The type of group measure space von Neumann algebras

1978 ◽  
Vol 85 (2) ◽  
pp. 149-162 ◽  
Author(s):  
Marc A. Rieffel
Keyword(s):  
Measure Space ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Group Measure

scholarly journals A remark on fullness of some group measure space von Neumann algebras

2016 ◽  
Vol 152 (12) ◽  
pp. 2493-2502 ◽  
Author(s):  
Narutaka Ozawa
Keyword(s):  
Lie Group ◽  
Measure Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Free Action ◽  
Von Neumann ◽  
Finite Center ◽  
Group Measure ◽  
Singular Action ◽  
Amenable Subgroup

Recently Houdayer and Isono have proved, among other things, that every biexact group $\unicode[STIX]{x1D6E4}$ has the property that for any non-singular strongly ergodic essentially free action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ on a standard measure space, the group measure space von Neumann algebra $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(X)$ is full. In this paper, we prove the same property for a wider class of groups, notably including $\text{SL}(3,\mathbb{Z})$. We also prove that for any connected simple Lie group $G$ with finite center, any lattice $\unicode[STIX]{x1D6E4}\leqslant G$, and any closed non-amenable subgroup $H\leqslant G$, the non-singular action $\unicode[STIX]{x1D6E4}\curvearrowright G/H$ is strongly ergodic and the von Neumann factor $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(G/H)$ is full.

scholarly journals Amalgamated free product type III factors with at most one Cartan subalgebra

2013 ◽  
Vol 150 (1) ◽  
pp. 143-174 ◽  
Author(s):  
Rémi Boutonnet ◽  
Cyril Houdayer ◽  
Sven Raum
Keyword(s):  
Free Product ◽  
Measure Space ◽  
Cartan Subalgebra ◽  
Von Neumann Algebras ◽  
Finite Subgroup ◽  
Standard Measure ◽  
Von Neumann ◽  
Cartan Subalgebras ◽  
Amalgamated Free Product ◽  
Normal States

AbstractWe investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras ${\mathop{M{}_{1} \ast }\nolimits}_{B} {M}_{2} $ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $({M}_{1} , {\varphi }_{1} )\ast ({M}_{2} , {\varphi }_{2} )$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established by A. Ioana [Cartan subalgebras of amalgamated free product ${\mathrm{II} }_{1} $factors, arXiv:1207.0054]. Next, we prove that any countable nonsingular ergodic equivalence relation $ \mathcal{R} $ defined on a standard measure space and which splits as the free product $ \mathcal{R} = { \mathcal{R} }_{1} \ast { \mathcal{R} }_{2} $ of recurrent subequivalence relations gives rise to a nonamenable factor $\mathrm{L} ( \mathcal{R} )$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors ${\mathrm{L} }^{\infty } (X)\rtimes \Gamma $ arising from nonsingular free ergodic actions $\Gamma \curvearrowright (X, \mu )$ on standard measure spaces of amalgamated groups $\Gamma = {\mathop{\Gamma {}_{1} \ast }\nolimits}_{\Sigma } {\Gamma }_{2} $ over a finite subgroup $\Sigma $.

scholarly journals Inner amenability for groups and central sequences in factors

2014 ◽  
Vol 36 (4) ◽  
pp. 1106-1129 ◽  
Author(s):  
IONUT CHIFAN ◽  
THOMAS SINCLAIR ◽  
BOGDAN UDREA
Keyword(s):  
Measure Space ◽  
Negative Curvature ◽  
Mapping Class ◽  
Discrete Groups ◽  
Class Groups ◽  
Curvature Condition ◽  
Von Neumann ◽  
Group Measure ◽  
Central Sequences ◽  
Inner Amenability

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].

Lectures on von Neumann Algebras

2019 ◽  
Author(s):  
Serban-Valentin Stratila ◽  
Laszlo Zsido
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann

scholarly journals Amalgamated free product rigidity for group von Neumann algebras

2018 ◽  
Vol 329 ◽  
pp. 819-850 ◽  
Author(s):  
Ionuţ Chifan ◽  
Adrian Ioana
Keyword(s):  
Free Product ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Amalgamated Free Product
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