Brown–Voiculescu entropy revisited

Abstract

Aided by the tools and outlook provided by modern classification theory, we take a new look at the Brown–Voiculescu entropy of endomorphisms of nuclear ${\mathrm{C}}^{\ast }$-algebras. In particular, we introduce ‘coloured’ versions of noncommutative topological entropy suitable for ${\mathrm{C}}^{\ast }$-algebras $A$ of finite nuclear dimension or finite decomposition rank. In the latter case, assuming further that $A$ is simple, separable, unital, satisfies the UCT, and has finitely many extremal traces, we prove a variational-type principle in terms of quasidiagonal approximations relative to this finite set of traces. Building on work of Kerr, we also show that infinite entropy occurs generically among endomorphisms and automorphisms of certain classifiable ${\mathrm{C}}^{\ast }$-algebras that function as noncommutative spaces of observables of topological manifolds.