DOCSLIB.ORG

14FUN Functional Analysis Commutative Banach Algebras Dr J. F. Feinstein March 5

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
14FUN Functional Analysis Commutative Banach Algebras Dr J. F. Feinstein March 5 G14FUN Functional Analysis Commutative Banach Algebras Dr J. F. Feinstein March 25, 2010 1 1 Introduction to normed algebras and Banach algebras Banach algebras may be thought of as Banach spaces with multiplication (in a sense made more formal below). The additional structure provided by the multiplication gives the theory of Banach algebras a rather different flavour from the more general theory of Banach spaces. Banach algebras may be real or complex. However the theory of complex Banach algebras is richer, and so this is what we will focus on. 2 Banach algebras may be commutative or non-commutative. In this module, we will focus mainly on commutative Banach algebras, including an introduction to the Gelfand theory. A key example of a commutative Banach algebra is CC(X), the algebra of continuous, complex-valued functions on a compact, Hausdorff space, with the usual pointwise operations and with the uniform norm. 3 A typical non-commutative Banach algebra is B(E), the algebra of all bounded linear operators from E to E for some Banach space E, with the usual vector space structure and operator norm, and with product given by composition of operators. In particular, for a Hilbert space H, B(H) and its subalgebras are of major interest. Just as the theory of Banach algebras should not be regarded as part of the theory of Banach spaces, the theory of B(H) and its subalgebras has its own flavour and a vast literature, including the quite distinct study of C∗-algebras and von Neumann algebras. 4 Definition 1.1 A complex algebra is a complex vector space A which is a ring with respect to an associative multiplication which is also a bilinear map, i.e., the distributive laws hold and, for all 2 C and a and b in A, we have ( a)b = a( b) = (ab) : The complex algebra A has an identity if there exists an element e 6= 0 2 A such that, for all a 2 A, we have ea = ae = a. The complex algebra A is commutative if, for all a and b in A we have ab = ba. Real algebras are defined similarly. 5 Note that, if A has an identity e, then this identity is unique, so we may call e the identity of A. We will often denote the identity by 1 rather than e, assuming that the context ensures that there is no ambiguity. The assumption above that e 6= 0 means that we do not count 0 as an identity in the trivial algebra f0g. From now on, all algebras will be assumed to be complex algebras, so we will be defining complex normed algebras and Banach algebras. However the corresponding definitions may be used to define real normed algebras and Banach algebras. 6 Definition 1.2 A (complex) normed algebra is a pair (A; k ⋅ k) where A is a complex algebra and k ⋅ k is a norm on A which is sub-multiplicative, i.e., for all a and b in A, we have kabk ≤ kakkbk : A normed algebra A is unital if it has an identity 1 and k1k = 1. A Banach algebra (or complete normed algebra) is a normed algebra which is complete as a normed space. 7 Notes. ∙ In our usual way, we will often call A itself a normed algebra, if there is no ambiguity in the norm used on A. ∙ We will mostly be interested in commutative, unital Banach algebras. ∙ The condition that the norm on A be sub-multiplicative is only slightly stronger than the requirement that multiplication be jointly continuous (or equivalently, that multiplication be a `bounded bilinear map') from A × A ! A. In fact (easy exercise) if we have kabk ≤ Ckakkbk, for some constant C > 1, then we can easily find an equivalent norm on A which is actually sub-multiplicative. 8 ∙ If a normed algebra (A; k ⋅ k) has an identity 1 such that k1k= 6 1, then we may again define another norm jjj ⋅ jjj on A as follows: ¯ jjjajjj = supfkabk : b 2 BA(0; 1) g : As another exercise, you may check that jjj ⋅ jjj is equivalent to k ⋅ k, that jjj ⋅ jjj is sub-multiplicative and that jjj1jjj = 1. So the assumption that k1k = 1 is a convenience, rather than a topological restriction. ∙ Every subalgebra of a normed algebra is a normed algebra, and every closed subalgebra of a Banach algebra is a Banach algebra. 