Approximate amenability of tensor products of Banach algebras

TL;DR

This paper explores the approximate amenability of tensor products of Banach algebras, showing that such properties do not always transfer and establishing conditions linking the amenability of individual algebras to their tensor products.

Contribution

It demonstrates that the tensor product of approximately amenable algebras may not be approximately amenable and identifies conditions under which approximate amenability is preserved or implied.

Findings

Tensor product of approximately amenable algebras need not be approximately amenable.

If the tensor product is amenable, then the individual algebras are also amenable.

The paper generalizes a 1996 result by Barry Johnson without additional assumptions.

Abstract

We show that the tensor product of approximately amenable algebras need not be approximately amenable, and investigate conditions under which $A$ and $B$ being approximately amenable implies, or is implied by, $A\hat{\otimes}B$ or $A^{\#}\hat{\otimes} B^{\#}$ being approximately amenable. Our methods also enable us to prove that if $A\hat{\otimes}B$ is amenable, then so are $A$ and $B$, a result proved by Barry Johnson in 1996 under an additional assumption.