Amenability and exactness for dynamical systems and their C*-algebras

In this survey, we study the relations between amenability (resp. amenability at infinity) of C*-dynamical systems and equality or nuclearity (resp. exactness) of the corresponding crossed products.Comment: 16 pages, Ams-Tex, minor grammatical change

Old and new about treeability and the Haagerup property for measured groupoids

This is mainly an expository text on the Haagerup property for countable groupoids equipped with a quasi-invariant measure, aiming to complete an article of Jolissaint devoted to the study of this property for probability measure preserving countable equivalence relations. We show that our definition is equivalent to the one given by Ueda in terms of the associated inclusion of von Neumann algebras. It makes obvious the fact that treeability implies the Haagerup property for such groupoids. For the sake of completeness, we also describe, or recall, the connections with amenability and Kazhdan property (T).Comment: 38 page

Pointwise limits for sequences of orbital integrals

In 1967, Ross and Str\"omberg published a theorem about pointwise limits of orbital integrals for the left action of a locally compact group $G$ onto $(G,\rho)$, where $\rho$ is the right Haar measure. In this paper, we study the same kind of problem, but more generally for left actions of $G$ onto any measured space $(X,\mu)$, which leaves the $\sigma$-finite measure $\mu$ relatively invariant, in the sense that $s\mu = \Delta(s)\mu$ for every $s\in G$, where $\Delta$ is the modular function of $G$. As a consequence, we also obtain a generalization of a theorem of Civin, relative to one-parameter groups of measure preserving transformations. The original motivation for the circle of questions treated here dates back to classical problems concerning pointwise convergence of Riemann sums relative to Lebesgue integrable functions

Exactness of locally compact groups

We give some new characterizations of exactness for locally compact second countable groups. In particular, we prove that a locally compact second countable group is exact if and only if it admits a topologically amenable action on a compact Hausdorff space. This answers an open question by Anantharaman-Delaroche.Comment: 18 pages, to appear in Adv. Mat

Examples of amalgamated free products and coupling rigidity

We present amalgamated free products satisfying coupling rigidity with respect to the automorphism group of the associated Bass-Serre tree. As an application, we obtain orbit equivalence rigidity for amalgamated free products of mapping class groups.Comment: 28 pages, 1 figure. This is an expansion of a part of v1 in arXiv:0902.288