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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



$ 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
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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Additional Information

Norio Nawata
Affiliation: Graduate School of Mathematics, Kyushu University, Motooka, Fukuoka, 819-0395, Japan

Keywords: Irrational rotation algebras, Morita equivalence, Cuntz-Pimsner algebras, real multiplication, real quadratic fields
Received by editor(s): June 11, 2009
Received by editor(s) in revised form: September 21, 2010, and April 1, 2011
Published electronically: February 2, 2012
Communicated by: Marius Junge
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.