Isomorphisms of $\ell^1$-crossed products of topological systems

Abstract

We give a complete description of the isometric isomorphisms between the $\ell^1$-crossed products of actions of discrete groups on nonempty compact Hausdorff spaces. In particular, two such crossed products are isometrically isomorphic as Banach algebras if and only if the actions are topologically isomorphic. It follows that $\ell^1(G\curvearrowright X)$ remembers not just $X$ but even the acting group and the action, and is not determined by the transformation groupoid $G\ltimes X$ alone. The proof recovers $C(X)$ from the hermitian elements and uses the geometry of $L^p$-spaces to identify the invertible isometries. We also obtain analogous results for measurable actions.