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Journal of Operator Theory

Volume 75, Issue 1, Winter 2016  pp. 49-73.

Rieffel proper actions

Authors:  Alcides Buss (1) Siegfried Echterhoff (2)
Author institution: (1) Departamento de Matem\'atica, Universidade Federal de Santa Catarina, 88.040-900 Florian\'opolis-SC, Brazil
(2) Mathematisches Institut, Westf\"alische Wilhelms-Universit\"at M\"unster, Einsteinstr. 62, 48149 M\"unster, Germany

Summary: In the late 1980's Marc Rieffel introduced a notion of properness for actions of locally compact groups on $C^*$-algebras which, among other things, allows the construction of generalised fixed-point algebras for such actions. In this paper we give a simple characterisation of Rieffel proper actions and use this to obtain several (counter) examples for the theory. In particular, we provide examples of Rieffel proper actions $\alpha:G\to\mathrm{Aut}(A)$ for which properness is not induced by a nondegenerate equivariant $*$-homomorphism $\phi:C_0(X)\to \mathcal M(A)$ for any proper $G$-space $X$. Other examples, based on earlier work of Meyer, show that a given action might carry different structures for Rieffel properness with different generalised fixed-point algebras.

Keywords: proper actions, crossed products, fixed-point algebras

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