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The Haagerup property for measure-preserving standard equivalence relations

Published online by Cambridge University Press:  22 December 2004

PAUL JOLISSAINT
PAUL JOLISSAINT
Affiliation:
Institut de Mathématiques, Université de Neuchâtel, Emile-Argand 11 CH-2000 Neuchâtel, Switzerland (e-mail: paul.jolissaint@unine.ch)
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Abstract

We define a notion of the Haagerup property for measure-preserving standard equivalence relations. Given such a relation R on X with finite invariant measure $\mu$, we prove that R has the Haagerup property if and only if the associated finite von Neumann algebra L(R) (see J. Feldman and C. C. Moore. Ergodic equivalence relations, cohomology and von Neumann algebras II. Trans. Amer. Math. Soc.234 (1977), 325–350) has relative property H in the sense of Popa with respect to its natural Cartan subalgebra $L^{\infty}(X,\mu)$. We also prove that if G is a countable group such that R = RG has the Haagerup property and if R is ergodic, then G cannot have Kazhdan's property T.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 25 , Issue 1 , February 2005 , pp. 161 - 174
Copyright
2004 Cambridge University Press

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