On the involutive Banach algebra associated to topologically free dynamical systems

Given an action $G\curvearrowright X$ of a discrete and countable infinite group $G$ on a compact and Hausdorff space $X$, we regard ${\mathrm{\ell }}^1(G\curvearrowright X)$ as the Banach *-algebra crossed product associated to the action. We characterize topological freeness of the action by showing that it is equivalent to every nontrivial closed ideal of ${\mathrm{\ell }}^1(G\curvearrowright X)$ intersecting $C(X)$ nontrivially. Most surprisingly, we show that when $G$ is torsion-free and abelian, ${\mathrm{\ell }}^1(G\curvearrowright X)$ can detect freeness of $G\curvearrowright X$: indeed, we show that $G\curvearrowright X$ is free if and only if every closed ideal of ${\mathrm{\ell }}^1(G\curvearrowright X)$ is self-adjoint, a property that is automatic in $C^{\ast }$-algebras. We also show with an example that this result does not hold beyond the torsion-free abelian case.