Exactness of Cuntz-Pimsner C*-algebras

TL;DR

This paper establishes that the exactness of Cuntz-Pimsner C*-algebras is equivalent to the exactness of the underlying algebra A, and explores implications for free product constructions and entropy in specific cases.

Contribution

It proves the equivalence of exactness between Cuntz-Pimsner algebras and their base algebras, providing new proofs and entropy results for special cases.

Findings

Cuntz-Pimsner algebra is exact iff the base algebra is exact

Alternative proofs for exactness of reduced amalgamated free products

Zero topological entropy for Bogljubov automorphisms when A is finite-dimensional

Abstract

Let H be a full Hilbert bimodule over a C*-algebra A. We show that the Cuntz-Pimsner C*-algebra associated to H is exact if and only if A is exact. Using this result, we give alternative proofs for exactness of reduced amalgamated free products of exact C*-algebras. In the case that A is a finite dimensional C*-algebra, we also show that the Brown-Voiculescu topological entropy of Bogljubov automorphisms of the Cuntz-Pimsner algebra associated to an A,A Hilbert bimodule is zero.