Recent Advances in Functional Analysis: Dedicated to the Memory of Joe Diestel and Victor Lomonosov
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Arxiv:1707.09546V1 [Math.GN] 29 Jul 2017 Ru,Sprbegop Rcmatgop Suoopc G Pseudocompact Group, Precompact Group, Separable Group, 54B15
THE SEPARABLE QUOTIENT PROBLEM FOR TOPOLOGICAL GROUPS ARKADY G. LEIDERMAN, SIDNEY A. MORRIS, AND MIKHAIL G. TKACHENKO Abstract. The famous Banach-Mazur problem, which asks if every infinite- dimensional Banach space has an infinite-dimensional separable quotient Ba- nach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Fr´echet spaces. In this paper the analogous problem for topological groups is investigated. Indeed there are four natural analogues: Does every non-totally disconnected topological group have a sep- arable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. All four questions are answered here in the negative. How- ever, positive answers are proved for important classes of topological groups including (a) all compact groups; (b) all locally compact abelian groups; (c) all σ-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all σ-compact pro-Lie groups; (f) all pseudocompact groups. Negative answers are proved for precompact groups. 1. Introduction It is natural to attempt to describe all objects of a certain kind in terms of basic building blocks of that kind. For example one may try to describe general Banach spaces in terms of separable Banach spaces. Recall that a topological space is said to be separable if it has a countable dense subset. -
CURRICULUM VITAE PISIER Gilles Jean Georges Born November 18 1950 in Nouméa, New Caledonia. French Nationality 1966-67 Mathéma
CURRICULUM VITAE PISIER Gilles Jean Georges Born November 18 1950 in Noum´ea,New Caledonia. French Nationality 1966-67 Math´ematiquesEl´ementaires at Lyc´eeBuffon, Paris. 1967 Baccalaur´eat,section C. 1967-68 Math´ematiquesSup´erieuresand Math´ematiquesSp´eciales 1968-69 at Lyc´eeLouis-le Grand, Paris. 1969-72 Student at Ecole´ Polytechnique. 1971 Ma^ıtrisede math´ematiquesUniversit´ePARIS VII. 1972 D.E.A. de Math´ematiquespures Universit´ePARIS VI. Oct. 72 Stragiaire de Recherche at C.N.R.S. Oct. 74 Attach´ede Recherche at C.N.R.S. Nov. 77 Th`esede doctorat d'´etat `es-sciencesmath´ematiques, soutenue le 10 Novembre 1977 from Universit´ePARIS VII under the supervision of L. Schwartz. Oct. 79 Charg´ede Recherche at C.N.R.S. Oct. 81 Professor at the University of PARIS VI. Oct. 84-Jan.85 Visiting Professor at IHES (Bures s. Yvette). Sept. 85 Distinguished Professor (Owen Chair of Mathematics), Texas A&M University. Nov. 88 Short term Visitor, Institute of Advanced Study (Princeton). Feb. 90-Apr.90 Visiting Professor at IHES (Bures s. Yvette). Feb. 1991 Professeur de Classe Exceptionnelle (Universit´eParis 6) 1 VARIOUS DISTINCTIONS { Salem Prize 1979. { Cours Peccot at Coll`egede France 1981. { Prix Carri`erede l'Acad´emiedes Science de Paris 1982. { Invited speaker at the International Congress of Mathematicians (Warsaw,1983). {Fellow of the Institute of Mathematical Statistics 1989. {Grands prix de l'Acad´emiedes Sciences de Paris: Prix Fond´epar l'Etat 1992 {Invited one hour address at A.M.S. meeting in College Station, october 93. {Faculty Distinguished Achievement Award in Research 1993, Texas A&M University (from the Association of Former Students) {Elected \Membre correspondant" by \Acad´emiedes Sciences de Paris", April 94. -
Manifestations of Nonlinear Roundness in Analysis, Discrete Geometry And
