Brown-Voiculescu entropy revisited

TL;DR

This paper revisits Brown-Voiculescu entropy for nuclear C*-algebras, introducing coloured entropy variants and establishing a variational principle under specific conditions, with results on generic infinite entropy.

Contribution

It introduces coloured noncommutative topological entropy for certain C*-algebras and proves a variational principle relating entropy to quasidiagonal approximations.

Findings

Infinite entropy occurs generically among certain classifiable C*-algebras.

Coloured entropy variants are suitable for C*-algebras with finite nuclear dimension.

A variational principle links entropy to quasidiagonal approximations in specific cases.

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 C*-algebras. In particular, we introduce `coloured' versions of noncommutative topological entropy suitable for C*-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 C*-algebras that function as noncommutative spaces of observables of topological manifolds.