Johnson pseudo-contractibility and pseudo-amenability of $ \theta $-Lau product of Banach algebras

TL;DR

This paper investigates the properties of Johnson pseudo-contractibility and pseudo-amenability in the context of the $\theta$-Lau product of Banach algebras, providing conditions and characterizations for these properties.

Contribution

It offers new characterizations and conditions for Johnson pseudo-contractibility and pseudo-amenability of the $\theta$-Lau product of Banach algebras, extending existing theory.

Findings

Johnson pseudo-contractibility of $A\times_{\theta} B$ implies $A$ and $B$ are Johnson pseudo-contractible.

Pseudo-amenability of $A\times_{\theta} B$ implies approximate amenability of $A$ and pseudo-amenability of $B$.

Complete characterizations of Johnson pseudo-contractibility in specific cases are provided.

Abstract

Given Banach algebras $ A $ and $ B $ with $ \theta\in\Delta(B) $. We shall study the Johnson pseudo-contractibility and pseudo-amenability of $ \theta $-Lau product $ A\times_{\theta} B $. We show that if $ A\times_{\theta} B $ is Johnson pseudo-contractible, then $ A $ is Johnson pseudo-contractible and has a bounded approximate identity and $ B $ is Johnson pseudo-contractible. In some particular cases complete characterization of Johnson pseudo-contractibility of $ A\times_{\theta} B $ are given. Also, we show that pseudo-amenability of $ A\times_{\theta} B $ implies approximate amenability of $ A $ and pseudo-amenability of $ B $.