9 Examples. (1) Let E be a complex Banach space of dimension > 1. Then (B(E); k ⋅ kop) is a non-commutative, unital Banach algebra, where the product on B(E) is composition of operators. (2) Let X be a non-empty, compact, Hausdorff topological space. Then CC(X) is a commutative, unital Banach algebra with pointwise operations and the uniform norm j ⋅ jX : recall that jfjX = supfjf(x)j : x 2 X g : As we are focussing on complex algebras, from now on we will denote CC(X) by C(X). 10 (3) With notation as in (2), every (uniformly) closed subalgebra of C(X) is also a commutative Banach algebra with respect to the uniform norm j ⋅ jX . A uniform algebra on X is a closed subalgebra A of C(X) which contains all the constant functions and which separates the points of X, i.e., whenever x and y are in X with x 6= y, there exists f 2 A with f(x) 6= f(y). Every uniform algebra on X is, of course, a commutative, unital Banach algebra, with respect to the uniform norm. Also, by Urysohn's Lemma, C(X) itself does separate the points of X, and so C(X) is a uniform algebra on X. 11 (4) In particular, let X be a non-empty, compact subset of C. Consider the following subalgebras of C(X): A(X) is the set of those functions in C(X) which are analytic (holomorphic) on the interior of X; P0(X) is the set of restrictions to X of polynomial functions with complex coefficients; R0(X) is the set of restrictions to X of rational functions with complex coefficients whose poles (if any) lie off X (so R0(X) = fp=q : p; q 2 P0(X); 0 2= q(X) g). 12 It is easy to see that these subalgebras contain the constant functions and separate the points of X. Indeed the polynomial function Z, also called the co-ordinate functional, defined by Z() = ( 2 C), clearly separates the points of X by itself, and is in all of these algebras. The algebra A(X) is closed in C(X), because uniform limits of analytic functions are analytic (see books for details), and so A(X) is a uniform algebra on X. 13 The algebras P0(X) and R0(X) are usually not closed in C(X) (exercise: investigate this). We obtain uniform algebras on X by taking their (uniform) closures: P (X) is the closure of P0(X) and R(X) is the closure of R0(X). The functions in P (X) are those which may be uniformly approximated on X by polynomials, and the functions in R(X) are those which may be uniformly approximated on X by rational functions with poles off X. We have P (X) ⊆ R(X) ⊆ A(X) ⊆ C(X). 14 (5) The Banach space C1[0; 1] of once continuously differentiable complex-valued functions on [0; 1] is a subalgebra of C[0; 1]. It is not uniformly closed, and so it is not a uniform algebra on [0; 1]. However, it is a Banach algebra when given 0 its own norm, kfk = jfjX + jf jX . This is a typical example of a Banach function algebra. Warning! In the literature, some authors call uniform algebras function algebras. (6) Every complex normed space E can be made into a (non-unital) normed algebra by defining the trivial multiplication xy = 0 (x; y 2 E). Thus every complex Banach space is also a Banach algebra. 15 (7) The following standard construction can be used to `unitize' Banach algebras. Let (A; k ⋅ kA) be a complex Banach algebra without an identity. We may form a Banach algebra A#, called the standard unitization of A, as follows. As a vector space, A# = A ⊕ C. This becomes a Banach space when given the norm k(a; )k = kakA + j j (a 2 A; 2 C) : We can then make A# into a unital Banach algebra using the following multiplication: for a and b in A and and in C, we define (a; )(b; ) = (ab + b + a; ) : 16 Exercise. Check the details of these claims. (What is the identity element of A#?) Show also that A# is commutative if and only if A is commutative. We can combine this construction with (6) to give examples of commutative, unital Banach algebras which have many non-zero elements whose squares are 0. 17 2 Characters and the character space for commutative Banach algebras In this section we will focus on commutative algebras. However, many of the definitions and results are valid (with some minor modifications) in the non-commutative setting too: see books for details. Definition 2.1 Let A be a commutative algebra with identity 1. An element a 2 A is invertible if there exists b 2 A with ab = 1. In this case the element b is unique: b is then called the inverse of a, and is denoted by a−1. The set of invertible elements of A is denoted by Inv A. 18 Notes. ∙ For non-commutative algebras, we would insist that both ab = 1 and ba = 1.
