Fereidoun Ghahramani

(Department of Mathematics, University of Manitoba)

Approximate identities in approximately amenable Banach algebras

Date Friday, January 18, 2008

The notion of approximate amenability was introduced in: [F.Ghahramani, and R.J. Loy, Generalized notions of amenability, J. Functional Analysis 208(2004) 229-260]. It was shown there that an approximately amenable Banach algebra has a right and a left approximate identity, but the existence of a two sided approximate identity remained open. Further sufficient results were obtained among the results of the paper [F. Ghahramani, R.J. Loy and Y.Zhang, Generalized notions of amenability, II, J. Functional Analysis, to appear.] In this talk I will show that every boundedly approximately amenable Banach algebra possessing a multiplier bounded left approximate identity and a multiplier bounded right approximate identity has a bounded approximate identity. In particular, every boundedly approximately contractible Banach algebra has a bounded approximate identity. This rules out the possibility of a Banach algebra in any of the following classes to be boundedly approximtely amenable: Fourier algebras of weakly amenable non- amenable groups, proper Segal algebras on locally compact groups,and certain operator algebras. This is joint work with Yemon Choi and Yong Zhang.