Circle maps and C*-algebras

TL;DR

This paper constructs C*-algebras from circle maps, generalizing existing frameworks, and provides criteria for their simplicity, classifies them as Kirchberg algebras, and develops methods for calculating their K-theory.

Contribution

It introduces a new construction of C*-algebras from circle maps that extends previous models and offers a systematic way to analyze their properties.

Findings

C*-algebras are simple iff the map is surjective and not locally injective.

The C*-algebras are classified as Kirchberg algebras under certain conditions.

An algorithmic method for computing K-theory groups of these algebras.

Abstract

We consider a construction of C*-algebras from continuous piecewise monotone maps on the circle which generalizes the crossed product construction for homeomorphisms and more generally the construction of Renault, Deaconu and Anantharaman-Delaroche for local homeomorphisms. Assuming that the map is surjective and not locally injective we give necessary and sufficient conditions for the simplicity of the C*-algebra and show that it is then a Kirchberg algebra. We provide tools for the calculation of the K-theory groups and turn them into an algorithmic method for Markov maps.