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Transactions of the American Mathematical Society

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C*-algebras from actions of congruence monoids on rings of algebraic integers


Author: Chris Bruce
Journal: Trans. Amer. Math. Soc. 373 (2020), 699-726
MSC (2010): Primary 46L05; Secondary 11R04
DOI: https://doi.org/10.1090/tran/7966
Published electronically: October 1, 2019
MathSciNet review: 4042889
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Abstract: Let $ K$ be a number field with ring of integers $ R$. Given a modulus $ \mathfrak{m}$ for $ K$ and a group $ \Gamma $ of residues modulo $ \mathfrak{m}$, we consider the semidirect product $ R\rtimes R_{\mathfrak{m},\Gamma }$ obtained by restricting the multiplicative part of the full $ ax+b$-semigroup over $ R$ to those algebraic integers whose residue modulo $ \mathfrak{m}$ lies in $ \Gamma $, and we study the left regular C*-algebra of this semigroup. We give two presentations of this C*-algebra and realize it as a full corner in a crossed product C*-algebra. We also establish a faithfulness criterion for representations in terms of projections associated with ideal classes in a quotient of the ray class group modulo $ \mathfrak{m}$, and we explicitly describe the primitive ideals using relations only involving the range projections of the generating isometries; this leads to an explicit description of the boundary quotient. Our results generalize and strengthen those of Cuntz, Deninger, and Laca and of Echterhoff and Laca for the C*-algebra of the full $ ax+b$-semigroup. We conclude by showing that our construction is functorial in the appropriate sense; in particular, we prove that the left regular C*-algebra of $ R\rtimes R_{\mathfrak{m},\Gamma }$ embeds canonically in the left regular C*-algebra of the full $ ax+b$-semigroup. Our methods rely heavily on Li's theory of semigroup C*-algebras.


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Additional Information

Chris Bruce
Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
Email: cmbruce@uvic.ca

DOI: https://doi.org/10.1090/tran/7966
Received by editor(s): February 1, 2019
Received by editor(s) in revised form: August 7, 2019
Published electronically: October 1, 2019
Additional Notes: The author’s research was supported by the Natural Sciences and Engineering Research Council of Canada through an Alexander Graham Bell CGS-D award. This work was done as part of the author’s Ph.D. project at the University of Victoria.
Article copyright: © Copyright 2019 American Mathematical Society