DOCSLIB.ORG

Notes on Noncommutative Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Notes on Noncommutative Geometry NOTES ON NONCOMMUTATIVE GEOMETRY Igor Nikolaev E-mail: [email protected] arXiv:1503.05411v1 [math.OA] 17 Mar 2015 Foreword These notes are neither an introduction nor a survey (if only a brief) of noncommutative geometry – later NCG; rather, they strive to answer some naive but vital questions: What is the purpose of NCG and what is it good for? Why a number theorist or an algebraic geometer should care about the NCG? Can NCG solve open problems of classical geometry inaccessible otherwise? In other words, why does NCG matter? What is it anyway? Good answer means good examples. A sweetheart of NCG called non- commutative torus captures classical geometry of elliptic curves because such a torus is a coordinate ring for elliptic curves. In other words, one deals with a functor from algebraic geometry to the NCG; such functors are at the heart of our book. What is NCG anyway? It is a calculus of functors on the classical spaces (e.g. algebraic, geometric, topological, etc) with the values in NCG. Such an approach departs from the tradition of recasting geometry of the classi- cal space X in terms of the C∗-algebra C(X) of continuous complex-valued functions on X, see the monograph by [Connes 1994] [39]. Boston, March 2014 Igor Nikolaev i ii Introduction It is not easy to write an elementary introduction to the NCG because the simplest non-trivial examples are mind-blowing and involve the KK-groups, subfactors, Sklyanin algebras, etc. Such a material cannot be shrink-wrapped into a single graduate course being the result of a slow roasting of ideas from different (and distant) mathematical areas. But this universality of NCG is a thrill giving the reader a long lost sense of unity of mathematics. In writing these notes the author had in mind a graduate student in (say) number theory eager to learn something new, e.g. a noncommutative torus with real multiplication; it will soon transpire that such an object is linked to the K-rational points of elliptic curves and the Langlands program. The book has three parts. Part I is preparatory: Chapter 1 deals with the simplest examples of functors arising in algebraic geometry, number the- ory and topology; the functors take value in a category of the C∗-algebras known as noncommutative tori. Using these functors one gets a set of non- commutative invariants for elliptic curves and Anosov’s automorphisms of the two-dimensional torus. Chapter 2 is a brief introduction to the cate- gories, functors and natural transformations; they will be used throughout the book. Chapter 3 covers an essential information about the category of C∗-algebras and their K-theory; we introduce certain important classes of the C∗-algebras: the AF-algebras, the UHF-algebras and the Cuntz-Krieger algebras. Our choice of the C∗-algebras is motivated by their applications in Part II. Part II deals with the noncommutative invariants obtained from the func- tors acting on various classical spaces. Chapter 4 is devoted to such functors on the topological spaces with values in the category of the so-called station- ary AF-algebras; the noncommutative invariants are the Handelman triples (Λ, [I],K), where Λ is an order in a real algebraic number field K and [I] an equivalence class of the ideals of Λ. Chapter 5 deals with the examples iii iv of functors arising in projective algebraic geometry and their noncommuta- tive invariants. Finally, Chapter 6 covers functors in number theory and the corresponding invariants. Part III is a brief survey of the NCG; the survey is cursory yet an extensive guide to the literature has been compiled at the end of each chapter. We hope that the reader can instruct himself by looking at the original publications; we owe an