Load more

Recommended publications
  • Noncommutative Uniform Algebras
    Noncommutative Uniform Algebras Mati Abel and Krzysztof Jarosz Abstract. We show that a real Banach algebra A such that a2 = a 2 , for a A is a subalgebra of the algebra C (X) of continuous quaternion valued functions on a compact setk kX. ∈ H ° ° ° ° 1. Introduction A well known Hirschfeld-Zelazkoú Theorem [5](seealso[7]) states that for a complex Banach algebra A the condition (1.1) a2 = a 2 , for a A k k ∈ implies that (i) A is commutative, and further° ° implies that (ii) A must be of a very speciÞc form, namely it ° ° must be isometrically isomorphic with a uniformly closed subalgebra of CC (X). We denote here by CC (X)the complex Banach algebra of all continuous functions on a compact set X. The obvious question about the validity of the same conclusion for real Banach algebras is immediately dismissed with an obvious counterexample: the non commutative algebra H of quaternions. However it turns out that the second part (ii) above is essentially also true in the real case. We show that the condition (1.1) implies, for a real Banach algebra A,thatA is isometrically isomorphic with a subalgebra of CH (X) - the algebra of continuous quaternion valued functions on acompactsetX. That result is in fact a consequence of a theorem by Aupetit and Zemanek [1]. We will also present several related results and corollaries valid for real and complex Banach and topological algebras. To simplify the notation we assume that the algebras under consideration have units, however, like in the case of the Hirschfeld-Zelazkoú Theorem, the analogous results can be stated for none unital algebras as one can formally add aunittosuchanalgebra.
    [Show full text]
  • Regularity Conditions for Banach Function Algebras
    Regularity conditions for Banach function algebras Dr J. F. Feinstein University of Nottingham June 2009 1 1 Useful sources A very useful text for the material in this mini-course is the book Banach Algebras and Automatic Continuity by H. Garth Dales, London Mathematical Society Monographs, New Series, Volume 24, The Clarendon Press, Oxford, 2000. In particular, many of the examples and conditions discussed here may be found in Chapter 4 of that book. We shall refer to this book throughout as the book of Dales. Most of my e-prints are available from www.maths.nott.ac.uk/personal/jff/Papers Several of my research and teaching presentations are available from www.maths.nott.ac.uk/personal/jff/Beamer 2 2 Introduction to normed algebras and Banach algebras 2.1 Some problems to think about Those who have seen much of this introductory material before may wish to think about some of the following problems. We shall return to these problems at suitable points in this course. Problem 2.1.1 (Easy using standard theory!) It is standard that the set of all rational functions (quotients of polynomials) with complex coefficients is a field: this is a special case of the “field of fractions" of an integral domain. Question: Is there an algebra norm on this field (regarded as an algebra over C)? 3 Problem 2.1.2 (Very hard!) Does there exist a pair of sequences (λn), (an) of non-zero complex numbers such that (i) no two of the an are equal, P1 (ii) n=1 jλnj < 1, (iii) janj < 2 for all n 2 N, and yet, (iv) for all z 2 C, 1 X λn exp (anz) = 0? n=1 Gap to fill in 4 Problem 2.1.3 Denote by C[0; 1] the \trivial" uniform algebra of all continuous, complex-valued functions on [0; 1].
