## Toeplitz algebras of semigroups

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- by Marcelo Laca and Camila F. Sehnem PDF
- Trans. Amer. Math. Soc.
**375**(2022), 7443-7507 Request permission

## Abstract:

To each submonoid $P$ of a group we associate a universal Toeplitz $\mathrm {C}^*$-algebra $\mathcal {T}_u(P)$ defined via generators and relations; $\mathcal {T}_u(P)$ is a quotient of Li’s semigroup $\mathrm {C}^*$-algebra $\mathrm {C}^*_s(P)$ and they are isomorphic iff $P$ satisfies independence. We give a partial crossed product realization of $\mathcal {T}_u(P)$ and show that several results known for $\mathrm {C}^*_s(P)$ when $P$ satisfies independence are also valid for $\mathcal {T}_u(P)$ when independence fails. At the level of the reduced semigroup $\mathrm {C}^*$-algebra $\mathcal {T}_\lambda (P)$, we show that nontrivial ideals have nontrivial intersection with the reduced crossed product of the diagonal subalgebra by the action of the group of units of $P$, generalizing a result of Li for monoids with trivial unit group. We characterize when the action of the group of units is topologically free, in which case a representation of $\mathcal {T}_\lambda (P)$ is faithful iff it is jointly proper. This yields a uniqueness theorem that generalizes and unifies several classical results. We provide a concrete presentation for the covariance algebra of the product system over $P$ with one-dimensional fibers in terms of a new notion of foundation sets of constructible ideals. We show that the covariance algebra is a universal analogue of the boundary quotient and give conditions on $P$ for the boundary quotient to be purely infinite simple. We discuss applications to a numerical semigroup and to the $ax+b$-monoid of an integral domain. This is particularly interesting in the case of nonmaximal orders in number fields, for which we show independence always fails.## References

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## Additional Information

**Marcelo Laca**- Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 2Y2, Canada
- MR Author ID: 335785
- ORCID: 0000-0002-0901-8165
- Email: laca@uvic.ca
**Camila F. Sehnem**- Affiliation: School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand
- ORCID: 0000-0002-5007-4230
- Email: camila.sehnem@vuw.ac.nz
- Received by editor(s): July 19, 2021
- Received by editor(s) in revised form: May 10, 2022, and May 12, 2022
- Published electronically: July 29, 2022
- Additional Notes: This research was partially supported by the Natural Sciences and Engineering Research Council of Canada, Discovery Grant RGPIN-2017-04052

The second author was supported by the Marsden Fund of the Royal Society of New Zealand, grant No. 18-VUW-056 - © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 7443-7507 - MSC (2020): Primary 46L55, 46L05; Secondary 20M30, 11R04, 47B35
- DOI: https://doi.org/10.1090/tran/8743