apology to the authors whose works are not on the list. There exist several excellent textbooks on the NCG. The first and fore- most is the monograph by A. Connes “G´eom´etrie Non Commutative”, Paris, 1990 and its English edition “Noncommutative Geometry”, Academic Press, 1994. The books by J. Madore “An Introduction to Noncommutative Dif- ferential Geometry & its Applications”, Cambridge Univ. Press, 1995, by J. M. Gracia-Bondia, J. C. Varilly and H. Figueroa “Elements of Noncom- mutative Geometry”, Birkh¨auser, 2000 and by M. Khalkhali “Basic Non- commutative Geometry”, EMS Series of Lectures in Mathematics, 2007 treat particular aspects of Connes’ monograph. A different approach to the NCG is covered in a small but instructive book by Yu. I. Manin “Topics in Non- commutative Geometry”, Princeton Univ. Press, 1991. Finally, a more spe- cialized “Noncommutative Geometry, Quantum Fields and Motives”, AMS Colloquium Publications, 2008 by A. Connes and M. Marcolli is devoted to the links to physics and number theory. None of these books treat the NCG as a functor [115]. I thank the organizers, participants and sponsors of the Spring Institute on Noncommutative Geometry and Operator Algebras (NCGOA) held annu- ally at the Vanderbilt University in Nashville, Tennessee; these notes grew from efforts to find out what is going on there. (I still don’t have an an- swer.) I am grateful to folks who helped me with the project; among them are D. Anosov, P. Baum, D. Bisch, B. Blackadar, O. Bratteli, A. Connes, J. Cuntz, G. Elliott, K. Goodearl, D. Handelman, N. Higson, B. Hughes, V. F. R. Jones, M. Kapranov, M. Khalkhali, W. Krieger, Yu. Manin, V. Manuilov, M. Marcolli, V. Mathai, A. Mishchenko, S. Novikov, N. C. Phillips, M. Ri- effel, W. Thurston, V. Troitsky, G. Yu and others. Contents I BASICS 3 1 Model Examples 5 1.1 Noncommutativetorus . .. 5 1.2 Elliptic curves . 10 1.3 Complex multiplication . 16 1.4 Anosovautomorphisms. 19 2 Categories and Functors 27 2.1 Categories............................. 27 2.2 Functors.............................. 31 2.3 Naturaltransformations . 34 3 C∗-Algebras 37 3.1 Basicdefinitions.......................... 37 3.2 Crossedproducts ......................... 41 3.3 K-theory of C∗-algebras ..................... 43 3.4 Noncommutativetori . 48 3.4.1 Generalcase........................ 48 3.4.2 Two-dimensionaltorus . 50 3.5 AF-algebras ............................ 52 3.5.1 Generalcase........................ 52 3.5.2 StationaryAF-algebras. 55 3.6 UHF-algebras ........................... 58 3.7 Cuntz-Kriegeralgebras . 59 v vi CONTENTS II NONCOMMUTATIVE INVARIANTS 63 4 Topology 65 4.1 Classificationofthesurfacemaps . 65 4.1.1 Pseudo-Anosovmapsofasurface . 66 4.1.2 Functorsandinvariants. 68 4.1.3 Jacobian of measured foliations . 70 4.1.4 Equivalent foliations . 73 4.1.5 Proofs ........................... 74 4.1.6 Anosovmapsofthetorus . 80 4.2 Torsioninthetorusbundles . 84 4.2.1 Cuntz-Kriegerfunctor . 84 4.2.2 Proofoftheorem4.2.1 . 85 4.2.3 Noncommutative invariants of torus bundles . 87 4.3 Obstruction theory for Anosov’s bundles . 89 4.3.1 FundamentalAF-algebra . 89 4.3.2 Proofs ........................... 94 4.3.3 Obstructiontheory . 99 5 Algebraic Geometry 107 5.1 Elliptic curves . 108 5.1.1 Noncommutative tori via Sklyanin algebras . 110 5.1.2 Noncommutative tori via measured foliations . 116 5.2 Highergenusalgebraiccurves . .122 5.2.1 ToricAF-algebras. .122 5.2.2 ProofofTheorem5.2.1. .124 5.3 Complexprojectivevarieties . .129 5.3.1 Serre C∗-algebras .....................130 5.3.2 Proofoftheorem5.3.3 . .133 5.3.3 Real multiplication revisited . 137 5.4 Application: Mapping class groups . 140 5.4.1 Harvey’sconjecture. .140 5.4.2 ProofofTheorem5.4.1. .141 6 Number Theory 147 6.1 Complex multiplication . 148 6.1.1 Functor on elliptic curves with complex multiplication . 