashhad, Mashhad, Iran. e-mail: [email protected].

192 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

PARSEVAL ADMISSIBLE VECTORS ON HYPERGROUPS

Seyyed Mohammad Tabatabaie and Soheila Jokar∗

Abstract In this paper we charactrize Parseval admissible vectors in L2(K), wherer K is a locally compact hypergroup.

2010 Mathematics subject classiﬁcation: Primary 43A62, Secondary 42C15. Keywords and phrases: locally compact hypergroup, frame, Parseval admissible vectors.

1. Introduction Hypergroups, as extensions of locally compact groups, were introduced in a series of papers by C. F. Dunkl, R. I. Jewett and R. Spector . In the last decade, the theory of frame and wavelet analysis has been extended in harmonic analysis on locally compact groups. In some other works we have initiated the concept of admissible vectors on some function spaces related to hypergroups and we have generalized basic properties of coorbit spaces. In this paper, we give a characterization of some special admissible vectors related to the left regular representation of hypergroups and really we extend the main results of [2].

2. Notation and preliminary results Deﬁnition 2.1. A locally compact Hausdorﬀ space, K, together with a bilinear mapping (µ, ν) 7→ µ ∗ ν from M(K) × M(K) into M(K), and an involutive homeomorphism x 7→ x− on K is called a hypergroups if: (i) for each µ, ν ∈ M+(K), µ ∗ ν ∈ M+(K). Also, the mapping (µ, ν) 7→ µ ∗ ν from M+(K) × M+(K) into M+(K) is continuous, where M+(K) is equipped with the cone topology. (ii) M(K) with ∗ is a complex associative algebra, and for all µ, ν ∈ M(K) we have ∫ ∫ ∫ ∫

f d(µ ∗ ν) = f d(δx ∗ δy)dµ(x)dν(y), K K K K

where f ∈ C0(K); ∗ speaker

193 S.M.Tabatabaie and S.Jokar

(iii) for all x, y ∈ K, δx ∗ δy is a compact supported probability measure; (iv) there exists an element e ∈ K (called identity) such that for all x ∈ K, δx ∗ δe = δe ∗ δx = δx; − (v) for all x, y ∈ K, (δx ∗ δy) = δy− ∗ δx− , where for each µ ∈ M(K), ∫ − − µ ( f ):= f (x )dµ(x), ( f ∈ C0(K)). K − Also, e ∈ supp(δx ∗ δy) if and only if x = y ; (vi) the mapping (x, y) 7→ supp(δx ∗ δy) from K × K into C(K) is continuous, where C(K) is the space of all non-empty compact subsets of K equipped with Michael topology.

A hypergroup K is called commutative if for all x, y ∈ K, δx ∗ δy = δy ∗ δx.A non-zero non-negative regular measure m on K is called left Haar measure if for each x ∈ K, δx ∗ m = m. Any commutative hypergroup has a left Haar measure m.

Deﬁnition 2.2. Let K be a commutative hypergroup. A non-zero complex-valued bounded continuous function ξ on K is called a character if for all x, y ∈ K, ξ(x ∗ y) = ξ(x)ξ(y) and ξ(x−) = ξ(x). The set of all characters of K equipped with the uniform convergence topology on compact subsets of K, is denoted by Kˆ and is called the dual of K. If Kˆ with the complex conjugation as involution and poinwise product, i.e. ∫

ξ(x)η(x) = χ(x)d(δξ ∗ δη)(χ), (x ∈ K and ξ, η ∈ Kˆ ), Kˆ as convolution is a hypergroup, then K is called a strong hypergroup.

Deﬁnition 2.3. A complex valued function f on K is called a∑ trigonometric polyno- ,..., ∈ C ξ , . . . , ξ ∈ ˆ = n ξ mial if for some a1 an and 1 n K we have f i=1 ai i. The set of all trigonometric functions on K is denoted by Trig(K).

The following well-known theorem is a useful tool in our proofs.

Theorem 2.4. If H is a normal compact subhypergroup of a hypergroup K, then H has the Weil’s property.

Deﬁnition 2.5. If H be a subhypergroup of a hypergroup K, then

H⊥ := {ξ ∈ Kˆ : ξ(x) = 1 for all x ∈ H} is called the annihilator of H in Kˆ .

Throughout this paper we assume that K is a compact Pontryagin commutative hypergroup and H is a normal compact subhypergroup of K.

194 Parseval admissible vectors on hypergroups

3. Main Results 2 − If f ∈ L (K) and x ∈ K, we put τx f (y):= f (x ∗ y), where y ∈ K. 2 Definition 3.1. For each φ ∈ L (K) we denote Aφ := linear span {τxφ : x ∈ K}, and ∥.∥2–closure of Aφ is denoted by Vφ. In this definition, if the elements x are considered from a subhypergroup H of K, then Vφ would be in respect to H. 2 2 Definition 3.2. ∫Let φ ∈ L (K). We denote by L (Hˆ , wφ) the space of all functions 2 r : Hˆ → C with |r(ξ)| wφ(ξ)dξ < ∞, where Hˆ ∫ 2 wφ(ξ):= |φˆ(ξ ∗ η)| dη. H⊥ In this case, the mapping

(∫ ) 1 2 2 2 ∥r∥φ := |r(ξ)| wφ(ξ)dξ (r ∈ L (Hˆ , wφ)) Hˆ 2 1 is a norm on L (Hˆ , wφ),and under above notations, we have wφ ∈ L (Hˆ ). 2 Lemma 3.3. Let φ ∈ L (K). Then f ∈ Aφ if and only if for some r ∈ Trig(Kˆ ), fˆ(ξ) = r(ξ)φ ˆ(ξ)(ξ ∈ Kˆ ). Lemma 3.4. Let Kˆ is bounded (i.e. there is a constant number M > 0 such that for 2 2 all ξ ∈ Kˆ , |ξ| ≤ M), and φ ∈ L (K). Then Trig(Kˆ ) ⊆ L (Hˆ , wφ). Here we recall the following theorem from [3]

Theorem 3.5. Let K be a locally compact commutative strong hypergroup∑ with Haar π ϵ > = { n λ ⟨ , ·⟩ ∈ measure m and associated Plancherel measure , 0, and A : j=1 j x j : n N, λ j ∈ C, x j ∈ K ( j = 1,..., n)}, where for each x ∈ K and ξ ∈ Kˆ , ⟨x, ξ⟩ := ξ(x). If 2 k1 ∈ L (Kˆ , π) has a null zeros set with respect to Plancherel measure π, then for each 2 k2 ∈ L (Kˆ , π) there exists an element ψ ∈ A such that ∥k2 − ψk1∥2 < ϵ. 2 Corollary 3.6. Let φ ∈ L (K), and φˆ , 0 a.e. on Kˆ . f ∈ Vφ if and only if 2 fˆ(ξ) = r(ξ)φ ˆ(ξ) for some r ∈ L (Hˆ , wφ). Definition 3.7. The mapping τ : M(K) → B(L2(K)) defined by τ(µ)( f ):= µ∗ f , where f ∈ L2(K) and µ ∈ M(K), is a representation of the hypergroup K called left regular representation. 2 Proposition 3.8. Let φ ∈ L (K). The set {τxφ : x ∈ K} is an orthogonal system in L2(K) if |φˆ| = 1 a.e. on Kˆ . Definition 3.9. Let K be a hypergroup with a (left) Haar measure m, H be a subhypergroup of K, and π : M(K) → B(Hπ) be a representation of K on a Hilbert space Hπ, and V ⊆ Hπ. A vector h0 ∈ Hπ is called a (π, V)–admissible vector with respect to H if there are constant numbers A, B > 0 such that for every h ∈ V, ∫ 2 2 2 A∥h∥ ≤ |⟨πx(h0), h⟩| dmH(x) ≤ B∥h∥ , H

195 S.M.Tabatabaie and S.Jokar where mH is a left Haar measure on H and πx := π(δx). If A = B = 1, h0 is called Parseval admissible. Deﬁnition 3.10. Let K be a hypergroup. The center of K is deﬁned as

Z(K):= {x ∈ K : δx ∗ δx− = δe = δx− ∗ δx}. = { }n ⊆ Remark 3.11. In particular, if H : xk k=1 be a subhypergroup of K and H Z(K), then h0 ∈ Hπ is a π–admissible vector with respect to H if and only if there exist constant numbers A, B > 0 such that for every h ∈ Hπ, ∑n ∥ ∥2 ≤ |⟨π , ⟩|2 ≤ ∥ ∥2, A h xk (h0) h B h k=1 since in this case mH is the counting measure. 2 Theorem 3.12. A function φ ∈ L (K) is a Parseval (τ, Vφ)–admissible vector if and φ = χ Ω = φ only if ˆ Ωφ a.e. on Kˆ , where φ : supp( ˆ).

References [1] W. R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, De Gruyter, Berlin, (1995). [2] R. A. Kamyabi Gol and R. Raisi Tousi, The structure of shift inveriant spaces on a locally compact abelian group, J. Math. Anal. Appl., 340 (2008), 219-225. [3] S. M. Tabatabaie, The problem of density on L2(G), Acta Mathematica Hungarica, 150 (2016), 339-345. [4] S. M. Tabatabaie, The problem of density on commutative strong hypergroup, to appear in Math. Reports

Seyyed Mohammad Tabatabaie, Department of Mathematics, University of Qom, Qom, Iran e-mail: [email protected]

Soheila Jokar, Department of Mathematics, University of Qom, Qom, Iran e-mail: [email protected]

196 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

HECKE ∗-ALGEBRAS ON HYPERGROUPS

Seyyed Mohammad Tabatabaie∗ and Bentolhoda Sadathoseyni

Abstract In this paper we initiate and study the Hecke ∗-algebra for a discrete hypergroup K and its m– subhypergroup H.

2010 Mathematics subject classiﬁcation: Primary 43A62, Secondary 46L55, 43A20. Keywords and phrases: Locally compact hypergroup, locally compact group, hecke algebra, measure algebra.

1. Introduction and Preliminaries In this work we extend the ideas in [5] and [2] for a hypergroup pair (K, H) in which K is a discrete hypergroup and H is an m-subhypergroup of K, where m is a left Haar measure of K. This new structure coincides classical Hecke algebra in the group case. We show that a property putting on elements of Ω, called condition (β), is equivalent with the associativity of the convolution product between elements of Ω. Our main references for hypergroups are [1], [3] and [4]. Deﬁnition 1.1. Suppose that K is a locally compact Hausdorﬀ space, (µ, ν) 7→ µ ∗ ν is a bilinear positive-continuous mapping from M(K) × M(K) into M(K) (called convolution), and x 7→ x− is an involutive homeomorphism on K (called involution) with the following properties: (i) M(K) with ∗ is a complex associative algebra; (ii) if x, y ∈ K, then δx ∗ δy is a probability measure with compact support; (iii) the mapping (x, y) 7→ supp(δx ∗ δy) from K × K into C(K) is continuous, where C(K) is the set of all non-empty compact subsets of K equipped with Michael topology; (iv) there exists a (necessarily unique) element e ∈ K (called identity) such that for all x ∈ K, δx ∗ δe = δe ∗ δx = δx; − (v) for all x, y ∈ K, e ∈ supp(δx ∗ δy) if and only if x = y ; Then K ≡ (K, ∗,− , e) is called a locally compact hypergroup.

∗ speaker

197 S. M. Tabatabaie and B. H. Sadathoseyni

Let f and g be complex-valued Borel measurable functions on K. For each x, y, z ∈ K we deﬁne ∫ y fx(y) = f (x) = f (x ∗ y):= f d(δx ∗ δy) and f (x ∗ y ∗ z):= fx(y ∗ z). K Throughout this paper, we assume that K is a discrete hypergroup with left Haar measure m, and H is a subhypergroup of K. We denote by C(K) the space of all ﬁnite supported complex-valued functions on K. We denote H\K := {H ∗ a : a ∈ K}, K/H := {a ∗ H : a ∈ K}, and H\K/H := {H ∗ a ∗ H : a ∈ K}. The set of all functions f ∈ C(K) such that f is constant on all left and right cosets of H is denoted by CT(H).

