proper action and amenable action - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:18:29Z http://mathoverflow.net/feeds/question/74450 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74450/proper-action-and-amenable-action proper action and amenable action m07kl 2011-09-03T17:36:26Z 2011-09-03T19:19:55Z <p>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.</p> http://mathoverflow.net/questions/74450/proper-action-and-amenable-action/74453#74453 Answer by Alain Valette for proper action and amenable action Alain Valette 2011-09-03T18:18:44Z 2011-09-03T18:18:44Z <p>Look at that paper by C. Anantharaman-Delaroche: <a href="http://www.univ-orleans.fr/mapmo/membres/anantharaman/publications/Exactness02.pdf" rel="nofollow">http://www.univ-orleans.fr/mapmo/membres/anantharaman/publications/Exactness02.pdf</a></p> <p>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.</p>