On the Banach \(^{\ast}\)-algebra crossed product associated with a topological dynamical system (Q425748)

The authors study the structure of the convolution Banach \({\ast}\)-algebra \(\ell^1(X,\sigma)\) associated to a dynamical system \((X,\sigma)\) consisting of a compact space \(X\) and a homeomorphism \(\sigma: X\to X\). They concentrate on the ideal structure of \(\ell^1(X,\sigma)\) and give in particular criteria for simplicity and primeness of these algebras. They also analyse the finite-orbit case and the case in which \(X\) is finite and prove that \(\ell^1(X,\sigma)\) is Hermitian in this case.NEWLINENEWLINEThe results of the paper show that \(\ell^1(X,\sigma)\) has a richer structure than its companion, the \(C^{\ast}\)-algebra crossed product \(C(X)\rtimes_\sigma \mathbb{Z}\) associated to \((X,\sigma)\), which is the enveloping \(C^{\ast}\)-algebra of \(\ell^1(X,\sigma)\). One indication for this is the fact that \(\ell^1(X,\sigma)\) has non-self-adjoint closed ideals (which happens if the dynamical system is not free), a feature that cannot happen for \(C^{\ast}\)-algebras.