Weak amenability of ultrapowers of Banach algebras

Document Type : Original Article

Authors
VANIA KHODAKARAMI 1
ESMAEIL FEIZI 2
1 Department of Department of Mathematics, Bu-Ali Sina University, Hamedan, Iran
2 Department of Department of Mathematics, Bu-Ali Sina University, Hamedan, Iran.
10.22034/maco.5.1.11
Abstract
We define ultra-weak amenability of Banach algebras and study the localization of weak amenability property. We study when weak amenability of Banach algebras are stable under the ultrapower constructions. We extend some general results of weak amenability of Banach algebras to their ultrapowers. Specifically, a Banach algebra $A$ is considered ultra-weakly amenable if, for every ultrafilter $U$, the ultrapower $(A)_U$ exhibits weak amenability. We further investigate the relationship between ultra-weak amenability, (weak)amenability, and ultra-amenability. We provide an example of an ultra-weakly amenable Banach algebra that is not amenable and vice versa. Finally, we show that for many abelian locally compact groups $G$, ultra-weak amenability of $L^1(G)$ is equivalent to $G$ being finite.

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