Approximate amenability of tensor products of Banach algebras

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.

There was a gap in one argument of the previous version and that result has been removed. Several other minor corrections have been made. 14 pages