MANIFESTATIONS OF NON LINEAR ROUNDNESS IN ANALYSIS, DISCRETE GEOMETRY AND TOPOLOGY STRATOS PRASSIDIS AND ANTHONY WESTON Abstract. Some forty years ago Per Enflo introduced the nonlinear notions of roundness and generalized roundness for general metric spaces in order to study (a) uniform homeomorphisms between (quasi-) Banach spaces, and (b) Hilbert's Fifth Problem in the context of non locally compact topological groups (see [23], [24], [25], and [26]). Since then the concepts of roundness and generalized roundess have proven to be particularly useful and durable across a number of important mathematical fields such as coarse geometry, discrete geometry, functional analysis and topology. The purpose of this article is to take a retrospective look at some notable applications of versions of nonlinear roundness across such fields, to draw some hitherto unpublished connections between such results, and to highlight some very intriguing open problems. 1. Nonlinear Roundness | Introduction and Background Nonlinear notions of roundness and generalized roundness (see Definition 1.1) were introduced by Enflo in the late 1960s in a series of concise but elegant papers [23], [24], [25] and [26]. The purpose of Enflo’s programme of study in these papers was to investigate Hilbert's fifth problem in the context of non locally compact topological groups and to address the nonlinear classification of topological vector spaces up to uniform homeomorphism. Therein, Enflo used both roundness and generalized roundness in order to expose decisive estimates on the distortion of certain nonlinear maps between metric spaces. Later, within the context of Banach spaces, it became clear that roundness could be viewed as a natural precursor of Rademacher type. -
L. Maligranda REVIEW of the BOOK by ROMAN
Математичнi Студiї. Т.46, №2 Matematychni Studii. V.46, No.2 УДК 51 L. Maligranda REVIEW OF THE BOOK BY ROMAN DUDA, “PEARLS FROM A LOST CITY. THE LVOV SCHOOL OF MATHEMATICS” L. Maligranda. Review of the book by Roman Duda, “Pearls from a lost city. The Lvov school of mathematics”, Mat. Stud. 46 (2016), 203–216. This review is an extended version of my two short reviews of Duda's book that were published in MathSciNet and Mathematical Intelligencer. Here it is written about the Lvov School of Mathematics in greater detail, which I could not do in the short reviews. There are facts described in the book as well as some information the books lacks as, for instance, the information about the planned print in Mathematical Monographs of the second volume of Banach's book and also books by Mazur, Schauder and Tarski. My two short reviews of Duda’s book were published in MathSciNet [16] and Mathematical Intelligencer [17]. Here I write about the Lvov School of Mathematics in greater detail, which was not possible in the short reviews. I will present the facts described in the book as well as some information the books lacks as, for instance, the information about the planned print in Mathematical Monographs of the second volume of Banach’s book and also books by Mazur, Schauder and Tarski. So let us start with a discussion about Duda’s book. In 1795 Poland was partioned among Austria, Russia and Prussia (Germany was not yet unified) and at the end of 1918 Poland became an independent country. -
L. Maligranda REVIEW of the BOOK BY
Математичнi Студiї. Т.50, №1 Matematychni Studii. V.50, No.1 УДК 51 L. Maligranda REVIEW OF THE BOOK BY MARIUSZ URBANEK, “GENIALNI – LWOWSKA SZKOL A MATEMATYCZNA” (POLISH) [GENIUSES – THE LVOV SCHOOL OF MATHEMATICS] L. Maligranda. Review of the book by Mariusz Urbanek, “Genialni – Lwowska Szko la Matema- tyczna” (Polish) [Geniuses – the Lvov school of mathematics], Wydawnictwo Iskry, Warsaw 2014, 283 pp. ISBN: 978-83-244-0381-3 , Mat. Stud. 50 (2018), 105–112. This review is an extended version of my short review of Urbanek's book that was published in MathSciNet. Here it is written about his book in greater detail, which was not possible in the short review. I will present facts described in the book as well as some false information there. My short review of Urbanek’s book was published in MathSciNet [24]. Here I write about his book in greater detail. Mariusz Urbanek, writer and journalist, author of many books devoted to poets, politicians and other figures of public life, decided to delve also in the world of mathematicians. He has written a book on the phenomenon in the history of Polish science called the Lvov School of Mathematics. Let us add that at the same time there were also the Warsaw School of Mathematics and the Krakow School of Mathematics, and the three formed together the Polish School of Mathematics. The Lvov School of Mathematics was a group of mathematicians in the Polish city of Lvov (Lw´ow,in Polish; now the city is in Ukraine) in the period 1920–1945 under the leadership of Stefan Banach and Hugo Steinhaus, who worked together and often came to the Scottish Caf´e (Kawiarnia Szkocka) to discuss mathematical problems. -
