Strong pseudo-amenability of some Banach algebras

TL;DR

This paper introduces the concept of strong pseudo-amenability for Banach algebras, characterizes it for certain matrix and semigroup algebras, and explores its distinctions from classical amenability.

Contribution

It defines strong pseudo-amenability, characterizes it for $ ext{ell}^1(S)$ with specific semigroup conditions, and compares it to existing notions of amenability.

Findings

Strong pseudo-amenability is characterized for $ ext{ell}^1(S)$ when $S$ is uniformly locally finite.

For Brandt semigroups, $ ext{ell}^1(S)$ is strong pseudo-amenable iff $G$ is amenable and $I$ is finite.

Examples demonstrate differences between strong pseudo-amenability and classical amenability.

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.