# 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.

**DOI: **http://dx.doi.org/10.7900/jot.2014oct28.2047

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

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