Approximate and pseudo-amenability of various classes of Banach algebras

TL;DR

This paper explores various notions of approximate amenability in Banach algebras, establishing new results about their properties, examples, and distinctions, including cases where approximate amenability does not imply amenability.

Contribution

It introduces new examples and results on approximate amenability, showing that bounded approximate amenability does not necessarily imply sequential approximate amenability, and analyzes specific classes of Banach algebras.

Findings

Boundedly approximately contractible Banach algebras have a bounded approximate identity.

The Fourier algebra of the free group on two generators is not approximately amenable.

Certain ${ m ext{ extlangle}} ext{ell}^1$-semigroup algebras are approximately amenable but not amenable.

Abstract

We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity. Among our other results, it is shown that the Fourier algebra of the free group on two generators is not approximately amenable. Further examples are obtained of ${\ell}^1$-semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate amenability need not imply sequential approximate amenability. Results are also given for Segal subalgebras of $L^1(G)$, where $G$ is a locally compact group, and the algebras $PF_p(\Gamma)$ of $p$-pseudofunctions on a discrete group $\Gamma$ (of which the reduced $C^*$-algebra is a special case).