Hermitian crossed product Banach algebras

Abstract

We show that the Banach *-algebra ${\ell }^1(G,A,\alpha )$, arising from a C*-dynamical system $(A,G,\alpha )$, is an hermitian Banach algebra if the discrete group $G$ is finite or abelian (or more generally, a finite extension of a nilpotent group). As a corollary, we obtain that ${\ell }^1(\mathbb{Z},C(X),\alpha )$ is hermitian, for every topological dynamical system $\Sigma =(X,\sigma )$, where $\sigma :X\to X$ is a homeomorphism of a compact Hausdorff space $X$ and the action is ${\alpha }_n(f)=f\circ {\sigma }^{-n}$ with $n\in \mathbb{Z}$.