The E-theoretic descent functor for groupoids

The paper establishes, for a wide class of locally compact groupoids $\Gamma$, the E-theoretic descent functor at the $C^{*}$-algebra level, in a way parallel to that established for locally compact groups by Guentner, Higson and Trout. The second section shows that $\Gamma$-actions on a $C_{0}(X)$-algebra $B$, where $X$ is the unit space of $\Gamma$, can be usefully formulated in terms of an action on the associated bundle $B^{\sharp}$. The third section shows that the functor $B\to C^{*}(\Gamma,B)$ is continuous and exact, and uses the disintegration theory of J. Renault. The last section establishes the existence of the descent functor under a very mild condition on $\Gamma$, the main technical difficulty involved being that of finding a $\Gamma$-algebra that plays the role of C_{b}(T,B)^{cont}$ in the group case.Comment: 21 page

Group amenability properties for von Neumann algebras

In his study of amenable unitary representations, M. E. B. Bekka asked if there is an analogue for such representations of the remarkable fixed-point property for amenable groups. In this paper, we prove such a fixed-point theorem in the more general context of a $G$-amenable von Neumann algebra $M$, where $G$ is a locally compact group acting on $M$. The F{\o}lner conditions of Connes and Bekka are extended to the case where $M$ is semifinite and admits a faithful, semifinite, normal trace which is invariant under the action of $G$