150 6.1.2 ProofofTheorem6.1.1. .151 CONTENTS vii 6.1.3 ProofofTheorem6.1.2. .154 6.1.4 ProofofTheorem6.1.3. .156 6.2 Ranks of the K-rational elliptic curves . 161 6.2.1 Arithmetic complexity of noncommutative tori . 162 6.2.2 Q-curves ..........................164 6.2.3 ProofofTheorem6.2.2. .165 6.2.4 Numericalexamples . .168 6.3 Non-commutativereciprocity. 169 6.3.1 L-functionofnoncommutative tori . 170 6.3.2 ProofofTheorem6.3.1. .171 6.3.3 Supplement: Gr¨ossencharacters, units and π(n) ....178 6.4 Langlands program for noncommutative tori . 182 6.4.1 L( 2n ,s).........................182 ARM 6.4.2 ProofofTheorem6.4.1. .186 6.4.3 Supplement: Artin L-function . .189 6.5 Projective varieties over finite fields . 191 6.5.1 Traces of Frobenius endomorphisms . 191 6.5.2 ProofofTheorem6.5.1 . .194 6.5.3 Examples .........................199 6.6 Transcendentalnumbertheory . 204 6.6.1 Algebraic values of transcendental functions . 204 6.6.2 ProofofTheorem6.6.1. .205 III BRIEF SURVEY OF NCG 211 7 Finite Geometries 213 7.1 Axiomsofprojectivegeometry. .213 7.2 Projective spaces over skew fields . 215 7.3 DesarguesandPappusaxioms . .216 8 Continuous Geometries 219 8.1 W ∗-algebras............................219 8.2 VonNeumanngeometry . .222 9 Connes Geometries 225 9.1 ClassificationoftypeIIIfactors . .225 9.1.1 Tomita-TakesakiTheory . .225 viii CONTENTS 9.1.2 ConnesInvariants. .227 9.2 Noncommutative differentialgeometry . 228 9.2.1 Hochschild homology . 228 9.2.2 Cyclic homology . 229 9.2.3 Novikov Conjecture for hyperbolic groups . 230 9.3 Connes’IndexTheorem . .231 9.3.1 Atiyah-Singer Theorem for families of elliptic operators 231 9.3.2 Foliatedspaces ......................232 9.3.3 IndexTheoremforfoliatedspaces . 233 9.4 Bost-Connesdynamicalsystem . 234 9.4.1 Hecke C∗-algebra .....................235 9.4.2 Bost-ConnesTheorem . .235 10 Index Theory 237 10.1 Atiyah-SingerTheorem. .237 10.1.1 Fredholmoperators . .237 10.1.2 Elliptic operators on manifolds . 239 10.1.3 IndexTheorem . .240 10.2K-homology ............................241 10.2.1 TopologicalK-theory . .241 10.2.2 Atiayh’s realization of K-homology . 243 10.2.3 Brown-Douglas-Fillmore Theory . 244 10.3 Kasparov’sKK-theory .
Load more

Recommended publications
  • Phase Transition on the Toeplitz Algebra of the Affine Semigroup Over the Natural Numbers
    University of Wollongong Research Online Faculty of Engineering and Information Faculty of Informatics - Papers (Archive) Sciences 1-1-2010 Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers Marcelo Laca University of Newcastle, Australia Iain F. Raeburn University of Wollongong, [email protected] Follow this and additional works at: https://ro.uow.edu.au/infopapers Part of the Physical Sciences and Mathematics Commons Recommended Citation Laca, Marcelo and Raeburn, Iain F.: Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers 2010, 643-688. https://ro.uow.edu.au/infopapers/1749 Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: [email protected] Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers Abstract We show that the group of orientation-preserving affine ansformationstr of the rational numbers is quasi- lattice ordered by its subsemigroup N⋊N×. The associated Toeplitz C∗-algebra T(N⋊N×) is universal for isometric representations which are covariant in the sense of Nica. We give a presentation of T(N⋊N×) in terms of generators and relations, and use this to show that the C∗-algebra QN recently introduced by Cuntz is the boundary quotient of in the sense of Crisp and Laca. The Toeplitz algebra T(N⋊N×) carries a natural dynamics σ, which induces the one considered by Cuntz on the quotient QN, and our main result is the computation of the KMSβ (equilibrium) states of the dynamical system (T(N⋊N×),R,σ) for all values of the inverse temperature β.
