Weak Amenability of Banach Algebras on Locally Compact Groups

✍ Scribed by A.T.-M. Lau; R.J. Loy

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 439 KB
• Volume: 145
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.3002

✦ Synopsis

Recent work by various authors has considered the implications of Banach algebra amenability for various algebras defined over locally compact groups, one of the basic tools being the fact that a continuous homomorphic image of an amenable algebra is again amenable. In the present paper we look at the consequences of weak amenability. Here the homomorphism property fails in general, however it remains true for suitable direct summands. It is this technique that we make much use of here.

1997 Academic Press

0. Introduction

Let A be a Banach algebra, X a Banach A-bimodule. Then X* is a Banach A-bimodule under the actions

A derivation D: A Ä X is a (bounded) linear map such that

The cohomology space H 1 (A, X ) is the quotient of the space of derivations by the inner derivations, and in many situations triviality of this space is of considerable importance. In particular, A is contractible if, for every Banach A-bimodule X, H 1 (A, X )=[0], amenable if, for every Banach A-bimodule X, H 1 (A, X*)=[0], and weakly amenable if H 1 (A, A*)=[0]. In the case that A is commutative, then weak amenability is equivalent to every derivation into a commutative bimodule being zero, [2, Theorem 1.5].