    [Show full text]
  • Uniform Algebras As Banach Spaces
    The Erwin Schrodinger International Boltzmanngasse ESI Institute for Mathematical Physics A Wien Austria Uniform Algebras as Banach Spaces TW Gamelin SV Kislyakov Vienna Preprint ESI June Supp orted by Federal Ministry of Science and Transp ort Austria Available via httpwwwesiacat UNIFORM ALGEBRAS AS BANACH SPACES TW Gamelin SV Kislyakov June Abstract Any Banach space can b e realized as a direct summand of a uniform algebra and one do es not exp ect an arbitrary uniform algebra to have an abundance of prop erties not common to all Banach spaces One general result concerning arbitrary uniform algebras is that no prop er uniform algebra is linearly homeomorphic to a C K space Nevertheless many sp ecic uniform algebras arising in complex analysis share or are susp ected to share certain Banach space prop erties of C K We discuss the family of tight algebras which includes algebras of analytic functions on strictly pseudo con vex domains and algebras asso ciated with rational approximation theory in the plane Tight algebras are in some sense close to C K spaces and along with C K spaces they have the Pelczynski and the DunfordPettis prop erties We also fo cus on certain prop erties of C K spaces that are inherited by the disk algebra This includes a dis p cussion of interp olation b etween H spaces and Bourgains extension of Grothendiecks theorem to the disk algebra We conclude with a brief description of linear deformations of uniform algebras and a brief survey of the known classication results Supp orted in part by the Russian
    [Show full text]
  • 1. Introduction
    FACTORIZATION IN COMMUTATIVE BANACH ALGEBRAS H. G. DALES, J. F. FEINSTEIN, AND H. L. PHAM Abstract. Let A be a (non-unital) commutative Banach algebra. We consider when A has a variety of factorization properties: we list the (ob- vious) implications between these properties, and then consider whether any of these implications can be reversed in various classes of commu- tative Banach algebras. We summarize the known counter-examples to these possible reverse implications, and add further counter-examples. Some results are used to show the existence of a large family of prime ideals in each non-zero, commutative, radical Banach algebra with a dense set of products. 1. Introduction Let A be a (non-unital) commutative Banach algebra. We wish to examine when A factors in a variety of senses. Our main results are counter-examples to a number of questions that have been raised. Indeed, we shall list seven such factorization properties, called (I)(VII), and note that each of these immediately implies the next one. We shall also, in x4, discuss two other `lo- cal' factorization properties, called (A) and (B) (where (A) ) (B)); these properties are relevant for Esterle's classication of commutative, radical Banach algebras that is given in [14]. We shall then discuss whether or not any of these implications can be reversed when we restrict attention to par- ticular classes of commutative Banach algebras. We shall show that several cannot be reversed, but we leave open other possible reverse implications. A summary in x6 describes our knowledge at the present time. We shall concentrate on two particular classes of commutative Banach algebras A: rst, on the case where A is semi-simple (so that A is a Banach function algebra), and in particular when A is a maximal ideal in a uniform algebra on a compact space, and, second, on the other extreme case where A 2010 Mathematics Subject Classication.
    [Show full text]
  • Order Structure, Multipliers, and Gelfand Representation of Vector-Valued Function Algebras
    ORDER STRUCTURE, MULTIPLIERS, AND GELFAND REPRESENTATION OF VECTOR-VALUED FUNCTION ALGEBRAS JORMA ARHIPPAINEN, JUKKA KAUPPI∗ AND JUSSI MATTAS Abstract. We continue the study begun by the third author of C∗-Segal algebra valued function algebras, with an emphasis on the order structure. Our main result is a characterization theorem for C∗-Segal algebra valued function algebras with an order unitization. As an intermediate step, we establish a function algebraic description of the multiplier module of arbitrary Segal algebra valued function algebras. We also consider Gelfand representation of these algebras in the commutative case. Introduction The concept of a Segal algebra originated in the work of Reiter on subalgebras of the L1-algebra of a locally compact group; cf. [19]. It was generalized to arbitrary Banach algebras by Burnham in [8]. A C∗-Segal algebra is a Banach algebra which is a dense (not necessarily self-adjoint) ideal in a C∗-algebra. Despite many important examples in analysis, such as the Schatten classes for instance, little has been known about the general structure and properties of C∗-Segal algebras. In particular, most results have relied on the additional assumption of an approximate identity. The multiplier algebra and the bidual of self-adjoint C∗-Segal algebras were described in [2, 14], and the form of the closed ideals of C∗-Segal algebras with an approximate identity was given in [6]. Recently, a new approach to Segal algebras was introduced in [5, 12], which allows a more detailed study of their structure and properties, particularly in the absence of an approximate identity ([5] the commutative case and [12] the noncommutative case).
