Maximality of dual coactions on sectional $C^*$-algebras of Fell bundles and applications

Alcides Buss ¹   ; Siegfried Echterhoff ²

¹ Departamento de Matemática Universidade Federal de Santa Catarina 88.040-900 Florianópolis, SC, Brazil
² Mathematisches Institut Westfälische Wilhelms-Universität Münster Einsteinstr. 62 48149 Münster, Germany

Studia Mathematica, Tome 229 (2015) no. 3, pp. 233-262

Résumé

We give a simple proof of the maximality of dual coactions on full cross-sectional $C^*$-algebras of Fell bundles over locally compact groups. This result was only known for discrete groups or for saturated (separable) Fell bundles over locally compact groups. Our proof, which is derived as an application of the theory of universal generalised fixed-point algebras for weakly proper actions, is different from these previously known cases and works for general Fell bundles over locally compact groups. As applications, we extend certain exotic crossed-product functors in the sense of Baum, Guentner and Willett to the category of Fell bundles and the category of partial actions, and we obtain results about the $K$-theory of (exotic) cross-sectional algebras of Fell bundles over $K$-amenable groups. As a bonus, we give a characterisation of maximal coactions of discrete groups in terms of maximal tensor products.

Détail

DOI : 10.4064/sm8361-1-2016

Keywords: simple proof maximality dual coactions full cross sectional * algebras fell bundles locally compact groups result only known discrete groups saturated separable fell bundles locally compact groups proof which derived application theory universal generalised fixed point algebras weakly proper actions different these previously known cases works general fell bundles locally compact groups applications extend certain exotic crossed product functors sense baum guentner willett category fell bundles category partial actions obtain results about k theory exotic cross sectional algebras fell bundles k amenable groups bonus characterisation maximal coactions discrete groups terms maximal tensor products