Some remarks on Reduced $C^*$-algebras of semigroup dynamical systems and product systems
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arXiv
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| Format: | Preprint |
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2026
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| author | Hossain, Md Amir Sundar, S. |
| author_facet | Hossain, Md Amir Sundar, S. |
| contents | 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 $(A, P,α)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03668 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Some remarks on Reduced $C^*$-algebras of semigroup dynamical systems and product systems Hossain, Md Amir Sundar, S. Operator Algebras 46L55, 46L05 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 $(A, P,α)$, 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. |
| title | Some remarks on Reduced $C^*$-algebras of semigroup dynamical systems and product systems |
| topic | Operator Algebras 46L55, 46L05 |
| url | https://arxiv.org/abs/2604.03668 |