$C^*$-algebras associated with real multiplication

TL;DR

This paper explores the structure and properties of certain $C^*$-algebras derived from noncommutative tori with real multiplication, linking them to number theory and class field theory.

Contribution

It introduces new $C^*$-algebras from special bimodules of irrational rotation algebras and analyzes their simplicity, K-theory, and connections to quadratic fields.

Findings

The $C^*$-algebras are simple and purely infinite.

K-groups are explicitly computed.

Connections to Pell's equation and units of quadratic fields.

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 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.