Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I

By Sorin Popa. 2003

351 citations recorded by the source corpus.

We consider crossed product II1 factors $M = N\rtimes_{\sigma}G$, with G discrete ICC groups that contain infinite normal subgroups with the relative property (T) and σ trace preserving actions of G on finite von Neumann algebras N that are “malleable” and mixing. Examples are the actions of G by Bernoulli shifts (classical and non-classical) and by Bogoliubov shifts. We prove a rigidity result for isomorphisms of such factors, showing the uniqueness, up to unitary conjugacy, of the position of the group von Neumann algebra L(G) inside M. We use this result to calculate the fundamental group of M, $\mathcal{F}(M)$, in terms of the weights of the shift σ, for $G=\mathbb{Z}^2\rtimes SL(2,\mathbb{Z})$ and other special arithmetic groups. We deduce that for any subgroup S⊂ℝ+* there exist II1 factors M (separable if S is countable or S=ℝ+*) with $\mathcal{F}(M)=S$. This brings new light to a long standing open problem of Murray and von Neumann.

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

View Paper: https://www.semanticscholar.org/paper/824be39e1549793128cd2e8f74f0703a4ea28827

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In Algebra and Number Theory

1. Adrian Ioana
2. S. Vaes
3. Sorin Popa
4. Cyril Houdayer
5. Thomas Sinclair
6. I. Chifan
7. A. Furman
8. J. Peterson

In Applied Mathematics

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

In Mathematical Physics

1. Adrian Ioana
2. S. Vaes
3. Sorin Popa
4. A. Furman
5. Thomas Sinclair
6. Cyril Houdayer
7. I. Chifan
8. J. Peterson

Citations in the corpus

• Cartan Subalgebras in $C^*$-Algebras (2008, background): “Popa has recently (see [37, 38]) constructed and studied a large class of type II1 factors (from Bernoulli actions of groups with property T ) which have a distinguished Cartan subalgebra, unique up to inner conjugacy; moreover, these factors satisfy remarkable rigidity properties: isomorphisms of t”
• A CLASS OF SUPERRIGID GROUP VON NEUMANN ALGEBRAS (2010, methodology): “Nevertheless, apart from the preliminary Section 2, where we also recall the notion of intertwining bimodules [Pop06c], our article is essentially selfcontained and Sections 4, 5 and 6 contain independent results, each having an interest of its own.”
• Ergodic Subequivalence Relations Induced by a Bernoulli Action (2008, background): “In a remarkable series of papers, S. Popa extensively investigated the equivalence relations associated with Bernoulli actions and showed that they have striking rigidity properties [ Po1 -4].”
• Unique prime factorization and bicentralizer problem for a class of type III factors (2015, positive): “For this strategy to work, we prove a generalization of Popa’s intertwining theorem [Po01, Po03] that allows us to ‘intertwine with expectation’ finite von Neumann subalgebras with expectation A ⊂ M into arbitrary von Neumann subalgebras with expectation B ⊂ M inside M (see Theorem 4.3).”
• W*‐superrigidity for group von Neumann algebras of left–right wreath products (2012, methodology): “We first use Popa’s malleable deformation for Bernoulli actions (see [22]) and his spectral gap rigidity (see [25]) to prove that the subalgebra L(Γ × Γ) ⊂ LG is invariant under Δ, up to unitary conjugacy.”