Partial crossed product description of the 𝐶*-algebras associated with integral domains

Boava, Giuliano; Exel, Ruy

doi:10.1090/S0002-9939-2013-11724-7

Partial crossed product description of the $C^*$-algebras associated with integral domains
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by Giuliano Boava and Ruy Exel

Proc. Amer. Math. Soc. 141 (2013), 2439-2451

DOI: https://doi.org/10.1090/S0002-9939-2013-11724-7

Published electronically: April 3, 2013

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Abstract:

Recently, Cuntz and Li introduced the $C^*$-algebra $\mathfrak {A}[R]$ associated to an integral domain $R$ with finite quotients. In this paper, we show that $\mathfrak {A}[R]$ is a partial group algebra of the group $K\rtimes K^\times$ with suitable relations, where $K$ is the field of fractions of $R$. We identify the spectrum of these relations and we show that it is homeomorphic to the profinite completion of $R$. By using partial crossed product theory, we reconstruct some results proved by Cuntz and Li. Among them, we prove that $\mathfrak {A}[R]$ is simple by showing that the action is topologically free and minimal.

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