Unique Cartan decomposition for II_1 factors arising from arbitrary actions of hyperbolic groups

By Sorin Popa, S. Vaes. 2012

155 citations recorded by the source corpus.

We prove that for any free ergodic probability measure preserving action $\Gamma \actson (X,\mu)$ of a non-elementary hyperbolic group, or a lattice in a rank one simple Lie group, the associated group measure space II_1 factor $L^\infty(X) \rtimes \Gamma$ has L^\infty(X) as its unique Cartan subalgebra, up to unitary conjugacy.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2ddf9fb2b911c2c447eb823dd6ea4f85d2d5dfa0

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

CCAP for Universal Discrete Quantum Groups (2013, background): “ the viewpoint of deformation rigidity theory, W∗CCAP and its variants are exploited to show the sparsity of Cartan subalgebras in the breakthrough papers by Ozawa and Popa [29], and by Popa and Vaes [31]. Regarding FOF, recent work of Isono [23] on a strengthened Akemann–Ostrand property establishe”
Type III factors with unique Cartan decomposition (2012, methodology): “Then, the amenability of L∞(X) ⋊ Γi is equivalent with the amenability of the action Γiy (X,µ) in the sense of Zimmer, see [Zi76, Theorem 2.4]. The proof of Theorem A relies heavily on the results in [PV12]. Given a Cartan subalgebra A⊂ M= L∞(X)⋊Γ, we apply the modular theory of Tomita, Takesaki and”
Examples of factors which have no Cartan subalgebras (2012, unknown): “After the work of Ozawa and Popa, there have been many non-existence results, and in the present paper we follow [27], in which Popa and Vaes proved remarkable uniqueness and non-existence results of Cartan subalgebras.”
W*‐superrigidity for group von Neumann algebras of left–right wreath products (2012, methodology): “rove that the subalgebra L(Γ×Γ) ⊂ LG is invariant under ∆, up to unitary conjugacy. We next use the recent results on normalizers of amenable subalgebras in crossed products by hyperbolic groups (see [PV12]), and in crossed products by arbitrary free product groups (see [Io12b]), to prove that also ”