    [Show full text]
  • Function Representation of a Noncommutative Uniform Algebra
    Function representation of a noncommutative uniform algebra Krzysztof Jarosz Abstract. We construct a Gelfand type representation of a real noncommu- tative Banach algebra A satisfying f 2 = f 2, for all f A: k k 2 1. Introduction A uniform algebra A is a Banach algebra such that (1.1) f 2 = f 2 , for all f A: k k 2 The commutative, complex uniform algebras constitute the most classical class of Banach algebras, one which has been intensely studied for decades. It is well known that any such algebra is isometrically isomorphic with a subalgebra of CC (X) - the algebra of all continuous complex valued functions de…ned on a compact set X and equipped with the sup-norm topology. Hirschfeld and Zelazko· [6] proved that the assumption that the algebra is commutative is super‡uous - commutativity al- ready follows from (1.1). Algebras of analytic functions serve as standard examples of complex uniform algebras. Commutative, real uniform algebras have also been studied for years [9]. Any such algebra A is isometrically isomorphic with a real subalgebra of CC (X) for some compact set X; furthermore X can be often divided into three parts X1;X2; and X3 such that A X1 is a complex uniform algebra, A X2 consists of complex conjugates j j of the functions from A X1 , and A X3 is equal to CR (X3). Any commutative, real uniform algebra can bej naturally identi…edj with a real subalgebra of df C (X; ) = f C (X): f ( (x)) = f (x) for x X C 2 C 2 n 2 o where : X X is a surjective homeomorphism with = idX .
    [Show full text]
  • Arxiv:Math/0203199V1 [Math.FA] 19 Mar 2002
    Amenability for dual Banach algebras Volker Runde Abstract ∗ We define a Banach algebra A to be dual if A = (A∗) for a closed submodule A∗ of A∗. The class of dual Banach algebras includes all W ∗-algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions an amenable dual Ba- nach algebra is already super-amenable and thus finite-dimensional. We then develop two notions of amenability — Connes-amenability and strong Connes-amenability — which take the w∗-topology on dual Banach algebras into account. We relate the amenability of an Arens regular Banach algebra A to the (strong) Connes-amenability of A∗∗; as an application, we show that there are reflexive Banach spaces with the approximation property such that L(E) is not Connes-amenable. We characterize the amenability of inner amenable locally compact groups in terms of their algebras of pseudo-measures. Finally, we give a proof of the known fact that the amenable von Neumann algebras are the subhomogeneous ones which avoids the equivalence of amenability and nuclearity for C∗-algebras. 2000 Mathematics Subject Classification: 43A10, 46B28, 46H25, 46H99 (primary), 46L10, 46M20. 1 Introduction arXiv:math/0203199v1 [math.FA] 19 Mar 2002 Amenable Banach algebras were introduced by B. E. Johnson in [Joh 2], and have since then turned out to be extremely interesting objects of research.
    [Show full text]
  • Mathematical Work of Franciszek Hugon Szafraniec and Its Impacts
    Tusi Advances in Operator Theory (2020) 5:1297–1313 Mathematical Research https://doi.org/10.1007/s43036-020-00089-z(0123456789().,-volV)(0123456789().,-volV) Group ORIGINAL PAPER Mathematical work of Franciszek Hugon Szafraniec and its impacts 1 2 3 Rau´ l E. Curto • Jean-Pierre Gazeau • Andrzej Horzela • 4 5,6 7 Mohammad Sal Moslehian • Mihai Putinar • Konrad Schmu¨ dgen • 8 9 Henk de Snoo • Jan Stochel Received: 15 May 2020 / Accepted: 19 May 2020 / Published online: 8 June 2020 Ó The Author(s) 2020 Abstract In this essay, we present an overview of some important mathematical works of Professor Franciszek Hugon Szafraniec and a survey of his achievements and influence. Keywords Szafraniec Á Mathematical work Á Biography Mathematics Subject Classification 01A60 Á 01A61 Á 46-03 Á 47-03 1 Biography Professor Franciszek Hugon Szafraniec’s mathematical career began in 1957 when he left his homeland Upper Silesia for Krako´w to enter the Jagiellonian University. At that time he was 17 years old and, surprisingly, mathematics was his last-minute choice. However random this decision may have been, it was a fortunate one: he succeeded in achieving all the academic degrees up to the scientific title of professor in 1980. It turned out his choice to join the university shaped the Krako´w mathematical community. Communicated by Qingxiang Xu. & Jan Stochel [email protected] Extended author information available on the last page of the article 1298 R. E. Curto et al. Professor Franciszek H. Szafraniec Krako´w beyond Warsaw and Lwo´w belonged to the famous Polish School of Mathematics in the prewar period.
