Reduced $C^{*}$-algebras of Product Systems -- an $E_0$-semigroup and a Groupoid perspective
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913962044424192 |
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| author | Hossain, Md Amir Sundar, S. |
| author_facet | Hossain, Md Amir Sundar, S. |
| contents | For Ore semigroups $P$ with an order unit, we prove that there is a bijection between $E_0$-semigroups over $P$ and product systems of $C^{*}$-correspondences over $P^{op}$. We exploit this bijection and show that the reduced $C^{*}$-algebra of a proper product system is Morita equivalent to the reduced crossed product of the associated semigroup dynamical system given by the corresponding $E_0$-semigroup. We appeal to the groupoid picture of the reduced crossed product of a semigroup dynamical system derived in [47] to prove that, under good conditions, the reduced $C^{*}$-algebra of a proper product system is nuclear/exact if and only if the coefficient algebra is nuclear/exact. We also discuss the invariance of $K$-theory under homotopy of product systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reduced $C^{*}$-algebras of Product Systems -- an $E_0$-semigroup and a Groupoid perspective Hossain, Md Amir Sundar, S. Operator Algebras For Ore semigroups $P$ with an order unit, we prove that there is a bijection between $E_0$-semigroups over $P$ and product systems of $C^{*}$-correspondences over $P^{op}$. We exploit this bijection and show that the reduced $C^{*}$-algebra of a proper product system is Morita equivalent to the reduced crossed product of the associated semigroup dynamical system given by the corresponding $E_0$-semigroup. We appeal to the groupoid picture of the reduced crossed product of a semigroup dynamical system derived in [47] to prove that, under good conditions, the reduced $C^{*}$-algebra of a proper product system is nuclear/exact if and only if the coefficient algebra is nuclear/exact. We also discuss the invariance of $K$-theory under homotopy of product systems. |
| title | Reduced $C^{*}$-algebras of Product Systems -- an $E_0$-semigroup and a Groupoid perspective |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2507.20319 |