# More on cyclic amenability of the Lau product of Banach algebras defined by a Banach algebra morphism

## TL;DR

This paper investigates the cyclic amenability of the $T$-Lau product of Banach algebras, providing necessary and sufficient conditions, extending previous results, and addressing open questions in the field.

## Contribution

It establishes comprehensive criteria for cyclic amenability of $T$-Lau products, extending known results and answering open questions about their properties.

## Key findings

- Derived necessary and sufficient conditions for cyclic amenability.
- Extended results on cyclic amenability of direct and $T$-Lau products.
- Resolved an open question on cyclic amenability of $A\times_T B$.

## Abstract

For two Banach algebras $A$ and $B$, the $T$-Lau product $A\times_T B$, was recently introduced and studied for some bounded homomorphism $T:B\to A$ with $\|T\|\leq 1$. Here, we give general nessesary and sufficent conditions for $A\times_T B$ to be (approximately) cyclic amenable. In particular, we extend some recent results on (approximate) cyclic amenability of direct product $A\oplus B$ and $T$-Lau product $A\times_T B$ and answer a question on cyclic amenability of $A\times_T B$.