    [Show full text]
  • Function Representation of a Noncommutative Uniform Algebra
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 2, February 2008, Pages 605–611 S 0002-9939(07)09033-8 Article electronically published on November 1, 2007 FUNCTION REPRESENTATION OF A NONCOMMUTATIVE UNIFORM ALGEBRA KRZYSZTOF JAROSZ (Communicated by N. Tomczak-Jaegermann) Abstract. We construct a Gelfand type representation of a real noncommu- tative Banach algebra A satisfying f 2 = f2, for all f ∈ A. 1. Introduction A uniform algebra A is a Banach algebra such that (1.1) f 2 = f2 , for all f ∈ A. The commutative, complex uniform algebras constitute the most classical class of Banach algebras, one which has been intensely studied for decades. It is well known that any such algebra is isometrically isomorphic with a subalgebra of CC (X), the algebra of all continuous complex-valued functions defined on a compact set X and equipped with the sup-norm topology. Hirschfeld and Zelazko˙ [6] proved that the assumption that the algebra is commutative is superfluous—commutativity already follows from (1.1). Algebras of analytic functions serve as standard examples of complex uniform algebras. Commutative, real uniform algebras have also been studied for years [9]. Any such algebra A is isometrically isomorphic with a real subalgebra of CC (X)forsome compact set X;furthermore,X can often be divided into three parts X1,X2, and X3 such that A|X1 is a complex uniform algebra, A|X2 consists of complex conjugates of the functions from A|X1 ,andA|X3 is equal to CR (X3). Any commutative, real uniform algebra can be naturally identified with a real subalgebra of df CC (X, τ) = f ∈ CC (X):f (τ (x)) = f (x)forx ∈ X , 2 where τ : X → X is a surjective homeomorphism with τ = idX .Inthecase of real uniform algebras the condition (1.1) no longer implies commutativity; the four-dimensional algebra of quaternions serves as the simplest counterexample.
    [Show full text]
  • Hermitian Operators and Isometries on Banach Algebras of Continuous Maps with Values in Unital C∗-Algebras
    Hermitian operators and isometries on Banach algebras of continuous maps with values in unital C∗-algebras . Osamu Hatori∗ (Niigata University) ∗ Supported by JSPS KAKENHI Grant Numbers JP16K05172, JP15K04921 «Joint work with Shiho Oi» RTOTA 3 May, 2018 University of Memphis 1/ 25 General Problem . Problem . Ej : Banach space, C(Xj ; Ej ) ⊃ Aj with a Banach space norm When is it true that every surjective isometrya U : A1 ! A2 will be of the form (canonical form) [ ] (Uf )(x) = Vx (f ('(x)) ; x 2 X2 for continuous (with respect to SOT) map x 7! Vx from X2 to the space of surjective isometries on E1 to E2, and a homeomorphism '? aIn this talk an isometry on a complex Banach space always means a .complex-linear isometry. 2/ 25 The origin of the problem . The Banach-Stone theorem (1932, 1937 and later) . Suppose that U : C(X1; C) ! C(X2; C) is a surjective isometry. () U(1) is a continous function with the unit modulus 9' : X2 ! X1 : homeomorphism s.t. U(f )(x) = U(1)(x)f ('(x)); x 2 X2 2 C .for every f C(X1; ). This canonical form is also called a weighted composition operator. 3/ 25 After the Banach-Stone theorem: . Several surjective isometries are of weighted composition operators Lp spaces (Isometries between Lp[0; 1] have been already studied by Banach in his book, Lamperti (1958) and several variations) Kadison’s theorem for a unital C∗-algebra 1951,1952 (later non-unital case by Paterson and Sinclair 1972) Nagasawa’s theorem for a uniform algebra (simple counterexample for non-unital case) Spaces of analytic functions (de Leeuw-Rudin-Wermer for H1 and for a uniform algebra (1960), Forelli for Hp (1964), and ··· ) Algebras of Lipschitz functions (de Leeuw(1961) and ········· ) ··· 4/ 25 Not always of a canonical form.