2. Main Results Deﬁnition 2.1. Let K be a discrete hypergroup with a Haar measure m, and H be a subhypergroup of K. We say that H satisﬁes condition (β) if for every f ∈ CT(H), every set A of representatives of distinct elements of H\K and each a ∈ K, there is a a a set B of representatives of distinct elements of H\K such that (χA f ) dm = χB f dm, i.e. for all g ∈ C(K), ∫ ∫

g(x)(χA f )(x ∗ a) dm(x) = g(x) f (x ∗ a) dm(x). K B Proposition 2.2. Let G be a discrete group and H be a subgroup of G. Then H satisﬁes in condition (β). Deﬁnition 2.3. Let K be a discrete hypergroup and H be a subhypergroup of K satisfying the condition (β). The set of all functions f ∈ C(K) satisfying following properties is denoted by Ω: 1. for each x ∈ K, f is constant on H ∗ x; 2. for each x ∈ K, f is constant on x ∗ H; c 3. for each x, y, z ∈ K, h ∈ H and µ, ν ∈ M+(K), if x ∗ y ⊆ x ∗ z ∗ h, then ∫ ∫

f d(µ ∗ δx ∗ δy ∗ ν) = f d(µ ∗ δx ∗ δz ∗ δh ∗ ν); K K c 4. for each x, y, z ∈ K, h ∈ H and µ, ν ∈ M+(K), if y ∗ x ⊆ h ∗ z ∗ x, then ∫ ∫

f d(µ ∗ δy ∗ δx ∗ ν) = f d(µ ∗ δh ∗ δz ∗ δx ∗ ν). K K Definition 2.4. Let K be a discrete hypergroup and m be a left Haar measure of K. A subhypergroup H of K is called m-subhypergroup if χAdm = χBdm, where A and B are two arbitrary sets of representatives of distinct elements of H\K. Definition 2.5. Let K be a discrete hypergroup with a left Haar measure m, and H be an m–subhypergroup of K. For each f, g ∈ Ω and x ∈ K, we define ∫ f ∗ g(x):= f (x ∗ y−)g(y) dm(y) and f ∗(x):= f (x−), A

198 Hecke ∗-Algebras on Hypergroups where in the above integral, A is a set of representatives of distinct elements of H\K. In Theorem 2.8 we will show that the above convolution does not depend on the choice of the set A. Lemma 2.6. Let K be a discrete hypergroup with a left Haar measure m, H be a subhypergroup of K satisfying the condition (β), f, g ∈ C(K), and a ∈ K. If A is a set of representatives of distinct elements of H\K, then there is a set B of representatives of distinct elements of H\K, such that

a a [χA ( f ∗ g)] = [(χB f ) ∗ g] . Proposition 2.7. Let K be a discrete hypergroup with a left Haar measure m, H be a subhypergroup of K satisfying the condition (β), f, g ∈ C(K), and a ∈ K. For every set B of representatives of distinct elements of H\K, there is a set E of representatives of distinct elements of H\K such that ∫ ∫ a− h(y)[χA ( f ∗ g)](y ∗ a) dm(y) = (χEh )(y)( f ∗ g)(y)dm(y). K K Theorem 2.8. Let K be a discrete hypergroup with a left Haar measure m, H be an m- subhypergroup of K satisfying the condition (β). Then with convolution and involution introduced in Definition 2.5, the space Ω is an ∗-algebra. Corollary 2.9. Let K be a discrete hypergroup and H be a subhypergroup of K. Ω with the convolution and involution introduced in Definition 2.5 is a ∗–algebra if and only if H satisfy in the condition (β). Definition 2.10. The set Ω with above convolution and involution is denoted by H(K, H) and is called hyper-Hecke ∗-algebra. Remark 2.11. Note that in the case that G is a discrete group, Γ is a subgroup of G, K := Γ\G/Γ, doubel cosets of Γ in G, and H := {ΓeΓ}, the trivial subhypergroup of K, then H(K, H) is the classical ∗-Hecke algebra. Theorem 2.12. If K is a discrete hypergroup and H is a subhypergroup of K, then H satisfies the condition (β) if and only if K is a group. In the other words, the space Ω related to (K, H) is an ∗–algebra if and only if K is a group.

References [1] W. R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, De Gruyter, Berlin, 1995. [2] J.-B. Bost and A. Connes, Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (N. S.), 1(1995), 411-457. [3] S. M. Tabatabaie, The problem of density on L2(G), Acta Math. Hungarica, 150 (2016), 339-345. [4] S. M. Tabatabaie, The problem of density on commutative strong hypergroup, to appear in Math. Reports [5] K. Tzanev, Hecke C∗-algebras and amenability, J. Operator Theory, 50(2003), 169-178.

199 S. M. Tabatabaie and B. H. Sadathoseyni

Seyyed Mohammad Tabatabaie, Department of Mathematics, University of Qom, Qom, Iran e-mail: [email protected]

Bentolhoda Sadathoseyni, Department of Mathematics, University of Qom, Qom, Iran e-mail: [email protected]

200 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

ON THE HIGHER DIMENSIONAL SHEARLET GROUP

Masoumeh Zare∗ and Rajab Ali Kamyabi Gol

Abstract In this paper we deﬁne standard higher dimension shearlet group S = R+ × Rn−1 × Rn for n ≥ 3 and determine the square-integrable sub-representations and the dual space of S.

2010 Mathematics subject classiﬁcation: Primary 22D10, Secondary 42C15. Keywords and phrases: Higher dimensional shearlet group, Irreducible representation.

1. Introduction Wavelet systems have been a popular method to analyze multidimensional data; however, these systems do not yield any information about directional components. To solve this problem, several approaches have been suggested in the context of directional signal analysis such as ridgelets, curvelets, contourlets and surfacelet, shearlets, and many others. Among all these approaches, the shearlet transform stands out because it is related to group theory, i.e., this transform can be derived from a square-integrable representation of the shearlet group (see [3]). Therefore, in the context of the shearlet transform, all the powerful tools of group representation theory can be exploited. For analyzing data in Rn, n ≥ 3, the shearlet transform has to be generalized to higher dimensions. Finding optimal representations of signals in higher dimensions is currently the subject of the researches by Dahlke et al. [2]. It is known that the standard unitary representations of shearlet group (2-D) and standard higher dimensional shearlet group are not square-integrable or even irreducible. In this paper we determine all irreducible and square-integrable sub- representations of these groups, where recently in [1] the authors characterize irreducible as well as square-integrable subrepresentations of the standard shearlet group representation in 2-D. Also we identify the dual of shearlet groups in arbitrary dimensional, with the structure of Mackey Theory [4].

2. Standart higher dimensional shearlet group For analysing data in Rn, n ≥ 3, Dahlke et al. [2] generalized two dimensional shearlet transform to higher dimensions, in the fallowing method. ∗ speaker

201 M. Zare, R. A. Kamyabi Gol

Let In denote the n × n identity matrix, also 0n, the vector with n entries. For a ∈ R∗ := R \{0} and s ∈ Rn−1 ( ) ( ) = a 0n−1 = 1 s . Aa T 1 and S s T | | n − 0n−1 sgn(a) a In−1 0n−1 In 1

The choice of S s lead shearlet transform to be a square integrable group representation. In order to have directional selectivity, the dilation factors at the diagonal of Aa is chosen in an anisotropic way, i.e., if the ﬁrst diagonal entry is a, the other ones should increase less than linearly in a as a → ∞. The set R∗ × Rn−1 × Rn endowed with the operation ′ ′ ′ ′ 1− 1 ′ ′ (a, s, t) ◦ (a , s , t ) = (aa , s + |a| n s , t + S sAat ) is a locally compact group, which is called full higher dimensional shearlet group. The left and right Haar measures on S are given by 1 1 dµ (a, s, t) = dadsdt and dµ (a, s, t) = dadsdt. l |a|n+1 r |a| For f ∈ L2(Rn) the map π : S → U(L2(Rn)), deﬁned by

1 − − − π , , = = | | 2n 1 1 1 − . (a s t) f (x) fa,s,t(x): a f (Aa S s (x t)) (1) is a unitary irreducible representation of full higher dimensional shearlet group on the 2 n Hilbert space L (R ), with respect to the Haar measure dµl. It is worthwhile to know that this representation is irreducible square integrable representation. Recall that a non trivial function ψ ∈ L2(Rn) is called admissible with respect to π if ∫ 2 | ⟨ψ, π(a, s, t)ψ⟩ | dµl(a, s, t) < ∞. (2) S If there exists at least one admissible vector ψ ∈ L2(Rn) with respect to the representation π , then π is called square-integrable. In the sequel, by a square-integrable representation, we mean irreducible square-integrable representation. + Consider( n × n dilation) matrices, depend on a parameter a ∈ R , deﬁned by a 0n−1 Aa = 1 , in which In is n × n identity matrix. The dilation factor T n 0n−1 a In−1 at the diagonal of Aa is chosen in an anisotropic way. This choice of Aa enables us to( detect special) directional information. The n × n shear matrices is deﬁned by = 1 s , ∈ Rn−1 Ss T where s . The set of shear matrices form a subgroup of 0n−1 In−1 GLn(R), all real n × n non-singular matrices . The similar calculation like full higher dimensional shearlet group [2], show that the set R+ × Rn−1 × Rn equipped with the group operations

′ ′ ′ ′ 1− 1 ′ ′ (a, s, t)o(a , s , t ) = (aa , s + a n s , t + S sAat ) and

202 On the higher dimensional shearlet group

− − 1 − − − , , 1 = 1, − n 1 , − 1 1 , (a s t) (a a s Aa S s t) is a locally compact group and we call it standard higher dimensional shearlet group 1 S. Its left and right Haar measures are dµ (a, s, t) = da ds dt and dµ (a, s, t) = l an+1 r 1 , ψ ∈ 2 Rn S a da ds dt respectively. For any L ( ), the quasi-regular representation of is deﬁned by

1 − − − π , , = = 2n 1 1 1 − , (a s t) f (x) fa,s,t(x): a f (Aa S s (x t)) (3) which is a unitary but not irreducible representation of S on U(L2(Rn)). Theorem 2.1. Square-integrable and irreducible sub-representations of higher dimensional shearlet group, which is deﬁned in (3), are precisely the following two sub-representations

2 π+ S −→ H , π+ , , ψ = ψ , , : U( A+ ) (a s t) (a s t) and

2 π− S −→ H , π− , , ϕ = ϕ , , : U( A− ) (a s t) (a s t),         x1 x1  .   .  respectively, where A+ = { .  , x > 0, x ∈ R, i = 2, ..., n}, A− = { .  , x < 0, x ∈   1 i   1 i xn xn R, = , ..., } Rcn H 2 i 2 n are the subsets of and A are the inverse Fourier transform of 2 L (A). In the sequel, by higher dimensional shearlet group S, we mean the standard higher dimensional shearlet group. We regard our higher dimensional{ shearlet group,} the semi n n direct product group S = H ⋉τ R , in the form S = (M, t); M ∈ H, t ∈ R , where H is the group of the matrices

n−1 + H = {S sAa; s ∈ R , a ∈ R } ⊆ GL(n, R), (4)

n and the homomorphism τ : H → Aut(R ) is deﬁned by τM(t) = Mt = S sAat. S Rcn , .χ = χ τ − Note that the action of on given by (M t) : { o} M 1 is free, that is, for any cn χ = (χ1, ..., χn) ∈ R , the stabilizer subgroup Hχ = In . The structure of the orbits associated with this action, depends on the choice of χi, i = 1, ..., n, to be positive, negative or zero. Theorem 2.2. Let S = R+ × Rn−1 × Rn be the standard higher dimensional shearlet group. Then the dual of S is the set bS = { S χ } ∪ { S ξ × χ ∈ \{ , ..., , , , ..., }, ξ ∈ ind n−1 (1,0,...,0) ind n−1 n ( x); x X (0 0) ( 1 0 0) Rχ (R ⋉R )χ (1,0,...,0) x d α Rn−1} ∪ {π ; α ∈ R} ∪ {πx; x ∈ Ω \ (0, ..., 0)},

203 M. Zare, R. A. Kamyabi Gol

+ − χ Rcn πα , , = iα πx , , = R ⋉Rn 1 γ , such that is character in , (a s t) a and (a s t) indRn−1 x for dn−1 γ ∈ R . Also X = {(1, 0, ..., 0), (−1, 0, ..., 0)} ∪ {(0, x1, x2, ..., xn−1); xi ∈ {0, −1, 1}, i = cn 1, ..., n − 1} and Ω = {(x1, ..., xn−1); xi ∈ {0, 1, −1}, i = 1, ..., n − 1} as the subsets of R d and Rn−1 respectively. As we know the dual of groups are introduced just as an equivalent class of irreducible representation.

Corollary 2.3. The representations π, which are deﬁned in Theorem 2.1, are unitarily S χ equivalent with the irreducible representations ind n−1 (1,0,...,0). Rχ (1,0,...,0)

References [1] V. Atayi and R. A. Kamyabi-Gol, On the characterization of subrepresentations of shearlet group,Wavelets and Linear Algebra, (2015) 1-9. [2] S. Dahlke and G. Teschke, The continuous shearlet transform in arbitrary space dimensions, J. Fourier Anal. App, (2010) 340 - 364. [3] D. Labate, W. Lim, G. Kutyniok and G. Weiss, Sparse multidimensional representation using shearlets, SPIE Proc. 5914, SPIE, Bellingham, (2005) 254 - 262. [4] G. W. Mackey, On induced representations of groups, Amer. J. Math, (1951) 576 - 592.

