C*-algebras from actions of congruence monoids on rings of algebraic integers
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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.References
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Additional Information
- Chris Bruce
- Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
- MR Author ID: 1322311
- Email: cmbruce@uvic.ca
- 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.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 699-726
- MSC (2010): Primary 46L05; Secondary 11R04
- DOI: https://doi.org/10.1090/tran/7966
- MathSciNet review: 4042889