Ergodic Theory and Dynamical Systems

Table of Contents - 11 July 2012 - Volume 32, Issue 04  

Research Article

The structure of the C *-algebra of a locally injective surjection

TOKE MEIER CARLSENa1 and KLAUS THOMSENa2

a1 Department of Mathematics, Norwegian University of Science and Technology, NO-7034 Trondheim, Norway (email: tokemeie@math.ntnu.no)

a2 Institut for matematiske fag, Ny Munkegade, DK-8000 Aarhus C, Denmark (email: matkt@imf.au.dk)

Abstract

In this paper we investigate the ideal structure of the C *-algebra of a locally injective surjection introduced by the second-named author. Our main result is that a simple quotient of the C *-algebra of a locally injective surjection on a compact metric space of finite covering dimension is either a full matrix algebra, a crossed product of a minimal homeomorphism of a compact metric space of finite covering dimension, or it is purely infinite and hence covered by the classification result of Kirchberg and Phillips. It follows in particular that if the C *-algebra of a locally injective surjection on a compact metric space of finite covering dimension is simple, then it is automatically purely infinite, unless the map in question is a homeomorphism. A corollary of this result is that if the C *-algebra of a one-sided subshift is simple, then it is also purely infinite.

(Received November 08 2010)

(Revised February 24 2011)

(Online publication June 10 2011)