Masoumeh Zare, Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran e-mail: [email protected]

Rajab Ali Kamyabi Gol, Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran e-mail: [email protected]

204 Posters The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

NUMERICAL STUDY OF RADIAL SCHRÖDINGER EQUATION FOR FREE PARTICLE, USING LEGENDER WAVELET

Mohammad Sadeghi, Fakhradin Mohammadi and Niloufar AAalipour∗

Abstract This paper deals with the Legendre wavelet (LW) collocation method for the numerical solution of the radial Schrödinger Equation for free particle (electron) in spherical coordinate. Approximately analytical also numerical results of the ground state mode l = 0, of wave function or probability density function R(r), has been presented and compared with the exact solution.

2010 Mathematics subject classiﬁcation: 46N50, 65M70, 65T60. Keywords and phrases: Schrödinger equation, Potential barrier, Legendre wavelet, Collocation method.

1. Introduction One of the most important eigenvalue equations in physics is the Schrödinger’s wave equation. For a particle mass m in the potential V(r) is: ℏ2 − ∇2ψ(r) + V(r) = Eψ(r) (1) 2m In which ψ(r) is the particle wave function and E is an energy eigenvalues [1]. For one-dimensional potential V(x), equation (1) is as follow; ℏ2 d2ψ(x) − + V(x) = Eψ(x) (2) 2m dx2 The Schrödinger’s wave function, must be convergent as x → ∞, because: ∫+∞ ψ(x)ψ∗(x)dx = 1, (3) −∞ which means that the particle must be somewhere in the x axes [1]. The ﬁrst derivative of the wave function also must be continuous as if, it’s second derivative which is appearing in the equation (2), could be exist.

∗ speaker

206 M. Sadeghi, F. Mohammadi and N. Aalipour

The numerical solution of the above equation is the subject of many research in the last two decades. This equation has the analytical answers for the few potential energy V(r). For many potentials, in physics and chemistry, it cannot be solved analytically. So in quantum mechanics, the numerical or approximately analytical solution of the Schrödinger’s wave equation is a real need. The aim of this paper is to study and obtain the results of the new approach of, Legendre wavelet expansion of the solution of Eq. (1). comparison will be made whit the other well known numerical solution’s methods.

2. Schrödinger equation for free particle In the spherical coordinate, Radial Schrödinger equation for the central potential V(r) as follows: [ ] [ ] d2 2 d 2µ l(l + 1)ℏ2 + R(r) + E + V(r) − R(r) = 0 (4) dr2 r dr ℏ2 2µr2 µ is the reduced mass of system, l is the orbital quantum number and R(r) is the radial wave function. In free space, particle moves without any potential energy V(r) = 0, thus: [ ] [ ] d2 2 d 2µ l(l + 1)ℏ2 + R(r) + E − R(r) = 0. (5) dr2 r dr ℏ2 2µr2 For simplicity, rewrite Eq. (5) in dimensionless form as [1]: d2R(ρ) 2 dR(ρ) l(l + 1) + − R(ρ) + R(ρ) = 0 (6) dρ2 ρ dρ ρ2 where: √ 2µ|E| ρ = r. (7) ℏ2 With assumuming that particle is electron, we have: 2µ 2m c2 2 × 0.511 × 106(eV) 1 = e ≃ ≃ 0.26246 ( ). (8) ℏ2 ℏ2c2 (1973)2(eV.A◦)2 eV.A◦2 From quantum mechanics it is known that, r2|R(r)|2 represents the probability density of ﬁnding the particle in place of r from the center of coordinate. State which particle l = have it’s minimum energy, called the ground state. In this state 0 and the initial R r | = dR = conditions ( ) r=0 1 and dr r=0 0, is imposed maximum presence of particle at the origin.

3. Method of solution Consider the the Schrödinger equation (5). First, we approximate R(t) in terms of the LWs [2, 3] as follows R(r) ≃ CT Ψ(r) = ΨT (r)C, (9)

207 Numerical study of Schrödinger Equation where C is the LWs coeﬃcient vector. By using the approximate R(t) ≃ CT Ψ(t) and operational matrix of derivative D, the residual function for the Schrödinger equation in the ground state l = 0, can be written as [ ] 2 2µ Res(r) = CT D2 + D Ψ(r) + CT [E − 0] Ψ(r), (10) r ℏ2 Hereafter, in order to approximate solution of the Schrödinger equation (5) with initial conditions, as in the typical collocation method, we generate 2k(M + 1) − 2 equations by applying

k Res(ri) = 0, i = 1, 2..., 2 M − r. (11) Moreover, by using following initial condition: CT Ψ(10−4) = 1, (12)

CT DΨ(10−4) = 0, (13) Eqs. (11) together with (12) and (13) generate a system of 2k(M + 1) algebraic equations for 2k(M + 1) unknown elements of the unknown vector C. This system can be solved for unknown coeﬃcient vector C and unknown function R(r) can be obtained by substituting the obtained vector C in Eq. (9). With the above considerations, approximate analytical LW expansion of wave function, R(r), with M = 8, k = 0 is obtained as follow:   0.0000000 r < 0   − . 8 + . 7−  0 00018597601r 0 00083651506r  − . 6 + . 5−  0 0015816998r 0 0016209070r ≃ − . 4 + . 3− ≤ < R(r)  0 00041023806r 0 00035908484r 0 r 1 (14)  − . 2 + . +  0 043819796r 0 00000875837r  + .  1 0000001 0.0000000 1 < r. Numerical results for the solution of the radial Schrödinger Eq. (5) in the case of l = 0, is shown in Table 1, for various methods.

4. Conclusion In this paper a new approach, Legendre wavelet (LW) is used and an approximate analytic expansion is derived for radial wave function of free electron in its ground state. Numerical results obtained from LW expansion, compared with exact and other well known numeric methods in Table 1. Runge Kutta Fehlberg and LW expansion methods, are shown accurate results. Forth order Runge Kutta method and Modiﬁed Euiler method (Heun) both with step size 0.1, are shown less accurate results than others.

208 M. Sadeghi, F. Mohammadi and N. Aalipour

Table 1. Comparison of the numerical solution for the radial Schrödinger equation in the ground state of the free electron R(r) r Exact RKF45 RK4 HUEN LW solution h = 0.1 h = 0.1 expansion 0.1 0.99956 0.99956 0.99869 0.99869 0.99956 0.2 0.99825 0.99825 0.99777 0.99738 0.99825 0.3 0.99607 0.99607 0.99574 0.99541 0.99607 0.4 0.99302 0.99302 0.99277 0.99251 0.99302 0.5 0.98910 0.98910 0.98891 0.98869 0.98910 0.6 0.98433 0.98433 0.98417 0.98398 0.98433 0.7 0.97870 0.97870 0.97857 0.97841 0.97870 0.8 0.97224 0.97224 0.97212 0.97198 0.97224 0.9 0.96494 0.96494 0.96484 0.96472 0.96494 1.0 0.95683 0.95683 0.95674 0.95662 0.95683

References [1] S. Gasiorowicz, Quantum Physics. 3rd Edn., JohnWiley and Sons, New York, (2003). [2] M. Razzaghi, S. Yousefi, The Legendre wavelets operational matrix of integration, International Journal of Systems Science (2001): 495-502. [3] F. Mohammadi, M. M. Hosseini, and S. T. Mohyud-Din. Legendre wavelet Galerkin method for solving ordinary differential equations with non-analytic solution, International Journal of Systems Science, (2011) 579-585. [4] F. Mohammadi, M. M. Hosseini, A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations, Journal of the Franklin Institute .

Mohammad Sadeghi, Department of Physics, University of Hormozgan, City Bandarabbas, Iran e-mail: [email protected]

Fakhradin Mohammadi, Department of Mathematics, University of Hormozgan, City Bandarabbas, Iran e-mail: [email protected]

Niloufar AAalipour, Department of Physics, University of Hormozgan, City Bandarabbas, Iran e-mail: [email protected]

209 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

WEIGHTED HARDY-TYPE INEQUALITIES WITH SHARP CONSTANTS IN LP SPACES

Omid Baghani∗ and Zohre Ghazvini

Abstract Error estimate and the rate of convergence are very important in the framework of numerical analysis. Without doubt having enough information of the most important inequalities play an important role in achieving better bound in numerical algorithms. This paper focuses on Hardy’s inequality associated with the Jacobi weight function ωα,β = (1 − x)α(1 + x)β with I := (−1, 1).

2010 Mathematics subject classiﬁcation: Primary 39B72, Secondary 26A33. Keywords and phrases: Hardy-type Inequalities, weighted spaces, jacobi weight function.

1. Introduction Hardy-type inequalities have attracted a lot of interest during all the years from the dramatic prehistory (see until Hardy discovered his famous inequality in 1925 [1]) to a still very active research (see [2, 3]). We now recall that∫ classical one-dimensional Hardy inequality [4]: let 1 < p < ∞, x f t ≥ F x = f t dt ( ) 0, and ( ) 0 ( ) , then ∫ ∫ ∞ ( ) ( ) ∞ F(x) p p p dx ≤ F′(x)dx. 0 x p − 1 0 In this work, we give an extension of this inequalities for the weight functions ω(x) = (x − a)α and ω(x) = (b − x)α.

2. Main result Let Ω be a Lebesgue-measurable subset of Rd (d = 1, 2, 3) with non-empty interior, and let f be a Lebesgue measurable function on Ω. Deﬁnition 2.1. For 1 ≤ p ≤ ∞ and the positive weight function ω(x) (a.e.), let

p Lω(Ω):= { f : f is measurable on Ω and ∥ f ∥p,ω < ∞},

∗ speaker

210 2 O. Baghani, Z. Ghazvini where for 1 ≤ p ≤ ∞, ∫ ( ) / p 1 p ∥ f ∥p,ω := | f (x)| ω(x)dx , Ω and ∥ f ∥∞ := ess supx∈X| f (x)|.

m Deﬁnition 2.2. The Sobolev space Hω (Ω) with m ∈ N is the space of functions 2 f ∈ Lω(Ω) such that all the distributional derivatives of order up to m can be 2 represented by functions in Lω(Ω). that is,

m 2 γ 2 Hω (Ω) = { f ∈ Lω(Ω): D f ∈ Lω(Ω) f or 0 ≤ |γ| ≤ m}, equipped with the norm ( ∑m ) γ 1/2 ∥ f ∥m,ω = ∥D f ∥2,ω . |γ|=0 m ∞ Ω m Ω Definition 2.3. The space H0,ω is defined as the closure of C0 ( ) in Hω ( ), where ∞ Ω ff Ω C0 ( ) is the set of all infinitely di erentiable functions with compact support in . < < < ∞ = p Theorem 2.4. Suppose that a b be two real numbers, 1 p and q p−1 . Let α < p ∈ p , ω = − α q . Then for any f L (a b) with (x) (x a) , we have the following Hardy inequalities: ∫ ∫ ∫ b ( x ) ( )( ) b 1 p 1 p p f (t)dt (x − a)αdx ≤ f p(x)(x − a)αdx. (1) − − q α − a x a a 1 p p 1 a Similarly, for any f ∈ Lp(a, b) with ω(x) = (b − x)α, we have ∫ ∫ ∫ b ( b ) ( )( ) b 1 p 1 p p f (t)dt (b − x)αdx ≤ f p(x)(b − x)α. (2) − − q α − a b x x 1 p p 1 a Proof. Firstly we prove the first inequality. The prove of the second inequality < β < 1 is similiar to the first∫ one. Pick 0 q , we will specifie it later. Define = − (α/p)−1 x F(x) (x a) a f (t)dt. We start using Hölder’s inequality ∫ ∫ x x |(x − a)1−(α/p)F(x)| = | f (t)dt| ≤ | f (t)(t − a)β||(t − a)−β|dt a ∫a ∫ x x ( )1/p( )1/q ≤ | f (t)|p(t − a)pβdt (t − a)−qβ ∫a a ( x ) / ( − 1−qβ ) / ≤ | |p − pβ 1 p (x a) 1 q f (t) (t a) dt − β a ∫ 1 q ( x )1/p = (1 − qβ)−1/q | f (t)|p(t − a)pβdt (x − a)1/q−β, a

211 Weighted Hardy-type Inequalities with Sharp Constants in Lp Spaces 3 hence ∫ ( x )1/p |F(x)| ≤ (1 − qβ)−1/q | f (t)|p(t − a)pβdt (x − a)−β−((1−α)/p). a Therefore ∫ ( x ) |F(x)|p ≤ (1 − qβ)−p/q | f (t)|p(t − a)pβdt (x − a)−pβ−(1−α). a Integrating and using Fubini’s theorem, we get ∫ ∫ ∫ b b ( x ) |F(x)|pdx ≤ (1 − qβ)−p/q | f (t)|p(t − a)pβ(x − a)−pβ−(1−α)dt dx a ∫a ∫a b ( b ) ≤ (1 − qβ)−p/q | f (t)|p(t − a)pβ(x − a)−pβ−(1−α)dx dt ∫a t ∫ b ( b ) ≤ (1 − qβ)−p/q | f (t)|p(t − a)pβ (x − a)−pβ−(1−α)dx dt ∫a t b ( 1 ) ≤ (1 − qβ)−p/q | f (t)|p(t − a)pβ ((b − a)α−pβ − (t − a)α−pβ) dt (α − pβ) ∫a ∫ (1 − qβ)−p/q ( b b ) = | f (t)|p(t − a)pβ(b − a)α−pβdt − | f (t)|p(t − a)pβ(t − a)α−pβdt (α − pβ) ∫a ∫ a (1 − qβ)−p/q ( b b ) = − | f (t)|p(t − a)αdt − | f (t)|p(t − a)pβ(t − a)α−pβdt (α − pβ) ∫a a (1 − qβ)−p/q ( b ) ≤ − | f (t)|p(t − a)αdt , (α − pβ) a and ﬁnally ∫ ∫ b (1 − qβ)−p/q ( b ) |F(x)|pdx ≤ | f (t)|p(t − a)αdt . a (pβ − α) a β = 1 < 1 Now for a sharper bound in above inequality, we pick : pq q in numerator and β = 1 : q in denominator, to get 1 (1 − qβ)−p/q(pβ)−1 = (1 − )−1/qq = qp/qq = q1+p(1−1/p) = qp, p and q (1 − (pβ)−1α) = 1 − α. p This completes the proof. □

3. Application In this section, we apply the above Hardy inequality to derive some useful inequalities associated with the Jacobi weight function ωα,β = (1−x)α(1+x)β with I := (−1, 1). The following inequalities can be found in [5].

