𝐶*-algebras associated with real multiplication

Nawata, Norio

doi:10.1090/S0002-9939-2012-11263-8

$C^*$-algebras associated with real multiplication

Author: Norio Nawata
Journal: Proc. Amer. Math. Soc. 140 (2012), 3409-3419
MSC (2010): Primary 46L05; Secondary 11D09, 11R11
DOI: https://doi.org/10.1090/S0002-9939-2012-11263-8
Published electronically: February 2, 2012
MathSciNet review: 2929010
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Abstract: Noncommutative tori with real multiplication are the irrational rotation algebras that have special equivalence bimodules. Y. Manin proposed the use of noncommutative tori with real multiplication as a geometric framework for the study of abelian class field theory of real quadratic fields. In this paper, we consider the Cuntz-Pimsner algebras constructed by special equivalence bimodules of irrational rotation algebras. We shall show that the associated $C^*$-algebras are simple and purely infinite. We compute the $K$-groups of the associated $C^*$-algebras and show that these algebras are related to the solutions of Pell’s equation and the unit groups of real quadratic fields. We consider the Morita equivalent classes of the associated $C^*$-algebras.

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