Rigidity results for wreath product II1 factors

By Adrian Ioana. 2006

62 citations recorded by the source corpus.

Abstract We consider II 1 factors of the form M = ( ⊗ ¯ G B ) ⋊ G , where either (i) B is a non-hyperfinite II 1 factor and G is an ICC amenable group or (ii) B is a weakly rigid II 1 factor and G is ICC group and where G acts on ⊗ ¯ G B by Bernoulli shifts. We prove that isomorphism of two such factors implies cocycle conjugacy of the corresponding Bernoulli shift actions. In particular, the groups acting must be isomorphic. As a consequence, we can distinguish between certain classes of group von Neumann algebras associated to wreath product groups.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2c5d0f73ff841ce645fd76453b358c2f536cd141

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In Algebra and Number Theory

1. S. Vaes
2. Adrian Ioana
3. Mihaita Berbec
4. Sorin Popa
5. A. Furman
6. Damien Gaboriau
7. J. Peterson
8. I. Chifan

In Applied Mathematics

1. S. Vaes
2. Adrian Ioana
3. Mihaita Berbec
4. Sorin Popa
5. Cyril Houdayer
6. J. Peterson
7. Yusuke Isono
8. I. Chifan

In Mathematical Physics

1. S. Vaes
2. Adrian Ioana
3. Mihaita Berbec
4. Sorin Popa
5. A. Furman
6. Damien Gaboriau
7. Cyril Houdayer
8. J. Peterson

Citations in the corpus

• A CLASS OF SUPERRIGID GROUP VON NEUMANN ALGEBRAS (2010, methodology): “Following [Ioa07] we consider the “tensor length deformation” θρ : M → M , which is roughly defined as θρ(Fug) = ρFug when g ∈ Γ and F ∈ L∞(XI 0 ) only depends on n variables in I.”
• Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa) (2006, background): “in this talk and they are applied in [29], in the study of amalgamated free products. These ideas may lead to many more applications in von Neumann algebra and ergodic theory (see e.g. the new papers [28, 58] written since this talk was given). 1The crossed product von Neumann algebra L∞(X,µ)⋊G cont”
• Classification and rigidity for von Neumann algebras (2012, background): “ry (see [Po06b, Section 6]) . Next, we review severalconstructions of malleable deformations. Example 3.4. A malleable deformation for Bernoulli actions related to the one in [Po03] was introduced in [Io06a]. Note that if (B,τ) is a tracial von Neumann algebra and I is a set, then one can construct ”
• W*‐superrigidity for group von Neumann algebras of left–right wreath products (2012, methodology): “We use the following variant, due to [8], of Popa’s malleable deformation for Bernoulli crossed products.”
• Rigidity for von Neumann algebras and their invariants (2010, background): “= τ(a)1. When t→0, we get a deformation of the identity on L∞(X) ⋊Γ that is at the heart of [Po03, Po04]. A variant of the malleable deformation for Bernoulli actions is the tensor length deformation [Io06]. Indeed, given a base probability space (X0,µ0) and a countable set I, put (X,µ) = (X0,µ0)I a”
• On the superrigidity of malleable actions with spectral gap (2006, background): “5.1, 5.2 allows us to prove that factors B ⋊ Λ associated to Bernoulli actions Λ y (B,τ) = (B0,τ0)Λ, with an arbitrary finite von Neumann algebra B0 6= C as base, are prime for any non-amenable Λ (see [I2] for related rigidity results on such factors). Note that Lemmas 5.1, 5.2 show that if Λ is an I”