212 4 O. Baghani, Z. Ghazvini

Lemma 3.1. If −1 < α, β < 1, then ′ 1 ∥ ∥ α− ,β− ≤ ∥ ∥ α,β , ∀ ∈ , f 2,ω 2 2 c f 2,ω f H0,ωα,β (I) (3) which implies the poincaré-type inequality: ′ ∥ ∥ α,β ≤ ∥ ∥ α,β , ∀ ∈ 1 , f 2,ω c f 2,ω f H0,ωα,β (I) (4) Proof. Taking a = −1, b = 1 and ϕ = d f in (2) yields that for α < 1, ∫ dx ∫ 1 1 f 2(x)(1 − x)α−2dx ≤ c f ′2(x)(1 − x)αdx. 0 0 hence, ∫ ∫ 1 1 f 2(x)(1 − x)α−2(1 + x)β−2dx ≤ c f 2(x)(1 − x)α−2dx 0 ∫0 1 ≤ c f ′2(x)(1 − x)αdx ∫0 1 ≤ c f ′2(x)(1 − x)α(1 + x)βdx. 0 Similarly, for β < 1, we use (2) to derive ∫ ∫ 0 0 f 2(x)(1 − x)α−2(1 + x)β−2dx ≤ c f ′2(x)(1 − x)α(1 + x)βdx. −1 −1 A combination of the above two inequalities leads to (3). In view of ωα,β(x) < ωα−2,β−2(x),(4) follows from (3). □

References [1] A. Kufner, L. Maligranda and L. E. Persson, The prehistory of the Hardy inequality, Amer. Math . Monthly 113 (2006) 715-732. [2] A. Kufner, L. Maligranda and L. E. Persson, The Hardy Inequality - About Its History and Some Related Results, Vydavetelsky Servis Publishing House, Pilsen, 2007. [3] A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientiﬁc, Singapore, 2003. [4] G. H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920) 314317. [5] J. Shen, T. Tang, L. L. Wang, Spectral Methods. Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 2011.

Omid Baghani, Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran e-mail: [email protected]

Zohre Ghazvini, Iran’s ministry of education-sabzevar e-mail: [email protected]

213 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

GENERALIZED SHIFT-INVARIANT SYSTEMS AND DUAL FRAMES FOR SUBSPACES

Nafiseh Gholami

Abstract

d 2 d Let Tk denote translation by k ∈ Z . Given countable collections of functions {ϕ j} j∈J , {ϕ˜j} j∈J ⊂ L (R ) and assuming that {Tkϕ j} j∈J,k∈Zd and {Tkϕ˜j} j∈J,k∈Zd are Bessel sequences, we are interested in expansions ∑ ∑ f = < f, Tkϕ˜ j > Tkϕ j, ∀ f ∈ span{Tkϕ j}k∈Zd . j∈J k∈Zd Our main result gives an equivalent condition for this to hold in a more general setting than described here, where translation by k ∈ Zd is replaced by translation via the action of a matrix. As special cases of our result we ﬁnd conditions for shift-invariant systems to generate a subspace frame with a corresponding dual having the same structure.In this paper we consider the important case of generalized shift-invariant systems and provide various ways of estimating the deviation from perfect reconstruction that occur when the systems do not form dual frames.The deviation from being dual frames will be measured either in terms of a perturbation condition or in terms of the deviation from equality in the duality conditions.

2010 Mathematics subject classiﬁcation: Primary 42C40, secondary 42C15. Keywords and phrases: Frames for subspaces, generalized shift-invariant systems, approximately dual frames .

1. Introduction Given a real and invertible d×d matrix C, we deﬁne for k ∈ ퟋd a generalized translation 2 d operator TCk acting on f ∈ L (R ) by

d (TCk f )(x) = f (x − Ck), x ∈ (R )

{ ϕ } d A generalized shift-invariant system is a system [1–3] of the type TC jk j j∈J,k∈Z where 2 d {C j} j∈J is a countable collection of real invertible d × d matrices, and {ϕ j} j∈J ⊂ L (R ). Generalized shift-invariant systems contain the classical wavelet systems and Gabor systems as special cases. Given the matrices {C j} j∈J, we are interested in functions 2 d {ϕ } {ϕ˜} ⊂ R { ϕ } d { ˜ } d j j∈J , j∈J L ( ) for which TC jk j j∈J,k∈Z and TC jk phi j j∈J,k∈Z are Bessel sequences and the expansions ∑ ∑

= < , ϕ˜ > ϕ , ∀ ∈ { ϕ } d f f TC jk j TC jk j f span TC jk j j∈J,k∈Z j∈J k∈Zd

214 N. Gholami

{ }∞ { }∞ H hold. If two sequences fk k=1 and gk k=1 in a separable Hilbert space form a pair of dual frames for H, each f ∈ H has a representation ∑∞ f = < f, gk > fk. (1) k=1 In signal processing terms this is expressed by saying that dual pairs of frames leads to perfect reconstruction.

2. Preliminaries and notations

{ ϕ } d we consider generalized shift-invariant systems of the type TC jk j j∈J,k∈Z , where 2 d {C j} j∈J is a countable collection of real invertible d × d matrices, and {ϕ j} j∈J ⊂ L (R ). Letting CT denote the transpose of an invertible matrix C, we use the notation C♯ = (CT )−1. For f ∈ L1(Rd) we denote the Fourier transform by ∫ F f (γ) = fˆ(γ) = f (x)e−2πix.γdx, Rd where x.γ denotes the inner product between x and γ . As usual, the Fourier 2 d transform [1] is extended to a unitary operator on L (R ). Furthermore, With Eb(x):= e2πib.x, b, x ∈ Rd , this yields the commutator relation

F TCk = E−CkF . While we present the general theory for the d-dimensional case, our discussion of shift-invariant systems will take place in L2(R). Generalized shift-invariant systems in L2(R) will be denoted by { ϕ } , > , ϕ ∈ 2 R Ta jk j j∈J,k∈Z where a j 0 j L ( ) Most of our calculations rely on Fourier transformation techniques rather than general Hilbert space results. { }∞ { }∞ { }∞ Two frames fk k=1 and gk k=1 are dual frames if (1) holds. Note that if fk k=1 and { }∞ { }∞ { }∞ gk k=1 are Bessel sequences and (1) holds, then fk k=1 and gk k=1 are automatically { }∞ frames. Given any Bessel sequence∑ fk k=1 one can deﬁne a bounded operator ℓ2 N −→ H { }∞ = T : ( ) by T ck k=1 : ck fk; the operator T is called the synthesis operator or preframe operator [2]. It is easy to see that the adjoint operator is given ∗ H −→ ℓ2 N ∗ = {< , >}∞ by T : ( ), T f f fk k=1. Denoting the synthesis operators for two { }∞ { }∞ { }∞ Bessel sequences fk k=1 and gk k=1 by T, respectively, U, it is clear that fk k=1 and { }∞ ∗ = gk k=1 are dual frames if and only if TU I. We note that in the summation over n ∈ Zd, the indices j, n always appear in the ♯ combination C jn; let us consider all possible outcomes, i.e., let Λ = { ♯ ∈ , ∈ Zd}. C jn : j J n (2)

215 Generalized SHift-Invariant Systems and Dual Frames for Subspaces

α ∈ Λ , ∈ × Zd α = ♯ Given , there might exist several pairs ( j n) J for which C jn; let = { ∈ ∃ ∈ Zd α = ♯ Jα j J : n such that C jn (3) Definition 2.1. A generalized shift invariant system [3] in L2(R) is a collection of { ϕ } {ϕ } ⊂ 2 R { } functions of the form Tc jk j k∈Z, j∈J, where j j∈J L ( ) and c j j∈J is a countable collection of positive numbers. { ϕ } { ϕ˜ } Definition 2.2. Consider two GSI-systems Tc jk j k∈Z, j∈J and Tc jk j k∈Z, j∈J (i) If ∑ ∑ ∫ 1 | ˆ γ + −1 ϕˆ γ |2 γ < ∞ f ( c j m) j( ) d c ˆ j∈Z m∈Z j supp f for all f ∈ D, (D := { f ∈ L2(R) | supp fîs compact and fˆ ∈ L∞(R)}), we say that { ϕ } Tc jk j k∈Z, j∈J satisfies the LIC condition. { ϕ } { ϕ˜ } α (ii) Tc jk j k∈Z, j∈J and Tc jk j k∈Z, j∈J satisfies the dual -LIC condition if ∑ ∑ ∫ 1 | ˆ γ + −1 ϕˆ γ ϕ˜ˆ γ + −1 | γ < ∞ f ( c j m) j( ) j( c j m) d c ˆ j∈Z m∈Z j supp f ∈ D { ϕ } for all f . We say that Tc jk j j∈Z satisfies the -LIC condition, if holds with ϕ j = ϕˆj.

3. Main results In this section we consider translates of a single function. The results obtained here will serve as starting point for the case of multiple generators. Lemma 3.1. Let ϕ, ϕ˜ ∈ L2(Rd), let C be a real and invertible d × d matrix, and 2 d assumethat {TCkϕ}k∈Zd and{TCkϕ˜}k∈Zd are Bessel sequences. Then for all f ∈ L (R ), ∑ ∑ 1 F ( < f, T ϕ˜ > T ϕ) = ϕˆ(γ) fˆ(γ + C♯n)ϕ˜ˆ(γ + C♯n). Ck Ck | det C| k∈Zd n∈Zd { ϕ } Our rst result below characterizes pairs of a frame TC jk j k∈Zd, j∈J and correspond- { ϕ˜ } ing generalized duals TC jk j k∈Zd, j∈J. The general classication is quite involved; the next sections show how it simplies in concrete cases.

Theorem 3.2. Let J be a countable index set and consider sequences {ϕ j} j∈J and 2 d {ϕ˜j} j∈J in L (R ). Let {C j} j∈J be a sequence of invertible real matrices, and assume that {T ϕ } ∈Zd are Bessel sequences. Then the following are equivalent: c jk ∑j j ∑ = < , ϕ˜ > ϕ , ∀ ∈ { ϕ } d (i) f j∈J k∈Zd f Tc jk j Tc jk j f span Tc jk j k∈Z , j∈J (ii) for all ℓ ∈ J , m ∈ Zd

∑ ∑ ♯ 1 −2πiCℓm.C n ♯ ♯ ϕˆ γ = ϕˆ γ j ϕˆ γ + ˆϕ γ + , ℓ( ) j( )( e ) ℓ( C jn)˜ ℓ( C jn) | det C j| j∈J n∈Zd

γ { ϕ } d holds for a.e. . If the conditions are satised, then Tc jk j j∈Z is a dual frame { ϕ } d for span Tc jk j k∈Z , j∈J.

216 N. Gholami

{ ϕ } { ϕ˜ } Theorem 3.3. Assume that the GSI-systems Tc jk j k∈Z, j∈J and Tc jk j k∈Z, j∈J are Bessel sequences and satisfy the dual α-LIC-condition for all f ∈ D; denote the associated preframe operators by T, resp., U. Then ∑ ∑ ∑ ∗ 1 ˆ ˆ ∥I − UT ∥ ≤ ∥ ϕˆj(γ)ϕ˜j(γ) − 1∥∞ + ∥ ϕˆj(γ)ϕ˜j(γ + α)∥∞ c j j∈J α∈Λ\{0} j∈Jα Λ = { −1 ∈ , ∈ Z} α ∈ Λ for : C j n : j J n and for , let = { ∈ ∃ ∈ Z α = −1 }. Jα : j J : n suchthat c j n

References [1] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, (2003). [2] O. Christensen, and R.Laugesen Approximately dual frames in Hilbert spaces and applications to Gabor frames, Sampling Theory in Signal and Image Processing, 9, (2011). [3] A. Ron. and Z. Shen, Generalized shift-invariant systemsConst. Appr. No. 1, 22, (2005), 1-45.

