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Canada Archives Canada Published Heritage Direction Du Branch Patrimoine De I'edition Amenability, Weak Amenability and Approximate Amenability of ^(S) by Filofteia Gheorghe A thesis submitted to the Faculty of Graduate Studies in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mathematics University of Manitoba Winnipeg, Manitoba Copyright © 2008 by Filofteia Gheorghe Library and Bibliotheque et 1*1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-48968-0 Our file Notre reference ISBN: 978-0-494-48968-0 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par Plntemet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non­ sur support microforme, papier, electronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in et des droits moraux qui protege cette these. this thesis. Neither the thesis Ni la these ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent etre imprimes ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission. In compliance with the Canadian Conformement a la loi canadienne Privacy Act some supporting sur la protection de la vie privee, forms may have been removed quelques formulaires secondaires from this thesis. ont ete enleves de cette these. While these forms may be included Bien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. Canada THE UNIVERSITY OF MANITOBA FACULTY OF GRADUATE STUDIES COPYRIGHT PERMISSION Amenability, Weak Amenability and Approximate Amenability of £*(S) BY Filofteia Gheorghe A Thesis/Practicum submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfillment of the requirement of the degree Of MASTER OF SCIENCE Filofteia Gheorghe © 2008 Permission has been granted to the University of Manitoba Libraries to lend a copy of this thesis/practicum, to Library and Archives Canada (LAC) to lend a copy of this thesis/practicum, and to LAC's agent (UMI/ProQuest) to microfilm, sell copies and to publish an abstract of this thesis/practicum. This reproduction or copy of this thesis has been made available by authority of the copyright owner solely for the purpose of private study and research, and may only be reproduced and copied as permitted by copyright laws or with express written authorization from the copyright owner. Abstract We are doing a survey on amenability, weak amenability and generalized notions of amenability of semigroup algebras ^(5). Based on the characterizations of amenability of a Banach algebra, F. Ghahramani, R.J. Loy and Y. Zhang have introduced approximate amenability and pseudo-amenability for Banach algebras. Since they have been introduced, many results concerning them have been obtained by many researchers. We focus on this topic regarding semigroup algebras. We give some new characterizations for a Banach alge­ bra to be approximately amenable. For a semigroup 5 with a generating set E, we also give necessary and sufficient conditions so that ^(5) is amenable, weakly amenable or bound- edly approximately amenable. It is known that if the semigroup algebra ^:(5) is approximately amenable then 5 must be a regular amenable semigroup. We prove that the converse is not true by examining the bicyclical semigroup Si which is an important semigroup and has been studied by many researchers from various aspects. Precisely, we show that, although Si is a regular amenable semigroup, I1 (Si) is not approximately amenable. In the appendix we also give a direct proof to the fact that l1^) is not approximately amenable, where 52 is the partially bicyclic semigroup denned by 52 =< 1, a, b, c \ ab — ac = 1 >. l Acknowledgements I would like to express my gratitude to my advisor, Dr. Yong Zhang whose expertise in the field, constructive comments and high expectations motivated me to achieve my goals. Because he is a hard worker and actively involved in research, he was an excellent role model for me. I also wish to thank Dr. F.Ghahramani for his clear explanations during my course work. His approachability and feedback to my thesis helped to make my graduate experience a positive one. I would also like to thank Dr. Liqun Wang for taking time to serve as my external advisor and Dr. Y.Choi whose help during the final stages of completion was most appreciated. I would like to acknowledge the Departament of Mathematics at the University of Manitoba for their generous financial support. I'm grateful to my friends in Winnipeg and family without whose help and encouragement this thesis would not have been possible. ii Contents Abstract i Acknowledgements ii 1 Preliminaries 1 1.1 i1- semigroup algebra 1 1.2 More about Banach algebras 5 2 Amenability of ll{S) 10 2.1 Amenability of semigroup algebras 10 2.2 Amenability of f(S,uj) 15 2.3 Amenability of ll{QS) 19 3 Weak amenability of I1 (S) 21 3.1 Weak amenability of discrete semigroup algebras 21 3.2 Weak amenability of ^(S, ui) where S is commutative 22 3.3 H1 (l1 (S), e°°(S)) and H1 (tx (5), l1 (S)) for some classes of semigroups . 25 4 Approximate amenability of I1 (S) 28 iii 1 Approximate amenability of a Banach algebra 28 2 Amenability of bicyclic and partially bicyclic semigroups 34 3 A characterization of approximate amenability of a Banach algebra 35 4 Approximate amenability of i1 (Si), where S\ is the bicyclic semigroup . 40 Chapter 1 Preliminaries 1.1 i1- semigroup algebra Terms and concepts of. basic real and functional analysis which we have not defined or discussed can be found in [8] and [49]. In this section we establish some notations and definitions. Definition 1.1.1. A complex algebra is a vector space A over the complex field C in which a multiplication is defined, called an algebra product, A x A -+ A; (a,b) -+ ab, that satisfies: a(bc) = (ab)c, (a + b)c = ac + be, a(b + c) = ab + ac (a, 6, c G A) (aa)b — a(ab) = a(ab) (a € C, a, b e A). We say that A is commutative if ab = ba for all a, b £ A. If in addition, A is a Banach space with respect to a norm that satisfies the submultiplicative inequality ||a6||<||o||||6|| (a,beA) (1.1) 1 2 then A is called a Banach algebra. If A contains a unit element e such that ae = ea'— a (a € A) then A is called a Banach algebra with identity. If || e ||= 1 then A is a unital Banach algebra. Suppose that the algebra A does not have a unit. Then we define the unitization of A to be A* = C 0 A. A* is a unital algebra, with unit (1,0), for the product (a, a)(j3, b) = (a/3, ab + /3a + ab) (a, (3 € C, a, b 6 A). If A is a Banach algebra, then so is A* for the norm ||(a,a)|| = \a\ + \\a\\. A subalgebra of an algebra A is a linear subspace B of A such that ab € 5 for all a, 6 € 5. A left ideal in an algebra A is a subalgebra I QA such that, if a £ A and 6 € /, then ab € /. Similarly, we can define a right ideal and a two-sided ideal for A. The radical of an algebra A is defined to be the intersection of the maximal left ideals of A#; it is denoted by rad A. The algebra A is semisimple if rad{A} = {0} (see [9]). We recall some definitions. For further details see [38]. Definition 1.1.2. A groupoid (S,n) is defined as a non-empty set S on which a binary operation /i: S x S —> S is defined. We say that (5, /x) is a semigroup if the operation -/x is associative, that is to say, if, for all x,y and z in 5, fi(n(x,y),z)) = n(x,n(y,z)). (1.2) We shall follow the usual algebraic practice of writing the binary operation as multiplication. Thus n{x,y) becomes xy, and formula (1.2) takes the simple form (xy)z = x(yz), the familiar associative law of elementary algebra. A non-empty subset T of S is a subsemi- group if T is a semigroup for the product in S. For s 6 S, we set (s) = {sn :n€N}, the semigroup generated by s; the subsemigroup of S generated by a subset T is denoted by{T). If the semigroup 5 has the property that, for all x, y in S, xy = yx, we shall say that S is a commutative semigroup. (The term abelian is also used, by analogy with the group theoretic term). If a semigroup S contains an element 1 with the property that, for all x in S, xl — lx, we say that 1 is an identity element (or just an identity) of S, and that 5 is a semigroup with identity or (more usually) a monoid.
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