$C^*$-algebras associated with real multiplication

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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 associated $C^*$-algebras are simple and purely infinite. We compute the K-groups of 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 associated $C^*$-algebras.
11pages

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Operator Algebras, Number Theory, 46L05 (Primary) 11D09, 11R11 (Secondary)

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http://salesiana.dossiersoluciones.com/handle/123456789/226097

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/0904.1085
 http://arxiv.org/abs/0904.1085

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arXiv