Asymptotic Abelianness, weak mixing, and property T
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David Kerr
Abstract
Let G be a second countable locally compact group and H a closed subgroup. We characterize the lack of Kazhdan's property T for the pair (G, H) by the genericity of G-actions on the hyperfinite II1 factor with a certain asymptotic Abelianness property relative to H, as well as by the genericity of measure-preserving G-actions on a nonatomic standard probability space that are weakly mixing for H. The latter furnishes a definitive generalization of a classical theorem of Halmos for single automorphisms and strengthens a recent result of Glasner, Thouvenot, and Weiss on generic ergodicity. We also establish a weak mixing version of Glasner and Weiss's characterization of property T for discrete G in terms of the invariant state space of a Bernoulli shift and show that on the CAR algebra a type of norm asymptotic Abelianness is generic for G-actions when G is discrete and admits a nontorsion Abelian quotient.
© Walter de Gruyter Berlin · New York 2008
Articles in the same Issue
- On the p-parts of quadratic Weyl group multiple Dirichlet series
- Characterization of SUq(ℓ + 1)-equivariant spectral triples for the odd dimensional quantum spheres
- The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field
- Lie ideals: from pure algebra to C*-algebras
- Strongly pseudoconvex homogeneous domains in almost complex manifolds
- The Cuntz semigroup as an invariant for C*-algebras
- Regular principal models of split semisimple Lie groups
- Asymptotic Abelianness, weak mixing, and property T
Articles in the same Issue
- On the p-parts of quadratic Weyl group multiple Dirichlet series
- Characterization of SUq(ℓ + 1)-equivariant spectral triples for the odd dimensional quantum spheres
- The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field
- Lie ideals: from pure algebra to C*-algebras
- Strongly pseudoconvex homogeneous domains in almost complex manifolds
- The Cuntz semigroup as an invariant for C*-algebras
- Regular principal models of split semisimple Lie groups
- Asymptotic Abelianness, weak mixing, and property T
