Amenability of ultraproducts of Banach algebras

TL;DR

This paper investigates the stability of properties like Arens regularity and amenability in ultrapowers of Banach algebras, linking these to structural features and tensor product behaviors, with implications for specific algebra classes.

Contribution

It provides new characterizations of when ultrapowers of Banach algebras retain properties like amenability and Arens regularity, especially relating to operator subalgebras and dual structures.

Findings

Ultrapowers of certain Banach algebras are Arens regular iff they embed into operators on super-reflexive spaces.

Conditions are identified under which ultrapowers of dual Banach algebras are well-defined.

Criteria for when ultrapowers of C*-algebras and group L^1 algebras are amenable are established.

Abstract

We study when certain properties of Banach algebras are stable under ultrapower constructions. In particular, we consider when every ultrapower of $\mc A$ is Arens regular, and give some evidence that this is if and only if $\mc A$ is isomorphic to a closed subalgebra of operators on a super-reflexive Banach space. We show that such ideas are closely related to whether one can sensibly define an ultrapower of a dual Banach algebra. We study how tensor products of ultrapowers behave, and apply this to study the question of when every ultrapower of $\mc A$ is amenable. We provide an abstract characterisation in terms of something like an approximate diagonal, and consider when every ultrapower of a C$^*$-algebra, or a group $L^1$-convolution algebra, is amenable.