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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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


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.
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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:
  • 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:
  • 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: