Purely infinite C*-algebras arising from dynamical systems

Abstract

We give a condition sufficient to ensure that the reduced C*-algebra associated with an r-discrete groupoid is purely infinite. As an application, we get many examples of purely infinite C*-algebras, from discrete groups of isometries of hyberbolic metric spaces, or of Hadamard manifolds, acting on their limit set. The expanding continuous surjective maps from a compact metric space onto itself provide also interesting examples. Actually, many of the examples considered here give purely infinite, simple, nuclear, separable C*-algebras, satisfying to the Universal Coefficient Theorem. Therefore, they are completely classified by their K-theory groups, thanks to the recent work of Kirchberg.