Nafiseh Gholami, Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran e-mail: naﬁ[email protected]

217 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

SAMPLING AND REPRODUCING KERNEL HILBERT SPACE

Mahereh Mehmandust

Abstract Reproducing kernel Hilbert spaces arise in a number of areas, including approximation theorys statistics, machine learning theory, group representation theory and various areas of complex analysis. In this talk the reproducing kernel Hilbert spaces are introduced and their general properties are investigated. For better understanding of these spaces some examples are given. Moreover, in this space, a sampling formula is introduced, which is an extension of a famous formula known as the Paley-Wiener (PW) of band-limited signals. In particular, a single channel sampling formula is discussed.

2010 Mathematics subject classiﬁcation: Primary 42C40; Secondary 43A65. Keywords and phrases: Sampling, reproducing kernel Hilbert space .

1. Introduction The theory of Hilbert spaces of entire functions was ﬁrst introduced by L. de Branges in the series of papers. These spaces, which are now called de Branges spaces, generalize the classical Paley-Wiener space which consists of the entire functions of exponential type which are square integrable on the real line. Recently, Ortega Cerd‘a and K. Seip provided a description of the exponential frames for the Paley-Wiener space, and a related study of sampling and interpolation, by connecting the problem to the de Branges spaces theory of entire functions. Let f be a band-limited signal with band region [−π, π], that is, a square integrable function on R of which the Fourier transform fˆ vanishes outside [−π, π].Then f can ∑∞ sinπ(t − n) be recovered by its uniformly spaced discrete values as f (t) = −∞ f (n) , π(t − n) which converges absolutely and uniformly over R. This series is called the cardinal series or the Whittaker-Shannon-Kotelnikov (WSK) sampling series. This formula tells us that once we know the values of a band-limited signal f at certain discrete points, we can recover f completely. In1941, Hardy [2] recognized that this cardinal series is actually an orthogonal expansion. WSK sampling series was generalized by Kramer [4] in 1957 as follows: Let k(ξ, t) be a kernel on I × X , where I is a bounded interval and X is a subset of R. Assume that k(., t) ∈ L2(I) for each t in X and there 2 are points {tn}n∈Z in X such that k(ξ, tn) is an orthonormal basis of L (I). Then any

218 Mehmandust ∫ = ξ ξ, ξ ξ ∈ 2 f (t) I F( )k( t)d with F( ) L (I) can be expressed as a sampling series. ∑ ∫ f (t) = f (tn) k(ξ, t)k(ξ, tn)dξ, n I which converges absolutely and uniformly over the subset D on which ∥k(., t)∥L2(I) is bounded. While WSK sampling series treats sample values taken at uniformly spaced points, Kramers series may take sample values at nonuniformly spaced points. Recently, A. G. Garcia and A. Portal [1] extended the WSK and Kramer sampling formulas further to a more general setting using a suitable abstract Hilbert space valued kernel. On the other hand, Papoulis introduced a multi-channel sampling formula for band-limited signals, in which a signal is recovered from discrete sample values of several transformed versions of the signal.

2. Preliminaries and notations We will consider Hilbert spaces over either the field of real numbers, R, or of complex numbers, C. We will use F to denote either R or C, so that when we wish to state a definition or result that is true for either the real or complex numbers, we will use F. Given a set X, if we equip the set of all functions from X to F, F (X, F) with the usual operations of addition, ( f + g)(x) = f (x) + g(x), and scalar multiplication, (λ. f )(x) = λ.( f (x)), then F (X, F) is a vector space over F. Definition 2.1. Given a set X, we will say that H is a reproducing kernel Hilbert space (RKHS) on X over F, provided that: (i) H is a vector subspace of F (X, F), (ii) H is endowed with an inner product, <, >, making it into a Hilbert space, (iii) for every y ∈ X, the linear evaluation functional, Ey : H −→ F, defined by Ey( f ) = f (y), is bounded If H is a RKHS on X, then since every bounded linear functional is given by the inner product with a unique vector in H, we have that for every y ∈ X, there exists a unique vector, ky ∈ H, such that for every f ∈ H, f (y) =< f, ky >.

Deﬁnition 2.2. The function ky is called the reproducing kernel for the point y. The 2-variable function dened by K(x, y) = ky(x) is called the reproducing kernel for H. 2 2 Note that we have, K(x, y) = ky(x) =< ky, kx > and ∥Ey∥ = ∥ky∥ =< ky, ky >= K(y, y). We now look at a few key examples 2 D Example 2.3. The Hardy∑ Space of the Unit Disk H ( ) 2 D = { = ∞ n { } ∈ ℓ2 N ∪ { } , ∈ D}, H ( ) f (z) n=0 anz : an ( 0 ) z with the inner product, ∑∞ 1 < f, g > 2 D = a b , and the reproducing kernel K(z, w) = . This function H ( ) n=0 n n 1 − wz is called the Szego kernel on the disk.

219 Sampling and Reproducing Kernel Hilbert Space

Example 2.4. Sobolev Spaces on [0,1] ′ 2 H = { f : [0, 1] −→ R, f ∈ L ∫[0, 1], f (0) = f (1) = 0, f is absolutely continuous}. 1 with the inner product, < f, g >= f ′(t)g′(t)dt, and the reproducing kernel 0   (1 − y)x, x ≤ y if k(x, y) = ky(x) =  . (1 − x)y, x ≥ y if

3. Main results Let X be any set and let H be a RKHS on X with kernel K. In this section, we will show that K completely determines the space H and characterize the functions that are the kernel functions of some RKHS. Proposition 3.1. Let H be a RKHS on the set X with kernel K . Then the linear span of the functions, ky(.) = K(., y) is dense in H.

Lemma 3.2. Let H be a RKHS on X and let { fn} ⊆ H. If limn ∥ fn − f ∥ = 0, then f (x) = limn fn(x) for every x ∈ X.

Proposition 3.3. Let Hi, i = 1, 2 be RKHSs on X with kernels, Ki(x, y), i = 1, 2. If K1(x, y) = K2(x, y) for all x, y ∈ X, then H1 = H2 and ∥ f ∥1 = ∥ f ∥2 for every f . H , { ∈ } Theorem 3.4. Let be a RKHS on X with reproducing∑ kernel, K(x y). If es : s S H , = is an orthonormal basis for , then K(x y) s∈S es(y)es(x) where this series converges pointwise. H , Theorem 3.5. Let be a RKHS on X with reproducing kernel ∑K(x y). Then { ∈ } ⊆ H H , = fs : s S is a Parseval frame for if and only if K(x y) s∈S fs(x) fs(y), where the series converges pointwise. Let H be a separable Hilbert space and k : X −→ H be an H-valued function on a subset X of the real line R. Deﬁne a linear operator T on H by

T(x)(t) = fx(t):=< x; k(t) >H ; t ∈ X We call k(t) the kernel of the linear operator T. Lemma 3.6. (a) T is one-to-one if and only if {k(t)|t ∈ X} is total in H. Assume {k(t)|t ∈ X} is total in H so that T : H −→ T(H) is a bijection. Then (b) < T(x), T(y) >T(H):=< x, y >H deﬁnes an inner product on T(H), with which T(H) is a Hilbert space and T : H −→ T(H) is unitary. Moreover, T(H) becomes an RKHS with the reproducing kernel k(s, t):=< k(t), k(s) >H . ⊆ ˜ { }∞ {˜ } Theorem 3.7. If ker T ker T and there exists a sequence tn n=1 in X such that k(tn) n is a basis of H, then T is one-to-one so that T(H) becomes an RKHS under the inner < , > =< , > { }∞ H product x y H T(x) T(y) T(H). Moreover,∑ there is a basis S n n=1 of T( ) = ∞ ˜ , . ∈ H with which we have the sampling expansion, fx(t) n=1 fx(tn)S n(t) ( fx( ) T( )) which converges not only in T(H) but also uniformly over any subset on which ∥k(t)∥ is bounded.

220 Mehmandust

Theorem 3.8.∩ (Asymmetric nonuniform multi-channel sampling formula) ⊆ N { , ≤ ≤ , ∈ Z} ⊂ If ker T i=1 ker Ti and there exist points ti,n 1 i N n X and constants {α j , 1 ≤ i ≤ N, 1 ≤ j ≤ M, n ∈ Z} for some integer M ⩾ 1 such that ∑ i,n { N α j ≤ ≤ , ∈ Z} H i,n i,nki(ti,n) : 1 j M n is an unconditional basis of , then there is a basis{S j,n(t) : 1 ≤ j ≤ M, n ∈ Z} of T(H) such that for any fx(t) = T(x)(t) ∈ T(H), ∑ ∑m = {α j 1 + α j 2 + ... + α j N } fx(t) 1,n fx (t1,n) 2,n fx (t2,n) N,n fx (tN,n) S j,n(t) (1) n∈Z j=1 which converges in T(H). Moreover, the series (1) converges absolutely and uniformly on any subset of X over which ∥k(t)∥H is bounded. Example 3.9. Sampling with Hilbert transform

Take k˜(t)(ξ) = i sgn(ξ)(k(t))(ξ) so that T˜( f )(t) = f˜x(t) is the Hilbert transform of e−inξ f (t) in PWπ . Choosing {t } ∈Z = {n} ∈Z, {x } ∈Z = {i sgnξ √ } ∈Z is an orthonormal n n n n n 2π n 2 −π, π { ∗} = { } basis of L [ ] so that xn n∈Z xn n∈Z We then have ∫ π −inξ π = ∗ = 1 ξ e itξ ξ = − 1 − − S n(t) T(xn)(t) √ i sgn( ) √ e d sinc (t n)sin (t n) 2π −π 2π 2 2 = sinπt where sinct : πt , Hence, we have ∑ 1 π f (t) = − f˜(n)sinc (t − n)sin (t − n), ( f ∈ PWπ) x 2 2 n∈Z ˜ = − ∈ Using the operational relation∑f (t) f (t) and the fact that if f PWπ, then so does ˜ ∈ ˜ = 1 − π − , ∈ f PWπ, we also have f (t) n∈Z f (n)sinc 2 (t n)sin 2 (t n) ( f PWπ).

References [1] A. G. Garcia, and A. Portal, Sampling in the functional Hilbert space induced by a Hilbert space valued kernel, Appl. Anal. 82, (2003), 1145-1158. [2] G. H. Hardy, Notes on special systems of orthogonal functions, IV: The orthogonal functions of Whittaker’s cardinal series. Proc. Camb. Phil. soc., 37, (1941), 331-348. [3] J. M. Kim, Y. M. Hong, and K. H. Kwon, Sampling theory in abstract reproducing kernel Hilbert space, Sample. Theory Signal Image Process. 6(1), (2007), 109-121. [4] H. P. Kramer, A generalised sampling theorem, J. Math. Phys., 63, (1957), 68-72.

Mahereh Mehmandust, Department of Pure Mathematics Ferdowsi University of Mashhad, Mashhad, Iran e-mail: [email protected]

221 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

CHARACTERIZATION OF VON NEUMANN-SCHATTEN Q-RIESZ BASES IN SEPARABLE BANACH SPACES

Sima Movahed∗ and Mehdi Choubin

Abstract The concept of von Neumann-Schatten p-frames and q-Riesz bases in a separable Banach space has been introduced by G. Sadeghi and A. Areﬁjamaal. In fact, a von Neumann-Schatten p-frame is a sequence of bounded linear operators from a Banach space X into the von Neumann-Schatten p-class Cp and also p-frames and g-frames can be considered as such frames. In this paper, we give some characterizations of von Neumann-Schatten q-Riesz bases.

2010 Mathematics subject classiﬁcation: Primary 46C50, Secondary 42C99. Keywords and phrases: von Neumann-Schatten p-frame, q-Riesz basis, von Neumann-Schatten q-Riesz basis.

