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

## TL;DR

This paper proves that the crossed product Banach algebra associated with a discrete amenable group and a strongly amenable C*-algebra is itself amenable, extending understanding of algebraic properties in dynamical systems.

## Contribution

It establishes the amenability of crossed product Banach algebras for a broad class of C*-dynamical systems, combining general results with Paterson's characterization.

## Key findings

- Crossed product Banach algebra is amenable under specified conditions
- Connection between amenability of algebra and group/algebra properties
- Extension of known results to a wider class of dynamical systems

## Abstract

We prove that the crossed product Banach algebra $\ell^1(G,A;\alpha)$ 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 strongly amenable ${\mathrm C}^\ast$-algebra. This is a consequence of the combination of a more general result with Paterson's characterisation of strongly amenable unital $\mathrm{C}^\ast$-algebras in terms of invariant means for their unitary groups.