We say that an action of a (discrete) group G on a locally compact space X is called proper if the map from $G\times X$ to $X\times X$ defined by $(g,x)\mapsto (gx,x)$ is proper. Why is a proper action amenable? (see On the Baum-Connes assembly map for discrete groups-Alain Valette, proof of lemma 2.13). If this is a case, the full crossed product and reduced product for $C\_0(X)$ are isomorphic.



proper action and amenable action
Look at that paper by C. Anantharaman-Delaroche: <http://www.univ-orleans.fr/mapmo/membres/anantharaman/publications/Exactness02.pdf>


In Prop. 2.2, point (2), you find an equivalent condition for amenability of the $G$-action on $X$, in terms of the existence of a net $(g\_i)$ of continuous, non-negative functions on $X\times G$. Now, if $X$ is a proper $G$-space, you find a Bruhat function on $X$, i.e. a continuous non-negative function $h$ on $X$ such that $\sum\_{t\in G}h(t^{-1}x)=1$. Define then $g\_i(x,t)=h(t^{-1}x)$. If I'm not mistaken, the conditions in Anantharaman's result are satisfied.


