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.
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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.
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$\begingroup$ thanks a lot. Bruhat function exists if proper space X is also G-compact? what about converse? Is amenable action proper? $\endgroup$– m07klCommented Sep 3, 2011 at 20:53
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$\begingroup$ OK, I see. For a locally compact paracompact Hausdorff space proper action is amenable. To see this if X is G-compact, then we can choose Bruhat function to be compactly supported. If X is not G-compact, Bruhat function comes from G-compact case by partition of unity since compact paracompact Hausdorff implies partition of unity. In this case Bruhat function is not compactly supported, but the intersection of support of Bruhat function with any G-compact set in X is compact. (We need also Usyhson's lemma to construct Bruhat function, but paracompact Hausdorff space is normal) $\endgroup$– m07klCommented Sep 3, 2011 at 21:47
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$\begingroup$ This implies also that for a proper G-C*-algebra the reduced and full crossed products coincide by Theorem 5.3 in C. Anantharaman-Delaroche $\endgroup$– m07klCommented Sep 3, 2011 at 21:59
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1$\begingroup$ About the converse: if $G$ is amenable and infinite, the action of $G$ on a point is amenable but not proper. $\endgroup$ Commented Sep 3, 2011 at 22:19