Hermitian crossed product Banach algebras

TL;DR

This paper demonstrates that certain Banach *-algebras derived from dynamical systems are hermitian when the underlying group has specific properties, extending the class of known hermitian Banach algebras.

Contribution

It establishes conditions under which the Banach *-algebra $ ext{l}^1(G,A, ext{α})$ is hermitian, including for finite, abelian, or certain extended nilpotent groups, and applies this to dynamical systems.

Findings

$ ext{l}^1(G,A, ext{α})$ is hermitian for finite or abelian groups

$ ext{l}^1( ext{Z},C(X), ext{α})$ is hermitian for any topological dynamical system

Extension of hermitian property to broader classes of groups and dynamical systems

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 $n\in\mathbb{Z}$.