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

Md Amir Hossain; S. Sundar

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arXiv · 2604.03668

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

Md Amir Hossain·S. Sundar

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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.

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Md Amir Hossain, S. Sundar. 2026-04-04. Some remarks on Reduced $C^*$-algebras of semigroup dynamical systems and product systems. https://arxiv.org/abs/2604.03668

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