Amenable actions and weak containment of certain representations of discrete groups

Kuhn, M. Gabriella

doi:10.1090/S0002-9939-1994-1209424-3

Amenable actions and weak containment of certain representations of discrete groups
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by M. Gabriella Kuhn

Proc. Amer. Math. Soc. 122 (1994), 751-757

DOI: https://doi.org/10.1090/S0002-9939-1994-1209424-3

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Abstract:

We consider a countable discrete group $\Gamma$ acting ergodically on a standard Borel space S with quasi-invariant measure $\mu$. Let $\pi$ be a unitary representation of $\Gamma$ on ${L^2}(S,d\mu ,\mathcal {H})$ "nicely" related with S. We prove that if $\Gamma$ acts amenably on S then $\pi$ is weakly contained in the regular representation.

References

Boundary amenability for hyperbolic groups and an application to smooth dynamic of simple groups

A. Hulanicki, Means and Følner condition on locally compact groups, Studia Math. 27 (1966), 87–104. MR 195982, DOI 10.4064/sm-27-2-87-104
Gabriella Kuhn and Tim Steger, More irreducible boundary representations of free groups, Duke Math. J. 82 (1996), no. 2, 381–436. MR 1387235, DOI 10.1215/S0012-7094-96-08218-6
Claudio Nebbia, Amenability and Kunze-Stein property for groups acting on a tree, Pacific J. Math. 135 (1988), no. 2, 371–380. MR 968619
John C. Quigg and J. Spielberg, Regularity and hyporegularity in $C^*$-dynamical systems, Houston J. Math. 18 (1992), no. 1, 139–152. MR 1159445
Robert J. Zimmer, On the von Neumann algebra of an ergodic group action, Proc. Amer. Math. Soc. 66 (1977), no. 2, 289–293. MR 460599, DOI 10.1090/S0002-9939-1977-0460599-3
Robert J. Zimmer, Hyperfinite factors and amenable ergodic actions, Invent. Math. 41 (1977), no. 1, 23–31. MR 470692, DOI 10.1007/BF01390162

Ergodic theory and semisimple groups