Hilbert C-modules and Amenable Actions

We study actions of discrete groups on Hilbert C-modules induced from topological actions on compact Hausdorff spaces. We prove that amenable actions give rise to proper affine isometric actions, provided there is a quasiinvariant measure which is sufficiently close to being invariant in a certain sense. This provides conditions on non-amenability of actions. The notion of amenability of group actions has found many applications in recent years, particularly in index theory. Yu proved [23] that the coarse BaumConnes conjecture and the Novikov conjecture hold for groups which satisfy property A, a weak version of amenability. Property A turned out to be equivalent to existence of a topologically amenable action on some compact space [13] and to exactness of the reduced group C-algebra C r (G) [11, 19]. Because of the interest of finding counterexamples to the the above conjectures, it is interesting to find conditions on topological actions which imply their non-amenability. Given a topological action of a non-amenable group on a compact space, the existence of a finite invariant measure implies that the action is not topologically amenable. However, apart from this fact there are practically no results which would give sufficient conditions for non-amenability of an action unless one assumes the existence of an invariant measure for the action. In this paper we study the situation in which we are given a topological action on a compact space X and a probability measure ν such that the action of G preserves the measure class. This means the translate gν and ν are absolutely continuous with respect to each other and, in particular, the Radon-Nikodym derivatives dgν/dν are defined almost everywhere for every g ∈ G. Our first result (Theorem 3) states that if the Radon-Nikodym derivatives of the translated measures satisfy some global integrability conditions, then a topologically amenable action gives rise to a proper, affine isometric action on a Hilbert space. The latter property, known as a-T-menability or the Haagerup property, was defined by Gromov [10]. As a consequence we get our main result, namely that for groups which do not admit such actions, e.g. groups with property (T), our condition on the Radon-Nikodym derivatives implies that the action of G cannot be topologically amenable. Date: 27 July 2009.