Approximate Cyclic Amenability of $T$-Lau Product of Banach Algebras

Abstract

Associated with two Banach algebras $\mathcal{A}$ and $\mathcal{B}$ and a norm decreasing homomorphism $T: \mathcal{B} \to \mathcal{A}$, there is a certain Banach algebra product $\mathcal{M}_T := \mathcal{A} \times_T \mathcal{B}$, which is a splitting extension of $\mathcal{B}$ by $\mathcal{A}$. In this paper, we investigate approximate cyclic amenability of $\mathcal{M}_T$ which has been introduced and studied by Esslamzadeh and Shojaee in [5]. In particular, apart from the characterization of all cyclic derivations on the Banach algebra $\mathcal{M}_T$, we improve the results of [1, 2] for cyclic amenability of $\mathcal{M}_T$. These results paves the way for obtaining new results for (approximate) cyclic amenability of the Banach algebra $\mathcal{A} \times \mathcal{B}$ equipped with the coordinatewise product algebra and $\ell^1$-norm.