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Circle maps and ${C}^{\ast } $-algebras

Published online by Cambridge University Press:  28 August 2013

THOMAS LUNDSGAARD SCHMIDT  and
KLAUS THOMSEN
THOMAS LUNDSGAARD SCHMIDT
Affiliation:
Institut for Matematik, Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark email tschmidt@imf.au.dkmatkt@imf.au.dk
KLAUS THOMSEN
Affiliation:
Institut for Matematik, Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark email tschmidt@imf.au.dkmatkt@imf.au.dk
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Abstract

We consider a construction of ${C}^{\ast } $-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}^{\ast } $-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.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 2 , April 2015 , pp. 546 - 584
Copyright
© Cambridge University Press, 2013 

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