On the Banach $*$-algebra crossed product associated with a topological dynamical system

Abstract

Given a topological dynamical system $\Sigma = (X, \sigma)$, where $X$ is a compact Hausdorff space and $\sigma$ a homeomorphism of $X$, we introduce the associated Banach $^*$-algebra crossed product $\ell^1 (\Sigma)$ and analyse its ideal structure. This algebra is the Banach algebra most naturally associated with the dynamical system, and it has a richer structure than its well studied $C^*$-envelope, as becomes evident from the possible existence of non-self-adjoint closed ideals. This paper initiates the study of these algebras and links their ideal structure to the topological dynamics. It is determined when exactly the algebra is simple, or prime, and when there exists a non-self-adjoint closed ideal. In addition, a structure theorem is obtained for the case when $X$ consists of one finite orbit, and the algebra is shown to be Hermitian if $X$ is finite. The key to these results lies in analysing the commutant of $C(X)$ in the algebra, which can be shown to be a maximal abelian subalgebra with non-zero intersection with each non-zero closed ideal.