Some remarks on Reduced $C^*$-algebras of semigroup dynamical systems and product systems

TL;DR

This paper investigates the conditions under which the reduced crossed product of a semigroup dynamical system is exact, establishing equivalence with the exactness of the underlying algebra under certain hypotheses.

Contribution

It extends previous results by removing the injectivity assumption on the action and compares different constructions of Fell bundles for product systems.

Findings

Reduced crossed product $A \rtimes_{red} P$ is exact iff $A$ is exact under certain conditions.

The groupoid crossed product and Fell bundle constructions are shown to be equivalent.

The result generalizes earlier work by relaxing the injectivity condition.

Abstract

We study the exactness of the reduced crossed product of a semigroup dynamical system and the reduced $C^{*}$-algebra of a product system. We show that for a semigroup dynamical system $(A, P,\alpha)$, under reasonable hypotheses (e.g., $P$ is abelian and finitely generated), the reduced crossed product $A \rtimes_{red} P$ is exact if and only if $A$ is exact. This strengthens our earlier result (\cite{Amir_Sundar-product-system}), where it was assumed that the action of $P$ on $A$ is by injective endomorphisms. We also compare the groupoid crossed product described in \cite{Amir_Sundar-product-system} and the Fell bundle constructed in \cite{Rennie_Sims} for a product system, and show that they are equivalent as Fell bundles.