On quantitative operator $K$-theory

[$K$-théorie quantitative pour les algèbres d’opérateurs]

Oyono-Oyono, Hervé ; Yu, Guoliang

Annales de l'Institut Fourier, Tome 65 (2015), p. 605-674 / Harvested from Numdam

Résumé

Dans cet article, nous développons une $K$-théorie quantitative pour les $C^*$-algèbres filtrées. Parmi les exemples les plus intéressants de telles $C^*$-algèbres figurent les algèbres de Roe, les $C^*$-algèbres de groupes et les $C^*$-algèbres de produits croisés. Nous établissons une version quantitative de la suite exacte à six termes en $K$-théorie ainsi que de la périodicité de Bott. Nous formulons en utilisant la $K$-théorie quantitative une version quantitative de la conjecture de Baum-Connes. Nous montrons que cette conjecture de Baum-Connes quantitative est vérifiée pour une large classe de groupes.

Abstract

In this paper, we develop a quantitative $K$-theory for filtered $C^*$-algebras. Particularly interesting examples of filtered $C^*$-algebras include group $C^*$-algebras, crossed product $C^\ast $-algebras and Roe algebras. We prove a quantitative version of the six term exact sequence and a quantitative Bott periodicity. We apply the quantitative $K$-theory to formulate a quantitative version of the Baum-Connes conjecture and prove that the quantitative Baum-Connes conjecture holds for a large class of groups.

Détail

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2940
Classification:  19K35,  46L80,  58J22
Mots clés: Conjecture de Baum-Connes, Géométrie à Grande Echelle, $C^*$-algèbres de Groupes et de Produits Croisés, Conjecture de Novikov, $K$-théorie pour les Algèbres d’Opérateurs

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