W*-superrigidity for Bernoulli actions of property (T) groups

By Adrian Ioana. 2010

80 citations recorded by the source corpus.

We consider group measure space II$_1$ factors $M=L^{\infty}(X)\rtimes\Gamma$ arising from Bernoulli actions of ICC property (T) groups $\Gamma$ (more generally, of groups $\Gamma$ containing an infinite normal subgroup with relative property (T)) and prove a rigidity result for *--homomorphisms $\theta:M\to M\bar{\otimes}M$. We deduce that the action $\Gamma\curvearrowright X$ is W$^*$--superrigid. This means that if $\Lambda\curvearrowright Y$ is {\bf any other} free, ergodic, measure preserving action such that the factors $M=L^{\infty}(X)\rtimes\Gamma$ and $L^{\infty}(Y)\rtimes\Lambda$ are isomorphic, then the actions $\Gamma\curvearrowright X$ and $\Lambda\curvearrowright Y$ must be conjugate. Moreover, we show that if $p\in M\setminus\{1\}$ is a projection, then $pMp$ does not admit a group measure space decomposition nor a group von Neumann algebra decomposition (the latter under the additional assumption that $\Gamma$ is torsion free). We also prove a rigidity result for *--homomorphisms $\theta:M\to M$, this time for $\Gamma$ in a larger class of groups than above, now including products of non--amenable groups. For certain groups $\Gamma$, e.g. $\Gamma=\Bbb F_2\times\Bbb F_2$, we deduce that $M$ does not embed in $pMp$, for any projection $p\in M\setminus\{1\}$, and obtain a description of the endomorphism semigroup of $M$.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c9372d6c70f5e59bbe1ef92f54f5fde3ceb312d6

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In Algebra and Number Theory

1. S. Vaes
2. Adrian Ioana
3. Mihaita Berbec
4. Sorin Popa
5. Xin Li
6. Rufus Willett
7. A. Furman
8. Ján Špakula

In Geometry and Topology

1. S. Vaes
2. Adrian Ioana
3. Sorin Popa
4. Mihaita Berbec
5. Xin Li
6. Rufus Willett
7. Ján Špakula
8. Damien Gaboriau

In Mathematical Physics

1. S. Vaes
2. Adrian Ioana
3. Mihaita Berbec
4. Sorin Popa
5. Xin Li
6. Ján Špakula
7. A. Furman
8. Rufus Willett

Citations in the corpus

• A CLASS OF SUPERRIGID GROUP VON NEUMANN ALGEBRAS (2010, methodology): “This gives its statement a W∗-superrigidity flavor, in the same spirit as the recent superrigidity results for group measure space II1 factors ([PV10], [Ioa11]), showing that certain classes of free ergodic probability measure preserving group actions G y (X,μ) can be completely recovered from their”
• Continuous orbit equivalence rigidity (2015, background): “lished itself as a key notion. The idea is to find classes of actions for which orbit equivalence already implies conjugacy. Indeed, impressive orbit equivalence rigidity results have been obtained in [28, 7, 15, 16, 17, 14, 13]. Viewing continuous orbit equivalence as the topological analogue of orb”
• W*‐superrigidity for group von Neumann algebras of left–right wreath products (2012, result): “These methods and results have been generalized in many subsequent works (see, for example, [9, 13, 26, 33]) and were in particular extended to cover certain generalized Bernoulli actions, associated with general group actions G I.”
• On rigidity of Roe algebras (2011, result): “The very recent results of Ioana [22] on superrigidity of Bernoulli shift actions have no obvious analogue in our context.”
• Unique Cartan decomposition for II_1 factors arising from arbitrary actions of hyperbolic groups (2012, methodology): “ce of virtually W∗-superrigid group actions was proven. In [PV09] we obtained the first concrete W∗-superrigidity theorem, for Bernoulli actions of a large class of amalgamated free product groups. In [Io10] it was shown that Bernoulli actions of icc property (T) groups are W∗-superrigid. In the pres”