1. Introduction Christensen and Stoeva [1] have deeply studied p-frames in separable Banach spaces. By using von Neumann-Schatten frames many basic properties of frames can be derived in a more general setting. A von Neumann-Schatten frame is a sequence of bounded linear operators from a Banach space X into Cp ⊆ B(H). Sadeghi and Arefijamaal [4] introduced the notion of a von Neumann-Schatten p-frame in separable Banach spaces. p-frames and g-frames can be considered as such frames. In the present paper we give som characterizations of von Neumann-Schatten q-Riesz bases in separable Banach spaces.Throughout this paper, H is Hilbert spaces and X denote separable Banach space with dual X∗. Also, we consider 1 < p, q < ∞ are real 1 + 1 = numbers such that p q 1. ∏ Suppose {Xi : i ∈ I} is a collection of normed spaces. Then {Xi : i ∈ I} is a vector space if the linear operations are defined coordinatewise. Let ∥.∥ denote the norm of each Xi. For 1 ≤ p < ∞, define      1  ⊕  ∏ ∑  p   p Xi ≡ x ∈ Xi : ∥x∥ ≡  ∥xi∥  < ∞ . p   i i ⊕ It is known that p Xi is a Banach space if and only if so is each Xi. ∗ speaker

222 S. Movahed and M. Choubin

Let B(H) denote the C∗-algebra of all bounded linear operators on a complex separable Hilbert space H. For a compact operator A ∈ B(H), let s1(A) ≥ s2(A) ≥ · · · ≥ 0 denote the singular values of A , i.e., the eigenvalues of the positive operator ∗ 1 |A| = (A A) 2 , arranged in a decreasing order and repeated according to multiplicity. ≤ < ∞ C For 1 p , the von Neumann-∑ Schatten p-class p is defined to be the set all ∞ p < ∞ ∈ C compact operators A for which i=1 si (A) . For A p, the von Neumann Schatten p-norm of A is defined by   ∞ 1 ∑  p  p  p 1 ∥ ∥C =   = | | p , A p  si (A) (tr A ) (1) = i 1 ∑ = ⟨ , ⟩ E where tr is the usual trace functional which defines as tr(A) e∈E A(e) e , where is any orthonormal basis of H. It is convenient to let C∞ denote the class of compact ∥ ∥ = ∥.∥ ∥.∥ operators, and in this⊕ case A C∞⊕s1(A) is the usual operator norm. Let p and ∞ C C C denote the norm of p p and ∞ ∞, respectively. 1 is called the trace class and C2 is called the Hilbert-Schmidt class. It is proved that Cp is a two sided ∗-ideal of B(H), that is, a Banach algebra under the norm (1) and the finite rank operators are C , ∥.∥ dense in ( p Cp ). For each S ∈ Cq the function ϑS (T) = tr(TS ), where T ∈ Cp, is a continuous linear functional on Cp. Moreover the mapping S → ϑS is an isometric isomorphism ( )∗ from Cq onto the dual space Cp of Cp.[3] ψ ∞ Definition 1.1. A countable family ( i)i=1 of bounded linear operators from X to Cp ⊆ B(H) is a von Neumann-Schatten p-frame for the Banach space X with respect to H if constants A, B > 0 exist such that   ∑∞ 1   p A∥ f ∥ ≤  ∥ψ ( f )∥p  ≤ B∥ f ∥ (2) X i Cp X i=1 for all f ∈ X. It is called a von Neumann-Schatten p-Bessel sequence with bound B if at least upper von Neumann-Schatten p-frame condition is satisfied.

Ψ = {ψ }∞ For von Neumann-Schatten p-Bessel sequences i i=1 for X with respect to H we can deﬁne the analysis operator ⊕ UΨ : X → Cp, f 7→ {ψi( f )} (3) and the synthesis operator ⊕ ∑∞ ∗ TΨ : Cq → X , {νi} 7→ νiψi. (4) q i=1 Ψ = {ψ }∞ ⊆ , C Proposition 1.2 ([4]). Let i i=1 B(X p). Then (1) If Ψ be a von Neumann-Schatten p-frame for X with respect to H, then the operator UΨ given by (3) has closed range, and X is reﬂexive.

223 Characterization of von Neumann-Schatten q-Riesz bases

(2) Ψ is a von Neumann-Schatten p-Bessel sequence for X with a bound B if and only if the operator TΨ defined by (4) is a well defined bounded operator with ∥TΨ∥ ≤ B. ∗ (3) If Ψ be a von Neumann-Schatten p-Bessel sequence for X, then UΨ = T and ∗ ∗ ∗ UΨ ⊆ TΨ, i.e., TΨ is an extension of UΨ. If X is reflexive, then UΨ = TΨ. (4) If X be a reflexive Banach space, then Ψ is a von Neumann-Schatten p-frame for X if and only if the operator TΨ defined by (4) is a well defined and onto.

2. Main results < < ∞ {ψ }∞ Deﬁnition 2.1. Let 1 q . A family i i=1 of bounded linear operators from X C ⊆ 1 + 1 = to p B(H), where p q 1, is called a von Neumann-Schatten q-Riesz basis for X∗ with respect to H if

(1) { f ∈ X : ψi( f ) = 0 ∀i ∈ N} = {0}, (2) there are positive⊕ constant A and B such that for any ﬁnite subset I ⊆ N and {ν }∞ ∈ C i i=1 q q

  1   1 ∑  q ∑ ∑  q  q   q  A  ∥νi∥C  ≤ νiψi ≤ B  ∥νi∥C  . q q i∈I i∈I i∈I ∑ ∞ ν ψ The assumption⊕ of latter deﬁnition implies that i=1 i i converges unconditionally {ν }∞ ∈ C for all i i=1 q q and

∑∞

A {ν }∞ ≤ ν ψ ≤ B {ν }∞ . i i=1 q i i i i=1 q i=1 {ψ }∞ ⊆ , C ∗ Thus i i=1 B(X p) is a von Neumann-Schatten q-Riesz basis for X with respect to H if and only if the operator TΨ defined in (4) is both bounded and bounded below. For q = 2 and H = C this definition is consistent with the standard definition of a Riesz basis for the closed span of its elements. {ϕ }∞ ⊆ ∗, C Theorem 2.2. Let i i=1 B(X p). The following are equivalent: (1) There exist constants A, B > 0 such that   ∞ 1 ∑  p  p  ∗ A∥g∥X∗ ≤  ∥ϕi(g)∥  ≤ B∥g∥X∗ , ∀g ∈ X . (5) Cp i=1 ∑ ⊕ {ν } 7→ ∞ ν ϕ C (2) X is reflexive and i i=1 i i is a well defined mapping of q q onto X. roof {ϕ }∞ ∗ P . i i=1 constitutes a von Neumann-Schatten p-frame for X By (1). Therefore X∗ is reflexive by Proposition 1.2(1), and thus X is reflexive. The rest follows by Proposition 1.2(4). □

224 S. Movahed and M. Choubin

Ψ = {ψ }∞ ⊆ , C By definition 2.1, a family i i=1 B(X p) is a von Neumann-Schatten ∗ ⊕q-Riesz basis for X if and only if the operator TΨ given by (4) is an isomorphism of C ∗ q q onto X . Ψ = {ψ }∞ ⊆ , C Theorem 2.3. Let i i=1 B(X p) be a von Neumann-Schatten p-frame for reflexive Banach space X with respect to H. Then the following are equivalent: (1) Ψ is a von Neumann-Schatten q-Riesz basis for X∗; ⊕ ∑ {ν } ∈ C ∞ ν ψ = ν = ∈ N (2) If i q q and i=1 i i 0, then i 0 for all i , i.e. the operator TΨ given by (4) is one-to-one; ⊕ = C (3) R(UΨ) p p, i.e., the operator UΨ given by (3) is onto. Proof. It is obvious that (1) ⇒ (2). If (2) is verified then by proposition 1.2, the operator TΨ given by (4) is bounded and onto. By (2), TΨ is also is one-to-one. Thus −1 ∗ TΨ has a bounded inverse TΨ , so Ψ is a von Neumann-Schatten q-Riesz basis for X and thus gives (i). For (1) ⇒ (3), we recall that the von Neumann-Schatten q-Riesz basis condition implies that TΨ ⊕has a bounded inverse on R⊕(T). By [2,⊕ Proposition ∗ ∗∗ → C C ∗ = C 97.1] the adjoint TΨ : X p p is onto (Note that ( q q) p p). X is reflexive by Proposition 1.2(1), so now by Proposition 1.2(3) implies (3). If (3) is ∗ verified then the operator UΨ is bijective, which implies that TΨ = UΨ is bijective and thus gives (1). □

References [1] O. Christensen and D. Stoeva, p-frames in separable Banach spaces, Adv. Comput. Math, 18(2003), 117-126. [2] H. Heuser, Functional Analysis, Wiley, New York, (1982). [3] J. R. Ringrose, Compact Non-Self-Adjoint Operators, Van Nostrand Reinhold Company, (1971). [4] G. Sadeghi and A. Arefijamaal, von Neumann-Schatten frames in separable Banach spaces, Mediterr. j. Math, 9(2012), 525-535.

Sima Movahed, Department of Mathematics, Velayat University, Iranshahr, Iran e-mail: [email protected]

Mehdi Choubin, Department of Mathematics, Velayat University, Iranshahr, Iran e-mail: [email protected]

225 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

FRAMES AND SINGLE WAVELETS FOR GROUPS

Atefeh Rahimi

Abstract Let π be a projective unitary representation of a discrete countable abelian group G on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We use Stone′s theorem and the theory of projection valued measure to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multiresolution analysis in term of the frame scaling vectors. If the set Bπ of Bessel vectors for π is dense in H, then for any vector 2 x ∈ H the analysis operator θx makes sense as a densely deﬁned operator from Bπ to l (G)-space.

2010 Mathematics subject classiﬁcation: Primary 42C40, Secondary 42C15 . Keywords and phrases: wavelet, multiresolution analysis, unitary group representation, frame.

1. Introduction The theory of wavelets and multiresolution analyses are inseparably interconnected. However, Journt’e introduced a wavelet that did not arise from a multiresolution analysis, so the standard techniques did not apply. We present an alternate technique for constructing wavelets. We will start with a cyclic GFMRA and construct an explicit formula for a wavelet associated with that GFMRA. Our methods are more functional analytic in nature: we use Stone′s theorem for representations of abelian groups and the theory of spectral multiplicity [1], [2], [3]. There is another important idea in the theory, that of frames. The paper is organized as follow, section 2 is devoted to analyzing frames that arise from the action of a discrete countable abelian group. Out of this analysis comes our construction technique Theorem 2.2. We consider special case, including the traditional case of translations by integers on R, and present theorems 2.3 and 2.4 which is presented in Section 3. Here we give some elementary deﬁnitions which are required in the sequel. For more details see [4, 5]. A frame for a Hilbert space H is a sequence {xn} in H with the property that there exist positive constants A, B > 0 such that ∑ 2 2 2 A∥x∥ ≤ |⟨x, xn⟩| ≤ B∥x∥ (1) g∈G

226 A. Rahimi holds for every x ∈ H. A tight frame refers to the case when A = B, and a Parseval frame refers to the case when A = B = 1. In the case that (1) holds only for all x ∈ span{xn}, then we say that {xn} is a frame sequence. If we require only the right- hand side of the inequality 1, then {xn} is called a Bessel sequence. A generalized multiresolution analysis of H is a sequence of closed subspaces {V j} j∈Z such that the following conditions hold: 1. V j ⊂ V j+1, 2. DV j = V j+1, 3. ∩ j∈ZV j = {0} and ∪ j∈ZV j has dense span in H, 4. the core subspace V0 is invariant under the action of π. By number 1 above, we can deﬁne a second sequence of subspaces {W j} given by V j+1 = V j ⊕ W j . For our purposes, we will assume that the subspace W0 is cyclic under the action of π(G); we shall call such a GMRA a cyclic GMRA. We shall assume a stronger condition than 4 above: suppose there exist vectors {ϕ j} j∈J ⊂ V0 such that {πgϕ j : g ∈ G, j ∈ J} is a frame for V0. We call such a structure a generalized frame multiresolution analysis. n A (orthonormal) wavelet is a vector ψ ∈ H such that the collection {D πgψ : n ∈ Z, g ∈ G} is an orthonormal basis of H. A frame wavelet is such that the same collection is a frame for H. A collection of vectors W = {w1, ··· , wn} will be called a wandering frame collection (or complete wandering frame collection) for π if the collection S = {πgwi : g ∈ G, i = 1, ··· , n} is a frame for its closed linear span (or for H). If X = {x1, ··· , xn} and Y = {y1, ··· , yn} are wandering frame collections for π on H, we will say that X and Y are complementary if X ∪ Y is a complete wandering frame collection for π on H. A vector ξ ∈ H is called a complete wandering tight frame vector, or a complete wandering Parseval frame vector for π if {π(g)ξ : g ∈ G} is a tight frame or Parseval frame, respectively for the whole Hibert space H. (Here, we view {π(g)ξ : g ∈ G} as a sequence indexed by G.) The vector ξ ∈ H is just called a frame vector, tight frame vector a Parseval frame vector for π if {π(g)ξ : g ∈ G} is a frame sequence, tight frame sequence or Parseval frame sequence, respectively. A Bessel vector for π is a vector ξ ∈ H such that {π(g)ξ : g ∈ G} is Bessel. We will use Bπ to denote the set of all the Bessel vectors of π.

2. Main Results In this section we present our main result as the following. First, we give a characterization of complete wandering frame, using the corresponding representation. Lemma 2.1. A representation admits a finite complete wandering frame collection if and only if the representation is unitarily equivalent to a sub-representation of a finite multiple of the regular representation. Next, the following theorem gives a sufficient condition for orthonormal wavelet.

