Semi-étale groupoids and applications

Thomsen, Klaus

Annales de l'Institut Fourier, Tome 60 (2010) no. 3, pp. 759-800.

Nous associons une $C^{*}$-algèbre à un groupoïde de Hausdorff localement compact avec la propriété pour l’application but d’être localement injective. La construction généralise celle de la $C^{*}$-algèbre réduite de J. Renault pour un groupoïde étale. Elle possède l’avantage de fonctionner pour le groupoïde résultant d’un système dynamique localement injectif par la méthode introduite par Renault, Deaconu et Anantharaman-Delaroche, en augmentant la généralité. Nous étudions les $C^{*}$-algèbres de tels groupoïdes et nous donnons des conditions nécessaires et suffisantes pour la simplicité. Nous montrons qu’un grand nombre d’entre elles contiennent une sous-algèbre de Cartan au sens de Renault. En particulier, cela est valable pour un système dynamique symbolique, auquel cas la $C^{*}$-algèbre coïncide avec celle introduite par Matsumoto et Carlsen.

We associate a $C^{*}$-algebra to a locally compact Hausdorff groupoid with the property that the range map is locally injective. The construction generalizes J. Renault’s reduced groupoid $C^{*}$-algebra of an étale groupoid and has the advantage that it works for the groupoid arising from a locally injective dynamical system by the method introduced in increasing generality by Renault, Deaconu and Anantharaman-Delaroche. We study the $C^{*}$-algebras of such groupoids and give necessary and sufficient conditions for simplicity, and show that many of them contain a Cartan subalgebra as defined by Renault. In particular, this holds when the dynamical system is a shift space, in which case the $C^{*}$-algebra coincides with the one introduced by Matsumoto and Carlsen.

Classification : 37B05, 46L55
Keywords: Groupoids, $C^*$-algebras, dynamical systems
Mot clés : groupoïde, $C^*$-algèbre, système dynamique

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