Characterizations of amenable Banach algebras

Abstract:

In this paper we show that a Banach algebra A is amenable if and only if A has any one of the following properties: (i) whenever X is a Banach A-bimodule and Y is an A-submodule of X, then for each $f \in {Y^ \ast }$ such that $a \cdot f = f \cdot a,a \in A$, there exists $\tilde f \in {X^ \ast }$ which extends f and $a \cdot \tilde f = \tilde f \cdot a$ for all $a \in A$; (ii) whenever X is a Banach A-bimodule, there exists a bounded projection P from ${X^ \ast }$ onto {$f \in {X^ \ast };a \cdot f = f \cdot a$ for all $a \in A$} such that $T \cdot P = P \cdot T$ for any $\mathrm {weak}^*$ continuous bounded linear operator T from ${X^ \ast }$ into ${X^\ast }$ commuting with the action of A on ${X^\ast }$. The class of ultraweakly amenable von Neumann algebras with separable predual can be similarly characterized.