Amenability of ultraproducts of Banach algebras

Details Amenability of ultraproducts of Banach algebras Amenability of ultraproducts of Banach algebras

2007-08-30

arxiv.org/abs/0708.4195v2

Proc. Edinb. Math. Soc. (2) 52 (2009), no. 2, 307--338.

Mathematics

Functional Analysis

Added appendix which contains an errata for Section 5

Scientific paper

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.

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