Amenable crossed product Banach algebras associated with a class of $\mathrm{C}^\ast$-dynamical systems

TL;DR

This paper proves that certain crossed product Banach algebras are amenable when associated with discrete amenable groups and specific types of C*-algebras, advancing understanding of their algebraic properties.

Contribution

It establishes conditions under which crossed product Banach algebras are amenable, specifically for discrete amenable groups and commutative or finite-dimensional C*-algebras.

Findings

Crossed product Banach algebra $ ext{ell}^1(A,G, ext{alpha})$ is amenable under specified conditions.

Amenability holds when $G$ is discrete and amenable, and $A$ is commutative or finite-dimensional.

Provides groundwork for further exploration of algebraic properties of these structures.

Abstract

We prove that the crossed product Banach algebra $\ell^1(A,G,\alpha)$ that is associated with a $\mathrm{C}^\ast$-dynamical system $(A,G,\alpha)$ is amenable if $G$ is a discrete amenable group and $A$ is a commutative or finite dimensional $\mathrm{C}^\ast$-algebra. Perspectives for further developments are indicated.