Approximate amenability of Schatten classes, Lipschitz algebras and second duals of Fourier algebras

TL;DR

This paper investigates the approximate amenability of Schatten classes, Lipschitz algebras, and second duals of Fourier algebras, providing new criteria and comprehensive results that extend understanding of their algebraic properties.

Contribution

It introduces a new criterion for non-approximate amenability of Banach algebras without bounded approximate identities and applies it to several classes of algebras, including Fourier and Segal algebras.

Findings

Complete solution for approximate amenability of Schatten classes and Lipschitz algebras.

Bounded approximate amenability of second duals of Fourier algebras implies finite-dimensionality.

Unified approach to existing non-approximate amenability results.

Abstract

Amenability of any of the algebras described in the title is known to force them to be finite-dimensional. The analogous problems for \emph{approximate} amenability have been open for some years now. In this article we give a complete solution for the first two classes, using a new criterion for showing that certain Banach algebras without bounded approximate identities cannot be approximately amenable. The method also provides a unified approach to existing non-approximate amenability results, and is applied to the study of certain commutative Segal algebras. Using different techniques, we prove that \emph{bounded} approximate amenability of the second dual of a Fourier algebra implies that it is finite-dimensional. Some other results for related algebras are obtained.