Ultrapowers of Banach algebras and modules

Details Ultrapowers of Banach algebras and modules Ultrapowers of Banach algebras and modules

2007-08-29

arxiv.org/abs/0708.4029v1

Glasg. Math. J. 50 (2008), no. 3, 539--555.

Mathematics

Functional Analysis

17 pages

Scientific paper

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.

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