Amenability, Group C -Algebras and Operator Spaces (WEP and LLP For
Amenability, Group C ∗-algebras and Operator spaces (WEP and LLP for C ∗(G)) Gilles Pisier Texas A&M University Gilles Pisier Amenability, Group C ∗-algebras and Operator spaces (WEP and LLP for C ∗(G)) Main Source... E. Kirchberg, On nonsemisplit extensions, tensor products and exactness of group C∗-algebras. Invent. Math. 112 (1993), 449{489. Gilles Pisier Amenability, Group C ∗-algebras and Operator spaces (WEP and LLP for C ∗(G)) C ∗-algebras A C ∗-algebra is a closed self-adjoint subalgebra A ⊂ B(H) of the space of bounded operators on a Hilbert space H An operator space is a (closed) subspace E ⊂ A of a C ∗-algebra I will restrict to unital C ∗-algebras The norm on the ∗-algebra A satisfies 8x; y 2 A kxyk ≤ kxkkyk kxk = kx∗k kx∗xk = kxk2 Such norms are called C ∗-norms After completion (or if A is already complete): there is a unique C ∗-norm on A Gilles Pisier Amenability, Group C ∗-algebras and Operator spaces (WEP and LLP for C ∗(G)) Then any such A can be written as A = span[π(G)] for some discrete group G and some unitary representation π : G ! B(H) of G on H Typical operator space E = span[π(S)] S ⊂ G Throughout I will restrict to discrete groups Gilles Pisier Amenability, Group C ∗-algebras and Operator spaces (WEP and LLP for C ∗(G)) Tensor products On the algebraic tensor product (BEFORE completion) A ⊗ B there is a minimal and a maximal C ∗-norm denoted by k kmin and k kmax and in general k kmin ≤6= k kmax C ∗-norms 8x; y 2 A kxyk ≤ kxkkyk kxk = kx∗k kx∗xk = kxk2 Gilles Pisier Amenability, Group C ∗-algebras and Operator spaces (WEP and LLP for C ∗(G)) Nuclear pairs Let A; B be C ∗-algebras Definition The pair (A; B) is said to be a nuclear pair if the minimal and maximal C ∗-norms coincide on the algebraic tensor product A ⊗ B, in other words A ⊗min B = A ⊗max B Definition A C ∗-algebra A is called nuclear if this holds for ANY B Gilles Pisier Amenability, Group C ∗-algebras and Operator spaces (WEP and LLP for C ∗(G)) Let A1 = span[π(G1)] ⊂ B(H1) A2 = span[π(G2)] ⊂ B(H2) for some unitary rep. -
Per Enflo to Return to Chagrin Series for Mozart Concertos on January 21 by Daniel Hathaway
Per Enflo to return to Chagrin Series for Mozart concertos on January 21 by Daniel Hathaway Most of us feel fortunate if we can make a dent in a single professional career. Thus it’s inspiring that Per Enflo has distinguished himself both as a theoretical mathematician and a concert pianist. His interest in those parallel but distinct disciplines dates from his childhood in Sweden, where he first showed an aptitude for mathematics and played his first full recital on a professional concert series at the age of 11. As a mathematician, Enflo has cracked several seemingly unsolvable problems in functional analysis while teaching at Berkeley, Stanford, the École Polytechnique in Paris, and the Royal Institute of Technology in Stockholm. In 1989, he was appointed one of three “University Professors” at Kent State. In addition to teaching mathematics, he also worked on such cross-disciplinary issues as the zebra mussel invasion and the phosphorus loading of Lake Erie, anthropology and human evolution, and acoustics and noise reduction. Upon retirement in 2012, he moved back to Sweden, but still makes regular appearances in the States. This weekend, Enflo will return to Northeast Ohio to play Mozart’s 17th and 21st Concertos with the Cleveland Virtuosi on the Chagrin Concert Series at Valley Lutheran Church in Chagrin Falls. The free 3:00 pm concert on Sunday, January 21 will be led by Enflo’s frequent collaborator, violinist and series artistic director Hristo Popov. I recently spoke with Per Enflo in a telephone conversation from his home in Östervåla near Uppsala and began by asking him how he spends his time these days. -
