Ultrapowers of Banach algebras and modules

TL;DR

The paper explores ultrapower methods for extending Banach algebra products, showing that Arens regularity allows recovery of the Arens product via ultrapowers, extending known results from C*-algebras.

Contribution

It demonstrates that ultrapowers can recover the Arens product in Arens regular Banach algebras, generalizing previous results beyond C*-algebras.

Findings

Ultrapowers can be used to recover the Arens product in Arens regular Banach algebras.

A Principle of Local Reflexivity for modules and algebras is developed.

The method extends known results from C*-algebras to a broader class.

Abstract

The Arens products are the standard way of extending the product from a Banach algebra $\mc A$ to its bidual $\mc A''$. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that if $\mc A$ is Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C$^*$-algebras. Our main tool is a Principle of Local Reflexivity result for modules and algebras.