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

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.

Categories: math.FA math.OA