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

Given a topological dynamical system Σ=(X,σ), where X is a compact Hausdorff space and σ a homeomorphism of X, we introduce the Banach ⁎-algebra crossed product ℓ1(Σ) most naturally associated with Σ and initiate its study. It has a richer structure than its well investigated C⁎-envelope, as becomes evident from the possible existence of non-self-adjoint closed ideals. We link its ideal structure to the dynamics, determining when the algebra is simple, or prime, and when there exists a non-self-adjoint closed ideal. A structure theorem is obtained when X consists of one finite orbit, and the algebra is shown to be Hermitian if X is finite. The key lies in analysing the commutant of C(X) in the algebra, which is shown to be a maximal abelian subalgebra with non-zero intersection with each non-zero closed ideal.