Strong pseudo-amenability of some Banach algebras

Abstract

In this paper we introduce a new notion of strong pseudo-amenability for Banach algebras. We study strong pseudo-amenability of some Matrix algebras. Using this tool, we characterize strong pseudo-amenability of ${\ell }^1(S)$, provided that $S$ is a uniformly locally finite semigroup. As an application we show that for a Brandt semigroup $S=M^0(G,I)$, ${\ell }^1(S)$ is strong pseudo-amenable if and only if $G$ is amenable and $I$ is finite. We give some examples to show the differences of strong pseudo-amenability and other classical notions of amenability.