Motivated by Popa’s seminal work Popa (Invent Math 165:409-45, 2006), in this paper, we provide a fairly large class of examples of group actions
Γ
↷
X
satisfying the extended Neshveyev–Størmer ...rigidity phenomenon Neshveyev and Størmer (J Funct Anal 195(2):239-261, 2002): whenever
Λ
↷
Y
is a free ergodic pmp action and there is a
∗
-isomorphism
Θ
:
L
∞
(
X
)
⋊
Γ
→
L
∞
(
Y
)
⋊
Λ
such that
Θ
(
L
(
Γ
)
)
=
L
(
Λ
)
then the actions
Γ
↷
X
and
Λ
↷
Y
are conjugate (in a way compatible with
Θ
). We also obtain a complete description of the intermediate subalgebras of all (possibly non-free) compact extensions of group actions in the same spirit as the recent results of Suzuki (Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems, To appear in Comm Math Phy. ArXiv Preprint:
arXiv:1805.02077
, 2020). This yields new consequences to the study of rigidity for crossed product von Neumann algebras and to the classification of subfactors of finite Jones index.
On page 43 in Adv. in Math. 50 (1983), pp. 27-48 Sorin Popa asked whether the following property holds: If \omega is a free ultrafilter on \mathbb{N} and \mathcal {R}_1\subseteq \mathcal {R} is an ...irreducible inclusion of hyperfinite II _1 factors such that \mathcal {R}'\cap \mathcal {R}^\omega \subseteq \mathcal {R}^\omega _1 does it follows that \mathcal {R}_1=\mathcal {R}? In this short note we provide an affirmative answer to this question.
We show that a large class of i.c.c., countable, discrete groups satisfying a weak negative curvature condition are not inner amenable. By recent work of Hull and Osin Groups with hyperbolically ...embedded subgroups. Algebr. Geom. Topol.13 (2013), 2635–2665, our result recovers that mapping class groups and $\text{Out}(\mathbb{F}_{n})$ are not inner amenable. We also show that the group-measure space constructions associated to free, strongly ergodic p.m.p. actions of such groups do not have property Gamma of Murray and von Neumann On rings of operators IV. Ann. of Math. (2) 44 (1943), 716–808.
We consider the following three properties for countable discrete groups \Gamma has an infinite subgroup with relative property (T), (2) the group von Neumann algebra L\Gamma does not have Haagerup's ...property. It is clear that (1) \Longrightarrow \Longrightarrow
On a question of D. Shlyakhtenko CHIFAN, IONUT; IOANA, ADRIAN
Proceedings of the American Mathematical Society,
03/2011, Volume:
139, Issue:
3
Journal Article
Peer reviewed
Open access
In this short paper we construct two countable, infinite conjugacy class (ICC) groups which admit free, ergodic, probability measure-preserving orbit equivalent actions but whose group von Neumann ...algebras are not (stably) isomorphic.
We show that any infinite collection (Γn)n∈N of icc, hyperbolic, property (T) groups satisfies the following von Neumann algebraic infinite product rigidity phenomenon. If Λ is an arbitrary group ...such that L(⊕n∈NΓn)≅L(Λ) then there exists an infinite direct sum decomposition Λ=(⊕n∈NΛn)⊕A with A icc amenable or trivial such that, for all n∈N, up to amplifications, we have L(Γn)≅L(Λn) and L(⊕k≥nΓk)≅L((⊕k≥nΛk)⊕A). The result is sharp and complements the previous finite product rigidity property found in 16. Using this we provide an uncountable family of restricted wreath products Γ≅Σ≀Δ of icc, property (T) groups Σ, Δ whose wreath product structure is recognizable, up to a normal amenable subgroup, from their von Neumann algebras L(Γ). Along the way we highlight several applications of these results to the study of rigidity in the C⁎-algebra setting.
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