    [Show full text]
  • Monoid Kernels and Profinite Topologies on the Free Abelian Group
    BULL. AUSTRAL. MATH. SOC. 20M35, 20E18, 20M07, 20K45 VOL. 60 (1999) [391-402] MONOID KERNELS AND PROFINITE TOPOLOGIES ON THE FREE ABELIAN GROUP BENJAMIN STEINBERG To each pseudovariety of Abelian groups residually containing the integers, there is naturally associated a profinite topology on any finite rank free Abelian group. We show in this paper that if the pseudovariety in question has a decidable membership problem, then one can effectively compute membership in the closure of a subgroup and, more generally, in the closure of a rational subset of such a free Abelian group. Several applications to monoid kernels and finite monoid theory are discussed. 1. INTRODUCTION In the early 1990's, the Rhodes' type II conjecture was positively answered by Ash [2] and independently by Ribes and Zalesskii [10]. The type II submonoid, or G-kernel, of a finite monoid is the set of all elements which relate to 1 under any relational morphism with a finite group. Equivalently, an element is of type II if 1 is in the closure of a certain rational language associated to that element in the profinite topology on a free group. The approach of Ribes and Zalesskii was based on calculating the profinite closure of a rational subset of a free group. They were also able to use this approach to calculate the Gp-kernel of a finite monoid. In both cases, it turns out the the closure of a rational subset of the free group in the appropriate profinite topology is again rational. See [7] for a survey of the type II theorem and its motivation.
    [Show full text]
  • (I) Eu ¿ 0 ; Received by the Editors November 23, 1992 And, in Revised Form, April 19, 1993
    proceedings of the american mathematical society Volume 123, Number 1, January 1995 TRIANGULAR UHF ALGEBRAS OVER ARBITRARYFIELDS R. L. BAKER (Communicated by Palle E. T. Jorgensen) Abstract. Let K be an arbitrary field. Let (qn) be a sequence of posi- tive integers, and let there be given a family \f¥nm\n > m} of unital K- monomorphisms *F„m. Tqm(K) —►Tq„(K) such that *¥np*¥pm= %im when- ever m < n , where Tq„(K) is the if-algebra of all q„ x q„ upper trian- gular matrices over K. A triangular UHF (TUHF) K-algebra is any K- algebra that is A'-isomorphic to an algebraic inductive limit of the form &~ = lim( Tq„(K) ; W„m). The first result of the paper is that if the embeddings ^Vnm satisfy certain natural dimensionality conditions and if £? = lim (TPn(K) ; Onm) is an arbitrary TUHF if-algebra, then 5? is A-isomorphic to ^ , only if the supernatural number, N[{pn)], of (q„) is less than or equal to the super- natural number, N[(p„)], of (pn). Thus, if the embeddings i>„m also satisfy the above dimensionality conditions, then S? is À"-isomorphic to F , only if NliPn)] = N[(qn)] ■ The second result of the paper is a nontrivial "triangular" version of the fact that if p, q are positive integers, then there exists a unital A"-monomorphism <P: Mq{K) —>MP(K), only if q\p . The first result of the paper depends directly on the second result. 1. Introduction Let K be a fixed but arbitrary field. In this paper we investigate a certain class of A.-algebras (i.e., algebras over K) called triangular UHF (TUHF) K- algebras.
    [Show full text]
  • Profinite Surface Groups and the Congruence Kernel Of
    Pro¯nite surface groups and the congruence kernel of arithmetic lattices in SL2(R) P. A. Zalesskii* Abstract Let X be a proper, nonsingular, connected algebraic curve of genus g over the ¯eld C of complex numbers. The algebraic fundamental group ¡ = ¼1(X) in the sense of SGA-1 [1971] is the pro¯nite completion of top the fundamental group ¼1 (X) of a compact oriented 2-manifold. We prove that every projective normal (respectively, caracteristic, accessible) subgroup of ¡ is isomorphic to a normal (respectively, caracteristic, accessible) subgroup of a free pro¯nite group. We use this description to give a complete solution of the congruence subgroup problem for aritmetic lattices in SL2(R). Introduction Let X be a proper, nonsingular, connected algebraic curve of genus g over the ¯eld C of complex numbers. As a topological space X is a compact oriented 2-manifold and is simply a sphere with g handles top added. The group ¦ = ¼1 (X) is called a surface group and has 2g generators ai; bi (i = 1; : : : ; g) subject to one relation [a1; b1][a2; b2] ¢ ¢ ¢ [ag; bg] = 1. The subgroup structure of ¦ is well known. Namely, if H is a subgroup of ¦ of ¯nite index n, then H is again a surface group of genus n(g ¡ 1) + 1. If n is in¯nite, then H is free. The algebraic fundamental group ¡ = ¼1(X) in the sense of SGA-1 [1971] is the pro¯nite completion top of the fundamental group ¼1 (X) in the topological sense (see Exp. 10, p. 272 in [SGA-1]).
