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

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.