Noncommutative Solenoids

TL;DR

This paper classifies noncommutative solenoid C*-algebras using noncommutative topology, determines their simplicity conditions, computes their K-theory, and describes their structure as bundles of matrices, revealing their algebraic properties.

Contribution

It provides a classification of noncommutative solenoids, establishes simplicity criteria, computes K-theory, and describes their structure as matrix bundles over solenoid groups.

Findings

Classified noncommutative solenoids up to *-isomorphism.

Derived necessary and sufficient conditions for simplicity.

Computed K-theory and described algebraic structure.

Abstract

A noncommutative solenoid is the C*-algebra $C^\ast(\Q_N^2,\sigma)$ where $\Q_N$ is the group of the $N$-adic rationals twisted and $\sigma$ is a multiplier of $\Q_N^2$. In this paper, we use techniques from noncommutative topology to classify these C*-algebras up to *-isomorphism in terms of the multipliers of $\Q_N^2$. We also establish a necessary and sufficient condition for simplicity of noncommutative solenoids, compute their K-theory and show that the $K_0$ groups of noncommutative solenoids are given by the extensions of $\Z$ by $\Q_N$. We give a concrete description of non-simple noncommutative solenoids as bundle of matrices over solenoid groups, and we show that irrational noncommutative solenoids are real rank zero AT C*-algebras.