    [Show full text]
  • One Dimensional Groups Definable in the P-Adic Numbers
    One dimensional groups definable in the p-adic numbers Juan Pablo Acosta L´opez November 27, 2018 Abstract A complete list of one dimensional groups definable in the p-adic num- bers is given, up to a finite index subgroup and a quotient by a finite subgroup. 1 Introduction The main result of the following is Proposition 8.1. There we give a complete list of all one dimensional groups definable in the p-adics, up to finite index and quotient by finite kernel. This is similar to the list of all groups definable in a real closed field which are connected and one dimensional obtained in [7]. The starting point is the following result. Proposition 1.1. Suppose T is a complete first order theory in a language extending the language of rings, such that T extends the theory of fields. Suppose T is algebraically bounded and dependent. Let G be a definable group. If G arXiv:1811.09854v1 [math.LO] 24 Nov 2018 is definably amenable then there is an algebraic group H and a type-definable subgroup T G of bounded index and a type-definable group morphism T H with finite kernel.⊂ → This is found in Theorem 2.19 of [8]. We just note that algebraic boundedness is seen in greater generality than needed in [5], and it can also be extracted from the proofs in cell decomposition [4]. That one dimensional groups are definably amenable comes from the following result. 1 Proposition 1.2. Suppose G is a definable group in Qp which is one dimen- sional.
    [Show full text]
  • MATHEMATICS Algebra, Geometry, Combinatorics
    MATHEMATICS Algebra, geometry, combinatorics Dr Mark V Lawson November 17, 2014 ii Contents Preface v Introduction vii 1 The nature of mathematics 1 1.1 The scope of mathematics . .1 1.2 Pure versus applied mathematics . .3 1.3 The antiquity of mathematics . .4 1.4 The modernity of mathematics . .6 1.5 The legacy of the Greeks . .8 1.6 The legacy of the Romans . .8 1.7 What they didn't tell you in school . .9 1.8 Further reading and links . 10 2 Proofs 13 2.1 How do we know what we think is true is true? . 14 2.2 Three fundamental assumptions of logic . 16 2.3 Examples of proofs . 17 2.3.1 Proof 1 . 17 2.3.2 Proof 2 . 20 2.3.3 Proof 3 . 22 2.3.4 Proof 4 . 23 2.3.5 Proof 5 . 25 2.4 Axioms . 31 2.5 Mathematics and the real world . 35 2.6 Proving something false . 35 2.7 Key points . 36 2.8 Mathematical creativity . 37 i ii CONTENTS 2.9 Set theory: the language of mathematics . 37 2.10 Proof by induction . 46 3 High-school algebra revisited 51 3.1 The rules of the game . 51 3.1.1 The axioms . 51 3.1.2 Indices . 57 3.1.3 Sigma notation . 60 3.1.4 Infinite sums . 62 3.2 Solving quadratic equations . 64 3.3 *Order . 70 3.4 *The real numbers . 71 4 Number theory 75 4.1 The remainder theorem . 75 4.2 Greatest common divisors .
    [Show full text]
  • Valuations, Orderings, and Milnor K-Theory Mathematical Surveys and Monographs
    http://dx.doi.org/10.1090/surv/124 Valuations, Orderings, and Milnor K-Theory Mathematical Surveys and Monographs Volume 124 Valuations, Orderings, and Milnor K-Theory Ido Efrat American Mathematical Society EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 12J10, 12J15; Secondary 12E30, 12J20, 19F99. For additional information and updates on this book, visit www.ams.org/bookpages/surv-124 Library of Congress Cataloging-in-Publication Data Efrat, Ido, 1963- Valuations, orderings, and Milnor if-theory / Ido Efrat. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 124) Includes bibliographical references and index. ISBN 0-8218-4041-X (alk. paper) 1. Valuation theory. 2. Ordered fields. 3. X-theory I. Title. II. Mathematical surveys and monographs ; no. 124. QA247.E3835 2006 515/.78-dc22 2005057091 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2006 by the American Mathematical Society.