Functional Analysis an Elementary Introduction
Functional Analysis An Elementary Introduction Markus Haase Graduate Studies in Mathematics Volume 156 American Mathematical Society Functional Analysis An Elementary Introduction https://doi.org/10.1090//gsm/156 Functional Analysis An Elementary Introduction Markus Haase Graduate Studies in Mathematics Volume 156 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 46-01, 46Cxx, 46N20, 35Jxx, 35Pxx. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-156 Library of Congress Cataloging-in-Publication Data Haase, Markus, 1970– Functional analysis : an elementary introduction / Markus Haase. pages cm. — (Graduate studies in mathematics ; volume 156) Includes bibliographical references and indexes. ISBN 978-0-8218-9171-1 (alk. paper) 1. Functional analysis—Textbooks. 2. Differential equations, Partial—Textbooks. I. Title. QA320.H23 2014 515.7—dc23 2014015166 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]. c 2014 by the American Mathematical Society. -
Grothendieck's Theorem, Past and Present UNCUT Updated and Still Expanding
Grothendieck's Theorem, past and present UNCUT updated and still expanding... by Gilles Pisier∗ Texas A&M University College Station, TX 77843, U. S. A. and Universit´eParis VI Equipe d'Analyse, Case 186, 75252 Paris Cedex 05, France October 13, 2015 Abstract Probably the most famous of Grothendieck's contributions to Banach space theory is the result that he himself described as \the fundamental theorem in the metric theory of tensor products". That is now commonly referred to as \Grothendieck's theorem" (GT in short), or sometimes as \Grothendieck's inequality". This had a major impact first in Banach space theory (roughly after 1968), then, later on, in C∗-algebra theory, (roughly after 1978). More recently, in this millennium, a new version of GT has been successfully developed in the framework of \operator spaces" or non-commutative Banach spaces. In addition, GT independently surfaced in several quite unrelated fields: in connection with Bell's inequality in quantum mechanics, in graph theory where the Grothendieck constant of a graph has been introduced and in computer science where the Grothendieck inequality is invoked to replace certain NP hard problems by others that can be treated by “semidefinite programming' and hence solved in polynomial time. This expository paper (where many proofs are included), presents a review of all these topics, starting from the original GT. We concentrate on the more recent developments and merely outline those of the first Banach space period since detailed accounts of that are already avail- able, for instance the author's 1986 CBMS notes. Warning: Compared to the published one in Bull Amer. -
Used the Idea of Metric Roundness to Investigate the Uniform Structure of Banach Spaces
ROUNDNESS PROPERTIES OF GROUPS JEAN-FRANC¸OIS LAFONT AND STRATOS PRASSIDIS∗ Abstract. Roundness of metric spaces was introduced by Per Enflo as a tool to study uniform structures of linear topological spaces. The present paper investigates geometric and topological properties detected by the roundness of general metric spaces. In particular, we show that geodesic spaces of roundness 2 are contractible, and that a compact Riemannian manifold with roundness > 1 must be simply connected. We then focus our investigation on Cayley graphs of finitely generated groups. One of our main results is that every Cayley graph of a free abelian group on ≥ 2 generators has roundness = 1. We show that if a group has no Cayley graph of roundness = 1, then it must be a torsion group with every element of order 2, 3, 5, or 7. 