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Claire Anantharaman-Delaroche
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ABSTRACT: 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).
05/2011;
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Claire Anantharaman-Delaroche
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ABSTRACT: 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.
03/2009;
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Claire Anantharaman-Delaroche
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ABSTRACT: To any action of a locally compact group $G$ on a pair $(A,B)$ of von Neumann algebras is canonically associated a pair $(\pi\_A^{\alpha}, \pi\_B^{\alpha})$ of unitary representations of $G$. The purpose of this paper is to provide results allowing to compare the norms of the operators $\pi\_A^{\alpha}(\mu)$ and $\pi\_B^{\alpha}(\mu)$ for bounded measures $\mu$ on $G$. We have a twofold aim. First to point out that several known facts in ergodic and representation theory are indeed particular cases of general results about $(\pi\_A^{\alpha}, \pi\_B^{\alpha})$. Second, under amenability assumptions, to obtain transference of inequalities that will be useful in noncommutative ergodic theory.
06/2006;
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Claire Anantharaman-Delaroche
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ABSTRACT: We extend in a noncommutative setting the individual ergodic theorem of Nevo and Stein concerning measure preserving actions of free groups and averages on spheres $s_{2n}$ of even radius. Here we study state preserving actions of free groups on a von Neumann algebra $A$ and the behaviour of $(s_{2n}(x))$ for $x$ in noncommutative spaces $L^p(A)$. For the Ces\`aro means $\frac{1}{n}\sum_{k=0}^{n-1} s_k$ and $p = +\infty$, this problem was solved by Walker. Our approach is based on ideas of Bufetov. We prove a noncommutative version of Rota ``Alternierende Verfahren'' theorem. To this end, we introduce specific dilations of the powers of some noncommutative Markov operators.
01/2005;
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Claire Anantharaman-Delaroche
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ABSTRACT: We extend the Delorme-Guichardet characterization of Kazhdan property $T$ groups to $r$-discrete measured groupoids. We give several applications, in particular to stability results of Kazhdan property $T$ and to the study of cocycles taking values in a group having the Haagerup property.
09/2003;
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Claire Anantharaman-delaroche
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ABSTRACT: We show that a measured G-space (X; ), where G is a locally compact group, is amenable in the sense of Zimmer if and only if the following two conditions are satis ed: the associated unitary representation X of G into L (X; ) is weakly contained into the regular representation G and there exists a G-equivariant norm one projection from L (X). We give examples of ergodic discrete group actions which are not amenable, although X is weakly contained into G .
11/2001;
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Claire Anantharaman-Delaroche
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ABSTRACT: These expository notes aim to introduce some weakenings of the notion of amenability for groups, and to develop their
connections with the theory of operator algebras as well as with recent remarkable applications.
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Claire Anantharaman-Delaroche
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ABSTRACT: We discuss properties and examples of discrete groups in connection with their operator algebras and related tensor products.