    [Show full text]
  • Math Student 2007
    ISSN: 0025-5742 THE MATHEMATICS STUDENT Volume 76, Numbers 1 - 4, (2007) copy- Edited by J. R. PATADIA circulation (Issued: October, 2008) Member'sfor not PUBLISHED BY THE INDIAN MATHEMATICAL SOCIETY (In the memory of late Professor M. K. Singal) THE MATHEMATICS STUDENT Edited by J. R. PATADIA In keeping with the current periodical policy, THE MATHEMATICS STUDENT (STUDENT) will seek to publish material of interest not just to mathematicians with specialized interest but to undergraduate and postgraduate students and teachers of mathematics in India. With this in view, it will publish material of the following type: 1. Expository articles and popular (not highly technical) articles. 2. Classroom notes (this can include hints on teaching certain topics or filling vital gaps in the usual treatments of topics to be found in text-books or how some execises can and should be made an integral part of teaching, etc.) 3. Problems and solutions, News and Views. 4. Information on articles of common interest published in other periodicals, 5. Proceedings of IMS Conferences. Expository articles, classroom notes, news and announcements and research papers are invited for publication in THE MATHEMATICS STUDENT. Manuscripts intended for publication should preferably be submitted online in the LaTeX .pdf file or in the standard word processing format (Word or Word Perfect) including figures and tables. In case of hard copy submission, three copies of the manuscripts along with the Compact Disc (CD) copy should be submitted to the Editor Dr. J. R. Patadia, Department of Mathematics, faculty of Science, The Maharaja Sayajirao University of Baroda, Vadodara-390 002 (Gujarat),copy- India.
    [Show full text]
  • Arxiv:2009.10022V2 [Math.OA] 28 Sep 2020 Bet,Isfra Osbe Yfiiedmninloe.Teie T Idea C of the Class Large Ones
    FINITE-DIMENSIONALITY IN THE NON-COMMUTATIVE CHOQUET BOUNDARY: PEAKING PHENOMENA AND C∗-LIMINALITY RAPHAEL¨ CLOUATREˆ AND IAN THOMPSON Abstract. We explore the finite-dimensional part of the non-commutative Choquet boundary of an operator algebra. In other words, we seek finite- dimensional boundary representations. Such representations may fail to exist even when the underlying operator algebra is finite-dimensional. Nevertheless, we exhibit mechanisms that detect when a given finite-dimensional representa- tion lies in the Choquet boundary. Broadly speaking, our approach is topologi- cal and requires identifying isolated points in the spectrum of the C∗-envelope. This is accomplished by analyzing peaking representations and peaking pro- jections, both of which being non-commutative versions of the classical notion of a peak point for a function algebra. We also connect this question with the residual finite-dimensionality of the C∗-envelope and to a stronger property that we call C∗-liminality. Recent developments in matrix convexity allow us to identify a pivotal intermediate property, whereby every matrix state is locally finite-dimensional. 1. Introduction In unraveling the structure of C∗-algebras, a fruitful paradigm is to model these objects, insofar as possible, by finite-dimensional ones. The idea that finer struc- tural properties of a large class of C∗-algebras can be detected upon approximation by matrix algebras has become a major trend in the field. Nuclearity, a notion at the center of recent capstone results in the classification program for some simple C∗-algebras (see [30],[51] and references therein), is an important example of an “internal” finite-dimensional approximation property.
    [Show full text]