1. Introduction In a series of papers Per Enflo ([6], [8], [9]) used the idea of metric roundness to investigate the uniform structure of Banach spaces. Later the same idea was used in [19] to compare uniform structures between normed and quasi-normed linear topological spaces. An extension of this prop- erty (generalized roundness) was used by Enflo in the solution of Smirnov’s problem ([7]). Also, if a metric space has non-trivial generalized roundness, then some positive power of the distance func- tion is a negative kernel on the space ([16]). Negative kernels on Cayley graphs of discrete groups were used for proving the coarse Baum–Connes Conjecture (and thus the Novikov Conjecture) for these groups ([14], [15]). We investigate the roundness and generalized roundness properties of general metric spaces. -
Three Non-Linear Problems on Normed Spaces A
THREE NON-LINEAR PROBLEMS ON NORMED SPACES A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Francisco J. Garc´ıa February, 2007 Dissertation written by Francisco J. Garc´ıa B.S., University of C´adiz, Spain, 2000 M.S., University of C´adiz, Spain, 2004 Ph.D., University of C´adiz, Spain, 2005 M.A., Kent State University, 2006 Ph.D., Kent State University, 2007 Approved by Richard M. Aron, Chair, Doctoral Dissertation Committee Andrew Tonge, Members, Doctoral Dissertation Committee Per H. Enflo, Johnnie W. Baker, (Outside Person) Paul S. Wang, (Graduate Representative) Accepted by Andrew Tonge, Chair, Department of Mathematical Sciences Jerry Feezel, Dean, College of Arts and Sciences ii TABLE OF CONTENTS ACKNOWLEDGEMENTS .............................. v INTRODUCTION ................................... 1 1 THE LINEABILITY PROBLEM FOR FUNCTIONALS ......... 7 1.1 Preliminaries . 7 1.2 Lineability of NA (X)............................... 16 1.3 Lineability of X∗ \ NA (X)............................ 21 1.4 Density of X∗ \ NA (X)............................. 24 2 THE MINIMUM-NORM PROBLEM FOR TRANSLATIONS ..... 31 2.1 Preliminaries . 31 2.2 Minimum-norm elements and norm-attaining functionals . 42 2.3 Non-complete normed spaces having only norm-attaining functionals . 45 2.4 Partial solutions . 48 3 THE BANACH-MAZUR CONJECTURE FOR ROTATIONS ...... 51 3.1 Preliminaries . 51 iii 3.2 Geometrical conditions . 58 3.3 Topological conditions . 60 3.4 Intermediate solutions . 63 BIBLIOGRAPHY . 68 iv ACKNOWLEDGEMENTS There are several people to whom I should be giving thanks for my stay at Kent State University: Richard Aron, Joe Diestel, Andrew Tonge, Artem Zvavitch, Per Enflo, and Juan Seoane. -
Hans Rådström and How to Define Smooth Functions on Any
The Bulletin of the Swedish Mathematical Society, 2019 May 15, pp. 20–23. Hans Rådström and how to define smooth functions on any set Christer Oscar Kiselman Hans Rådström was born in Stockholm on 1919 March 26, a little more than a century ago. He died in Linköping on 1970 November 05. He was Associate Professor (in Swedish laborator) at the Royal Institute of Technology from 1954 through 1964. During his tenure at the Royal Institute of Technology Hans advised two students to the PhD (at the time called Tekn Dr) in mathematics. Lars Ingelstam defended his thesis Studies in Real Normed Algebras in 1964 (he later became Professor of Technol- ogy and Social Change at Linköping University). In 1967, Johan Philip presented his thesis Reconstruction of intensity distributions.1 Hans Rådström then moved to a similar position at Stockholm University, where he stayed from 1964 up to 1969. It was there that I came to know him. Part of Hans’ wide interests in different branches of mathematics and applications led him to approach non-zero-sum games in a novel way. His results, however, were never published. For a similar situation, but with a different outcome, see the section on smooth functions below. Hans acted for the creation of a single Department of Mathematics in Stockholm, joint for the Royal Institute of Technology and Stockholm University. He based his arguments on his experience from both institutions. That this was seriously considered by Stockholm University is illustrated by the fact that Göran Björck and several others visited Copenhagen, Aarhus and Oslo to see how such a “Mathematical Center” could function.