Notes on amenability

TL;DR

This paper investigates the relationships between amenability properties of Banach algebras and their tensor products, establishing equivalences and implications under certain conditions.

Contribution

It proves the equivalence of amenability of a Banach algebra and its tensor products with itself and its opposite algebra, and links weak amenability of ideals to that of larger algebras.

Findings

Amenability of A, A⊗A, and A⊗A^{op} are equivalent for Banach algebra A with bounded approximate identity.

Weak amenability of A in a larger algebra B implies weak amenability of A if A is a closed ideal in B.

Establishes conditions under which amenability properties are preserved or equivalent in tensor product constructions.

Abstract

We show that for a Banach algebra $A$ with a bounded approximate identity, the amenability of the projective tensor product of A with A, the amenability of the projective tensor product of A with A^{op}and the amenability of A are equivalent. Also if A is a closed ideal in a commutative Banach algebra B, then the weak amenability of the projective tensor product of A and B implies the weak amenability of A.