Amenability, weak amenability and approximate amenability of L1(S)
Filofteia. Gheorghe · Mspace (University of Manitoba) · 2008
We are doing a suvey on amenability, weak amenability and generalized notions of amenability of semigroup algebras ¿16).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 introd.uced,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 ,9 with a generating set E, we also give necessarY and sufficient conditions so that /1(S) is amenable, weakl¡r amenable or bound.-edly approximately amenable.It is known that if the semigroup algebra Z1(S) is approximately amenable then ,9 must be a regulal amenable semigroup.!tr'e prove that the converse is not true by examining the bicyclical semigroup ,Sr which is an important semigroup and has been studied by many reseilchers from various aspects.Precisely, we show that, although ^91 is a regular amenable semigroup, tt(Sr) is not approximately amenable.In the appendix we also give a direct proof to the fact that lt(Sz) is not approximately amenable, where ^92 is the partially bicyclic semigroup defined by ,S2 :q L, a,b, c I ab : o,c : 7 ).His approachability and feedback to my thesis helped to make my graduate experience a positive one.I w.ould also llke fs fhanÞ Tlr T.in,,-\Ã/onr for ta.king timC to Sci,v,e as m¡.external advisor and Dr. Y.Choi whose help during the final stages of completion was most appreciated.