    [Show full text]
  • P-Adic L-Functions and Modular Forms
    p-adic L-functions and modular forms Alexei PANCHISHKIN Institut Fourier, Université Grenoble-1 B.P.74, 38402 St.Martin d'Hères, FRANCE Lectures for the International Summer School and Conference p-adic Analysis and its Applications August 31th to September 18th, 2009 (Trieste - ITALY) September, 2009 Alexei PANCHISHKIN (Grenoble) p-adic L-functions and modular forms ICTP, September,2009 1 / 1 Abstract 1) Congruences and p-adic numbers. Hensel's lemma. The Tate eld 2) Continuous and p-adic analytic functions. Mahler's criterion. Newton polygons Zeroes of analytic functions. The Weierstrass preparation theorem and its generalizations. 3) Distributions, measures, and the abstract Kummer congruences. The Kubota and Leopoldt p-adic L-functions and the Iwasawa algebra 4) Modular forms and L-functions. Congruences and families of modular forms. 5) Method of canonical projection of modular distributions. Examples of construction of p-adic L-functions Alexei PANCHISHKIN (Grenoble) p-adic L-functions and modular forms ICTP, September,2009 2 / 1 Related topics (for discussions, not included in the text of these materials) 6) Other approaches of constructing p-adic L-functions by Mazur-Tate-Teitelbaum, J.Coates, P.Colmez, H.Hida ... (using modular symbols by Manin-Mazur and their generalizations; Euler systems, work of D.Delbourgo, T.Ochiai, L.Berger, ..., overconvergent modular symbols by R.Pollack, G.Stevens, H.Darmon, ...) 7) Relations to the Iwasawa Theory 8) Applications of p-adic L-functions to Diophantine geometry 9) Open questions and problems in the theory of p-adic L-functions (Basic sources: Coates 60th Birthday Volume, Bourbaki talks by P.Colmez, J.Coates ...) Alexei PANCHISHKIN (Grenoble) p-adic L-functions and modular forms ICTP, September,2009 3 / 1 Lecture N◦1.
    [Show full text]
  • A Finiteness Theorem for Odd Perfect Numbers
    A finiteness theorem for odd perfect numbers Paul Pollack University of Georgia June 23, 2020 1 of 23 Ex: 5 is deficient (s(5) = 1), 12 is abundant (s(12) = 16), and 6 is perfect (s(6) = 6). Three kinds of natural numbers Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed one to the other; as for those that occupy the middle point between the two, they are said to be perfect. { Nicomachus (ca. 100 AD), Introductio Arithmetica P Let s(n) = djn;d<n d be the sum of the proper divisors of n. Abundant: s(n) > n. Deficient: s(n) < n. Perfect: s(n) = n. 2 of 23 Three kinds of natural numbers Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed one to the other; as for those that occupy the middle point between the two, they are said to be perfect. { Nicomachus (ca. 100 AD), Introductio Arithmetica P Let s(n) = djn;d<n d be the sum of the proper divisors of n. Abundant: s(n) > n. Deficient: s(n) < n. Perfect: s(n) = n. Ex: 5 is deficient (s(5) = 1), 12 is abundant (s(12) = 16), and 6 is perfect (s(6) = 6). 2 of 23 . In the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort | of which the most exemplary form is that type of number which is called perfect.
    [Show full text]
  • ON CERTAIN P-ADIC BANACH LIMITS of P-ADIC TRIANGULAR MATRIX ALGEBRAS
    International Journal of Pure and Applied Mathematics Volume 85 No. 3 2013, 435-455 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v85i3.1 ijpam.eu ON CERTAIN p-ADIC BANACH LIMITS OF p-ADIC TRIANGULAR MATRIX ALGEBRAS R.L. Baker University of Iowa Iowa City, Iowa 52242, USA Abstract: In this paper we investigate the class of p-adic triangular UHF (TUHF) Banach algebras.A p-adic TUHF Banach algebra is any unital p-adic Banach algebra of the form = n, where ( n) is an increasing sequence T T T T of p-adic Banach subalgebras of such that each n contains the identity of T S T T and is isomorphic as an Ωp-algebra to Tpn (Ωp) for some pn, where Tpn (Ωp) is the algebra of upper triangular pn pn matrices over the p-adic field Ωp. The main × result is that the supernatural number associated to a p-adic TUHF Banach algebra is an invariant of the algebra, provided that the algebra satisfies certain local dimensionality conditions. AMS Subject Classification: 12J25, 12J99, 46L99, 46S10 Key Words: p-adic Banach algebras, p-adic Banach limits, p-adic UHF algebras, p-adic Triangular UHF algebras 1. Introduction In the article, Triangular UHF algebras [2], the author of the present paper introduced a very concrete class of non-selfadjoint operator algebras which are currently called standard triangular UHF operator algebras. Briefly: a standard triangular UHF (TUHF) operator algebra is any unital Banach algebra that is isometrically isomorphic to a Banach algebra inductive limit of the form = T lim(Tp ; σp p ).
