Center of Banach algebra valued Beurling algebras

TL;DR

This paper characterizes the center of Banach algebra valued Beurling algebras for certain groups and weights, and explores conditions for properties like weak amenability and semisimplicity.

Contribution

It establishes an isomorphism for the centers of these algebras and discusses their algebraic properties under specific conditions.

Findings

Center of Banach algebra valued Beurling algebras is characterized.

Isomorphism between centers of tensor product and individual centers is proven.

Conditions for properties like weak amenability and semisimplicity are discussed.

Abstract

We prove that for a Banach algebra $A$ having a bounded $\mathcal{Z}(A)$-approximate identity and for every $\bf[IN]$ group $G$ with weight $w$ which is either constant on conjugacy classes or $w \geq 1$, $\mathcal{Z}\big(L^1_w(G) \otimes^\gamma A\big) \cong \mathcal{Z}(L^1_w(G)) \otimes^\gamma \mathcal{Z}(A)$. As an application, we discuss the conditions under which $\mathcal{Z}\big(L^1_w(G,A)\big)$ enjoys certain Banach algebraic properties, for example, weak amenability, semisimplicity etc.