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Zbl 0896.46044
Anantharaman-Delaroche, Claire
Purely infinite $C^*$-algebras arising from dynamical systems. (English)
[J] Bull. Soc. Math. Fr. 125, No.2, 199-225 (1997). ISSN 0037-9484

Summary: We give a sufficient condition 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 hyperbolic 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 the universal coefficient theorem. Therefore, they are completely classified by their $K$-theory groups, thanks to the recent work of Kirchberg.

MSC 2000:: *46L55 Noncommutative dynamical systems
46L35 Classifications and factors of C*-algebras
54H20 Topological dynamics
46L80 K-theory and operator algebras

Keywords: reduced $C^*$-algebra; $r$-discrete groupoid; purely infinite $C^*$-algebras; discrete groups of isometries; hyperbolic metric spaces; Hadamard manifolds; universal coefficient theorem; $K$-theory groups

Cited in: Zbl 0993.46033

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