    [Show full text]
  • MATHEMATICS Algebra, Geometry, Combinatorics
    MATHEMATICS Algebra, geometry, combinatorics Dr Mark V Lawson October 24, 2014 ii Contents 1 The nature of mathematics 1 1.1 What are algebra, geometry and combinatorics? . .1 1.1.1 Algebra . .1 1.1.2 Geometry . .5 1.1.3 Combinatorics . .7 1.2 The scope of mathematics . .8 1.3 Pure versus applied mathematics . .9 1.4 The antiquity of mathematics . 11 1.5 The modernity of mathematics . 12 1.6 The legacy of the Greeks . 14 1.7 The legacy of the Romans . 15 1.8 What they didn't tell you in school . 15 1.9 Further reading and links . 16 2 Proofs 19 2.1 How do we know what we think is true is true? . 20 2.2 Three fundamental assumptions of logic . 22 2.3 Examples of proofs . 23 2.3.1 Proof 1 . 23 2.3.2 Proof 2 . 26 2.3.3 Proof 3 . 28 2.3.4 Proof 4 . 29 2.3.5 Proof 5 . 31 2.4 Axioms . 37 2.5 Mathematics and the real world . 41 2.6 Proving something false . 41 2.7 Key points . 42 2.8 Mathematical creativity . 43 i ii CONTENTS 2.9 Set theory: the language of mathematics . 43 2.10 Proof by induction . 52 3 High-school algebra revisited 57 3.1 The rules of the game . 57 3.1.1 The axioms . 57 3.1.2 Indices . 63 3.1.3 Sigma notation . 66 3.1.4 Infinite sums . 68 3.2 Solving quadratic equations . 70 3.3 Order . 76 3.4 The real numbers .
    [Show full text]
  • Relative Abelian Kernels of Some Classes of Transformation Monoids
    BULL. AUSTRAL. MATH. SOC. 20M05, 20M20, 20K45, 20MO7 VOL. 73 (2006) [375-404] RELATIVE ABELIAN KERNELS OF SOME CLASSES OF TRANSFORMATION MONOIDS E. CORDEIRO, M. DELGADO AND V.H. FERNANDES We consider the symmetric inverse monoid Zn of an n-element chain and its in- verse submonoids VOIn, VOVIn, VOVln and VO1Hn of all order-preserving, order-preserving or order-reversing, orientation-preserving and orientation-preserving or orientation-reversing transformations, respectively, and give descriptions of their Abelian kernels relative to decidable pseudovarieties of Abelian groups. INTRODUCTION The computability of kernels of finite monoids became a popular problem in Finite Semigroup Theory after a paper of Rhodes and Tilson [23]. Independent solutions were given by Ash [3] and Ribes and Zalesskii [24]. Both solutions contains deep results which led to the development of the theory in various directions. See for instance [19, 2]. Com- puting kernels relative to other pseudovarieties had then interested various researchers; for examples, see Ribes and Zalesskii [25], the second author [5] or Steinberg [26]. As computing kernels are directly connected with the Mal'cev product of pseudovarieties of monoids where the second factor is a pseudovariety of groups (see [18]), decidability results can be obtained in this way. The interest of the second author in using in practice the algorithm to compute the Abelian kernel of a finite monoid obtained in [5] led him to work towards a concrete implementation. Having this idea in mind, by detailing some aspects and improving the efficiency of that algorithm (see [6]) a concrete implementation in GAP [27] was then produced.
    [Show full text]