Module approximate amenability of Banach algebras

Document Type: Research Paper

Authors

1 Tabriz University, Tabriz, Iran

2 Islamic Azad University of Garmsar, Garmsar, Iran

Abstract

In the present paper, the concepts of module (uniform) approximate amenability and contractibility of Banach algebras that are modules over another Banach algebra, are introduced. The general theory is developed and some hereditary properties are given. In analogy with the Banach algebraic approximate amenability, it is shown that module approximate amenability and contractibility are the same properties. It is also shown that module uniform approximate (contractibility) amenability and module (contractibility, respectively) amenability for commutative Banach modules are equivalent. Applying these results to l^1 (S) as an l^1 (E)-module, for an inverse semigroup S with the set of
idempotents E, it is shown that l^1(S) is module approximately amenable (contractible) if and only if it is module uniformly approximately amenable if and only if S is amenable.
Moreover, l^1(S)^{**} is module (uniformly) approximately amenable if and only if an appropriate group homomorphic image of S is finite.

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Volume 39, Issue 6
November and December 2013
Pages 1137-1158
  • Receive Date: 29 October 2011
  • Revise Date: 29 October 2012
  • Accept Date: 07 November 2012