227 Frames and Single Wavelets for Groups

Theorem 2.2. Suppose the vectors {ϕ j} j∈J are the frame scaling vectors for a GFMRA {V j}. For γ ∈ Γ, let πg ∈ π(G) be a representative of the γ coset of G1 ⊂ π(G). For γ ∈ Γ and j ∈ J, deﬁne the vectors. φ = π ϕ γ, j Pw0 D gγ j

For appropriately chosen sets Eγ, j ⊂ Gˆ, the vector ψ defined by ∑ ∑ χ ξ Eγ, j ( ) ψˆ(ξ) = φˆγ, j(ξ) |φˆγ, (ξ)| γ∈Γ j∈J j is a normalized tight frame wavelet. If ∪γ∈Γ ∪ j∈J Eγ, j = Gˆ, then ψ is an orthonormal wavelet. Special Cases: We now apply our results√ to classical wavelet theory on L2(R). The dilation operator D is given by D f (x) = 2 f (2x). The group in question is the 2 l l integers; the representation on L (R) given by πl = T , where T f (x) = f (x − l). We shall normalize the Fourier transform on L2(R) as follows; for f ∈ L1(R) ∩ L2(R) ∫ fˆ(ξ) = f (x)e−2πixξdx. R A projective unitary representation π for a countable group G is a mapping g −→ π(g) from G into the group U(H) of all the unitary operators on a separable Hilbert space H such that π(g)π(h) = µ(g, h)π(gh) for all g, h ∈ G, where µ(g, h) is a scalar-valued function on G × G taking values in the circle group T. This function µ(g, h) is then called a multiplier of π. In this case we also say that π is a µ-projective unitary representation. It is clear from the definition that we have i)µ(g1, g2g3)µ(g2, g3) = µ(g1g2, g3)µ(g1, g2) for all g1, g2, g3 ∈ G, (ii) µ(g, e) = µ(e, g) for all g ∈ G, where e denotes the group unit of G. Any function µ : G × G −→ T satisfying (i)-(ii) above will be called a multiplier or 2-cocycle of G. It follows from (i) and (ii) that we also have (iii) µ(g, g−1) = µ(g−1, g) for all g ∈ G. For any projective representation π of a countable group G on a Hilbert space H ∈ θ θ ⊆ 2 and x ∑H, the analysis operator x for x from D( x)(∑ H) to l (G) is defined by θ = ⟨ , π ⟩χ θ = { ∈ |⟨ , π ⟩|2 < ∞} x(y) g∈G y (g)x g, where D( x) y H : g∈G y (g)x is the domain space of θx. Clearly, Bπ ⊆ D(θx) holds for every x ∈ H. In the case that Bπ is dense in H, we have θx as a densely defined and closable linear operator from Bπ to l2(G). Let M be a subset of a Hilbert space H and let A be a subset of the space B(H) of all the bounded linear operators on H. In what follows we will use [M] to denote the closed ′ linear span of M, and A to denote the commutant {T ∈ B(H): TA = AT, ∀A ∈ A} of A. Two vectors x and y are called π-orthogonal if the range spaces of θx and θy are orthogonal, and they are π-weakly equivalent if the closures of the ranges of θx and θy

228 A. Rahimi are the same. The above π-orthogonality definition can be extended in an obvious way to a set of several (or even infinitely many) vectors. The following theorem characterizes the π-orthogonality and π-weak equivalence in terms of the commutant of π(G). Theorem 2.3. Let π be a projective representation of a countable group G on a Hilbert space H such that Bπ is dense in H, and let x, y ∈ H. Then ′ ′ (i) vectors x and y are π-orthogonal if and only if [π(G) x]⊥[π(G) y] (or equivalently, ′ x⊥[π(G) y]). ′ ′ (ii) vectors x and y are π-weakly equivalent if and only if [π(G) x] = [π(G) y]. For a projective representation π of a countable group G on a Hilbert space H, we define the decomposition space of π to be the subspace Dπ = span{θξ(H): ξ ∈ Bπ} of l2(G). Let N be a positive integer. We call N the orthogonality index of π if N is the smallest natural number such that there exist N strongly disjoint vectors ξi ∈ Bπ such {θ = , ··· , } π that ξi (Bπ): i 1 N generates Dπ. We say that has the orthogonality index ∞ if no such a finite integer N exists. Theorem 2.4. Let π be a projective representation of a countable group G on a Hilbert space H such that Bπ is dense in H. Then π has the orthogonality index N if and only ′ ′ if π(G) has the cyclic multiplicity N, where π(G) denotes the commutant of π(G).

References [1] L. Baggett, H. Medina and K. Merril, Generalized Multiresolution Analyses, and a Construction Procedure for All Wavelet Sets in Rn. J. Fourier Anal. Appl. (6) 5 (1999), 563-573. [2] P. Halmos, Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Chelsea, New York, (1957). [3] E. Weber, Frames and single wavelets for unitary groups, Canad. J. Math., (2002). [4] D.Han and D.Larson, Frame duality properties for projective unitary representations, Bull. London Math. Soc., (2008), 685-695. [5] D.Han, A note on the density theorem for projective unitary representations, Article electronically published on October 26, 2016.

Atefeh Rahimi, Department of Pure Mathematic, Ferdowsi university of Mashhad, Mashhad, Iran e-mail: [email protected]

229 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

CAMPARING SOME PROPERTIES OF FRAMES AND K-FRAMES IN HILBERT SPACES

Nasimeh Salahzehi∗ and Mehdi Choubin

Abstract

K-frames, as a new generalization of frames which were recently introduced by Gavru¸tain˘ Hilbert spaces. They are more general than frames in the sence that the lower frame bound only holds for the elements in the rang of K, where K is a bounded linear operator in a separable Hilbert space H. Many properties for frames may not hold for K-frames. In this paper we will discuss some diﬀerences between K-frames and frames.

2010 Mathematics subject classiﬁcation: Primary 42C15; Secondary 41A58. Keywords and phrases: Frame, K-frame, dual K-frame.

1. Introduction

Frames in Hilbert spces were introduced by J. Duffin and A.C. Schaffer in 1952. They are a redundant set of vectores which yield are presentation for each vector in the space. Gavru¸tarecently˘ presented a generalization of frames with a linear bounded operator K. K-frames posses higher generality than frames in the sense that the lower frame bound condition holds only for the elements in the range of K and that they allow the reconstruction of the elements from the range of K in a stable way and, in general, the range is not even a closed space. Hence K-frames provides more flexibility and thus make the study of them interesting. First introduce basic definitions as follow: { }∞ ⊂ Definition 1.1. A family of elements fn n=1 H is called a frame of H if there exist constant A, B > 0 such that ∑∞ 2 2 2 A∥ f ∥ ≤ |⟨ f, fn⟩| ≤ B∥ f ∥ , ∀ f ∈ H. n=1 , { }∞ The constant A B are called frame bound. The sequence fn n=1 is said to be Bessel sequence for H if we only require the right-hand inequality.

∗ speaker

230 N. Salahzehi and M. Choubin

{ }∞ ⊂ Definition 1.2. A sequence fn n=1 H is called a K-frame for H if there exist two constants 0 < A ≤ B < ∞ such that ∑∞ ∗ 2 2 2 A∥K f ∥ ≤ |⟨ f, fn⟩| ≤ B∥ f ∥ , ∀ f ∈ H. n=1 The numbers A, B are called K-frame bounds. { }∞ Definition 1.3. Let fn n=1 is a K-frame for H. Obviously it is∑ a Bessal sequence, so 2 → , { }∞ = ∞ , ∀{ }∞ ∈ 2, we can define the following operator T : l H T cn n=1 n=1 cn fn cn n=1 l ∗ → 2, ∗ = {⟨ , ⟩}∞ , ∈ = ∗, then we have T : ∑H l T f f fn n=1 for all f H. Let S TT we obtain → , = ∞ ⟨ , ⟩ , ∈ , ∗, S : H H S f n=1 f fn fn for all f H. We call T T S the synthesis operator, { }∞ analysis operator and frame operator for K-frame fn n=1, respectively. By a simple calculation we have ∑∞ 2 ∗ 2 ∗ ⟨S f, f ⟩ = |⟨ f, fn⟩| ≥ A∥K f ∥ = ⟨AKK f, f ⟩, ∀ f ∈ H, (1) n=1 it follows that S ≥ AKK∗.

2. Main results We will discuss some differances between frames and K-frames as follow: (1) Frame operator of ordinary frames is invertible, but frame operator of a K-frame is not invertible on H general. This problem can be solvable with some conditions, we can show that it is invertible on a subspace R(K) ⊂ H. In fact, since R(K) is closed, there exists a pseudo-invers K† of K such that KK† f = f, ∀ f ∈ R(K), namely † ∗ † ∗ ∗ KK = IR(K), so we have I = (K ) K . Hence for any f ∈ R(K), we obtain |R(K) R(K) |R(K) † ∥ f ∥ = ∥(K )∗K∗ f ∥ ≤ ∥K†∥.∥K∗ f ∥, that is, ∥K∗ f ∥2 ≥ ∥K†∥−2∥ f ∥2. Combined with |R(K) (1) we have ⟨S f, f ⟩ ≥ A∥K∗ f ∥2 ≥ A∥K†∥−2∥ f ∥2, ∀ f ∈ R(K). So, from the definition of K-frame we have A∥K†∥−2∥ f ∥ ≤ ∥S f ∥ ≤ B∥ f ∥, ∀ f ∈ R(K), which implies that S : R(K) → S (R(K)) is a homomorfism, forthermore, we have B−1∥ f ∥ ≤ ∥S −1 f ∥ ≤ A−1∥K†∥2∥ f ∥, ∀ f ∈ S (R(K)). Then, we follow that the frame operator of K-frames with condition of subspace R(K) ⊂ H is invertible. { }∞ { }∞ (2) Let fn n=1 be a K-frame for H. We call a Bessel sequence gn n=1 for H a dual { }∞ K-frame of fn n=1 if ∑∞ K f = ⟨ f, gn⟩ fn, ∀ f ∈ H. (2) n=1 It is well Known that, in classical frame theory, the duals of a frame are necessarily frames. But, there is not an analogue for K-frames. As shown in the following example.

231 Short title of the paper for running head

{ }∞ ∈ Example 2.1. Let en n=1 be an orthonormal basis for H and deﬁne K L(H) as follows: Ke2n = e2n + e2n−1, Ke2n−1 = 0, n = 1, 2, ... Then for each f ∈ H we have ∑∞ ( ∑∞ ∑∞ ) K f = K ⟨ f, en⟩en = K ⟨ f, e2n⟩e2n + ⟨ f, e2n−1⟩e2n−1 n=1 n=1 n=1 ∑∞ = ⟨ f, e2n⟩(e2n + e2n−1). n=1 ∗ → ∗ = It∑ is easy to check that the adjoint operator K : H H is given by K f ∞ ⟨ , + ⟩ , ∀ ∈ . n=1 f e2n e2n−1 e2n f H But, since

∑∞ 2 ∑∞ ∗ 2 2 ∥K f ∥ = ⟨ f, e2n + e2n−1⟩e2n = |⟨ f, e2n + e2n−1⟩| n=1 n=1 ∑∞ ∑∞ 2 2 2 ≤ 2 |⟨ f, e2n⟩| + 2 |⟨ f, e2n−1⟩| ≤ 4∥ f ∥ , n=1 n=1 { }∞ = { + }∞ { }∞ = { }∞ it follows that fn n=1 e2n e2n−1 n=1 is a K-frame for H. Clearly, gn n=1 e2n n=1 > ∥ ∗ ∥2 ≤ ∑is a Bessel sequence for H. If there exists a constant C 0 such that C K f ∞ |⟨ , ⟩|2 ∈ n=1 f gn for all f H, then we have ∑∞ ∑∞ 2 2 ∗ 2 2 |⟨e1, gn⟩| = |⟨e1, e2n⟩| = 0 ≥ C∥K e1∥ = C∥e2∥ = C, n=1 n=1 { }∞ a contradiction. thus gn n=1 is not a K-frame for H. But there is can show that a dual K-frame is necessarily a K∗-frame for H. { }∞ { }∞ { }∞ Lemma 2.2. Let fn n=1 and gn n=1 be two Bessel sequences in (2). Then fn n=1 and { }∞ ∗ gn n=1 are K-frame and K -frame, respectively. { }∞ (3) In the bellow proposition, will show that a K-frame fn n=1 for H has a dual { }∞ frame on the closed subspace R(K) which is derived from of fn n=1. { }∞ { }∞ Proposition 2.3. Supppose that fn n=1 and gn n=1 are as in (2). Then there exists a ∞ † ∗ ∞ ∞ sequance {hn} = {(K ) gn} derived by {gn} such that n=1 |R(K) n=1 n=1 ∑∞ f = ⟨ f, hn⟩ fn, ∀ f ∈ R(K). n=1 { }∞ { }∞ ∈ Moreover, hn n=1 and fn n=1 are interchangable for any f R(K). Unlike the frames, a K-frame not admit a dual frame on the whole space H. It shown in following example:

232 N. Salahzehi and M. Choubin

{ }∞ { }∞ Example 2.4. Let en n=1 be an orthonormal basis for H and let fn n=1 be deﬁned in = = en form fn en when n is even, and∑ fn n when n is odd. Deﬁne a linear bounded → , = ∞ ⟨ , ⟩ . ∈ operator as K : H H K f n=1 f e2n e2n For any f H we compute that ∑∞ ∑∞ ∑∞ ∗ 2 2 2 2 ∥K f ∥ = |⟨ f, e2n⟩| ≤ |⟨ f, fn⟩| = |⟨ f, e2n⟩| n=1 n=1 n=1 ∑∞ ∑∞ 1 2 2 2 + |⟨ f, e − ⟩| ≤ |⟨ f, e ⟩| = ∥ f ∥ . (2n − 1)2 2n 1 n n=1 n=1 { }∞ { }∞ { }∞ Hence fn n=1 is a K-frame for H. Suppose that fn n=1 has s dual frame gn n=1. For any k ∈ N, taking e2n−1 ∈ H, we have ∑∞ ∑∞ e2k−1 = ⟨e2k−1, gn⟩ fn = ⟨e2k−1, fn⟩gn n=1 n=1 ∑∞ ∑∞ ⟨ ⟩ e2n−1 = ⟨e − , e ⟩g + e − , g − . 2k 1 2n 2n 2k 1 2n − 1 2n 1 n=1 n=1 = g2k−1 = − Thus e2k−1 2k−1 , and g2k−1 (2k 1)e2k−1 as a consequence. Now ∑∞ 2 2 2 |⟨e2k−1, gn⟩| ≥ |⟨e2k−1, g2k−1⟩| = (2k − 1) → ∞, n=1 → ∞ { }∞ as k , which contradict the fact that gn n=1 is a Bessel sequence for H.

References [1] O. Christensen, An Introduction to Frames and Riesz Bases., Birkhauser, Boston (2003) [2] F. A. Neyshaburi and A. A. Arefijamaal, Some construction of K-frames and their duals., To appear in Rocky Mountain J. Math. [3] X. C. Xiao, Y. C. Zhu and L. Gavru˘ ta¸ , Some properties of K-frames in Hilbert spaces., Results. Math., (2013). [4] Z. Q. Xiang and Y. M. Li, Frame sequences and dual frames for operators., ScieceAsia, (2016).

Nasimeh Salahzehi, Department of Mathematics, Velayet University, Iranshahr, Iran e-mail: [email protected]

Mehdi Choubin, Department of Mathematics Velayat University Iranshahr, Iran e-mail: [email protected]

233 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

THE CONDITION OF UNIQUENESS DUAL FOR FRAMES AND K-FRAMES

Nasimeh Salahzehi

Abstract Frames in Hilbert spaces are a redundant of vectores which yield a representation for each vector in the space. K-frames were recently introduced by Gavru¸tain˘ Hilbert spaces. They respect to a bounded linear operator K. In this paper, we will present some conditions of uniqueness dual frame and dual K-frame in cases.

2010 Mathematics subject classiﬁcation: Primary 42C15; Secondary 41A58. Keywords and phrases: Frame, K-frame, dual K-frame, K-minimal frame, uniqueness.

1. Introduction Frames in Hilbert spaces were introduced by J. Duffin and A.C. Schaffer in 1952. One of the essential applications of frames is that they provide basis-like but generally nonunique decompositions, dual frames play a key role. Gavru¸tarecently˘ presented a generalization of frames with a linear bounded operator K. There are many dual for frames and K-frames, in case the canonical dual frame. But we can find conditions that make this existance in a unique way. First introduce basic definitions as follow: { }∞ { }∞ Definition 1.1. A Riesz basis for H is a family of the form Uen n=1, where en n=1 is an orthonoamal basis for H and U : H → H is a bounded bijective operator. { }∞ ⊂ Definition 1.2. A family of elements fn n=1 H is called a frame of H if there exist constant A, B > 0 such that ∑∞ 2 2 2 A∥ f ∥ ≤ |⟨ f, fn⟩| ≤ B∥ f ∥ , ∀ f ∈ H. n=1 , { }∞ The constant A B are called frame bound. The sequence fn n=1 is said to be Bessel sequence for H if we only require the right-hand inequality. { }∞ Definition 1.3. A Bessel sequence gn n=1 in H is called a dual frame for the frame { }∞ fn n=1 if ∑∞ f = ⟨ f, gn⟩ fn, ∀ f ∈ H. n=1

234 N. Salahzehi

{ }∞ ⊂ Definition 1.4. A sequence fn n=1 H is called a K-frame for H if there exist two constants 0 < A ≤ B < ∞ such that ∑∞ ∗ 2 2 2 A∥K f ∥ ≤ |⟨ f, fn⟩| ≤ B∥ f ∥ , ∀ f ∈ H. n=1 The numbers A, B are called K-frame bounds. { }∞ { }∞ Definition 1.5. Let fn n=1 be a K-frame for H. We call a Bessel sequence gn n=1 for { }∞ H a dual K-frame of fn n=1 if ∑∞ K f = ⟨ f, gn⟩ fn, ∀ f ∈ H. n=1 { }∞ ∈ Definition 1.6. A K-frame fn n=1 of H is called a K-exact frame if for every m I ( I { } is the countable index set) the sequence fn n,m is not a K-frame for∑ H. Also we call { }∞ { } ∈ 2 ∞ = fn n=1 a K-minimal frame whenever for each cn l such that n=1 cn fn 0 then cn = 0 for all n.

2. Main results We present condition on frames and K-frames to have duals. In bellow oﬀer theorem and equivalent conditions for existance uniqeness dual for frames and present relationship between frame and Riesz basis. { }∞ Theorem 2.1. Let fn n=1 be a frame for H. Then the following are equivalent: { }∞ (1) fn∑n=1 is a Riesz basis for H. ∞ = { }∞ ∈ 2 N = ∀ ∈ N (2) If n=1 cn fn 0 for some cn n=1 l ( ), then cn 0, n . { }∞ { }∞ Theorem 2.2. If fn n=1 is a Riesz basis for H, then fn n=1 is a Bessel sequence. { }∞ Furthermore, there exists a unique sequence gn n=1 in H such that ∑∞ f = ⟨ f, gn⟩ fn, ∀ f ∈ H. n=1 { }∞ The sequence gn n=1 is also a Riesz basis, and the series converges unconditionally for all f ∈ H. { }∞ Proposition 2.3. Let fn n=1 be a frame for Hilbert space H. Then the following are equevalent: { }∞ (1) fn n=1 is a Riesz basis for H. { }∞ (2) fn n=1 is an exact frame, i.e. It ceases to be a frame when an orbitrary element is removed. { }∞ < { , } (3) fn∑n=1 is minimal, i.e. fn span nm : m n for all n. ∞ = { } ∈ 2 = (4) If n=1 cn fn 0 for some cn l , then cn 0 for all n. { }∞ (5) fn n=1 is a basis.

235 Short title of the paper for running head

Remark 2.4. A frame that is not a Riesz basis is said to be overcomplete; in fact, if { }∞ ﬃ { }∞ ∈ 2 N \{ } fn n=1 is a frame that is not a Riesz basis, there exist coe cient cn n=1 l ( ) 0 for which ∑∞ cn fn = 0. n=1 That is, for such frames there is some dependancy between the frame elements. With some attention on which said above about frame and Riesz basis, we can present this corollary: { }∞ Corollary 2.5. Let a frame fn n=1 be a Riesz basis, there is a unique Bessel sequence { }∞ { }∞ ∀ ∈ gn n=1 that be unique dual frame for fn n=1. Then f H ∑∞ f = ⟨ f, gn⟩ fn. n=1 In K-frames the unique dual exists in the condition, that present bellow:

Lemma 2.6. Every K-exact frame is a K-minimal frame, the convers does not hold in general. With the following example, we can show that the convers is not hold. = C3 { }∞ → Example 2.7. Let H and en n=1 is the orthogonal basis of H. Deﬁne K : H H by ∑∞ K cnen = c1e1 + c2e1 + c3e2. n=1 ∈ { }∞ { , } Then K B(H) and fn n=1 is a K-minimal frame. Easily, we can see that e1 e2 is = 1 = also a K-frame with bounds A 8 and B 1. The Riesz basis are in particular frames. Diﬀerent from frames, each Riesz basis has a unique dual which is the canonical dual. Theorem 2.8 shows that the K-minimal frames also have such property. { }∞ Theorem 2.8. A K-frame fn n=1 has a unique K-dual if and only if it is a K-minimal frame.

References [1] Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhauser, Boston (2003) [2] F. A. Neyshaburi, A. A. ArefijamaalSome construction of K-frames and their duals. To appear in Rocky Mountain J. Math. [3] X. C. Xiao, Y. C. Zhu and L. Gavru˘ ta¸ . Some properties of K-frames in Hilbert spaces. Results. Math. 63 (2013), 1243-1255. [4] Z. Q. Xiang, Y. M. Li.Frame sequences and dual frames for operators. ScieceAsia42 (2016) : 222-230

236 N. Salahzehi

Nasimeh Salahzehi, Department of Mathematics, Velayet University, Iranshahr, Iran e-mail: [email protected]

237 The 5th Seminar on Harmonic Analysis and Applications Organized by the Iranian Mathematical Society January 18–19, 2017, Ferdowsi University of Mashhad, Iran

ON THE PROXIMINALITY OF SUBSETS OF SOME BANACH SPACES

Sanaz Zebarjad

Abstract In this paper we seek some conditions under which proximinality of a set can be obtained. Then we show that, as an example of our results, the set of constant polynomials is proximinal in C[0, 1].

2010 Mathematics subject classiﬁcation: Primary 41A50, Secondary 41A52. Keywords and phrases: proximanal, nearest point, farthest point.

1. Introduction The problem of finding the shortest distance to a set from a point is an important matter in many applications. Determining the proximinality of subsets of normed spaces have applications in issues such as: Solution to an over-determined system of equations, best least squares polynomial approximation to a function and etc. In this theory the functions to be approximated and the approximating functions are regarded as elements of certain normed linear (or, more generally, of certain metric) spaces of functions and best approximation amounts to finding nearest points. The advantages and a brief history of this point of view can be found in [3]. Furthermore, a recent work of this theory can be find in [1] Let G be a nonempty subset of the normed linear space X. For any x ∈ X, define d(x, G) = inf{∥x − z∥ : z ∈ G}.

A point z0 ∈ X is called a best approximation for x ∈ X if

∥x − z0∥ = d(x, G). The set of best approximations x from G is deﬁned by

PG(x) = {z0 ∈ G : ∥x − z0∥ = d(x, G)}. The set G is said to be, i) proximinal if each point of X has a at least one best approximation in G. ii) chebyshev if each point of X has a unique best approximation in G. iii) semi-chebyshev, If for every x ∈ X the set PG(x) contains at most one element.

238 S. Zebarjad

From [2], using Hahn-Banach theorem, the following basic theorem can be obtained: Theorem 1.1. Let G be a subspace of a normed space X, x ∈ X \ G and z ∈ G. We ∗ have z ∈ PG(x) if and only if there exists an f ∈ X with ∥ f ∥ = 1 such that f |G = 0 and | f (x − z)| = ∥x − z∥. Although the above theorem is obatined by a simple application of a corollary of the Hahn-Banach theorem, it gives some basic results in the theory of Best approximation.

Theorem 1.2. For the compact set A, let X = CR(A) , G a linear subspace of X, x ∈ X ∖G¯ and g0 ∈ G. We have g0 ∈ PG(x) if and only if there exist two∫ disjoint closed +, − µ |µ| = µ = ∈ subsets G G of A and a Radon measure such that (A) 1, A gd 0, g G and µ is non-negative on G+ and non-positive on G−.

2. Main results Theorem 1. Let X be a normed space and Y a subspace of X. If each member of a basis for X has best approximation to Y, then Y is proximinal. Lemma 2.1. Let X be a normed space and A a dense subspace of X and let Y be a subspace of X. If each member of A has best approximation to Y, then Y is proximinal. The above lemma and theorem prior to it lead us to the following example: Example 2.2. The set of all conestant functions in C[0, 1] is proximinal.

References [1] H. Mazaheri, T. D. Narang and H. khademzadeh, nearest and farthest points in normed spaces, yazd university, (2014). [2] I. Singer, Best approximation in normed linear spaces, Centro internazionale mathematico estivo 57 (1971) 683-792. [3] I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Publ. House Acad.Soc. Rep. Romania, Bucharest, (1967).

Sanaz Zebarjad, Department of Mathematics, University of Yazd, Yazd, Iran e-mail: [email protected]

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