Some rigidity results for non-commutative Bernoulli shifts☆

By Sorin Popa. 2006

104 citations recorded by the source corpus.

Abstract We introduce the outer conjugacy invariants S ( σ ) , S s ( σ ) for cocycle actions σ of discrete groups G on type II 1 factors N, as the set of real numbers t > 0 for which the amplification σ t of σ can be perturbed to an action, respectively, to a weakly mixing action. We calculate explicitly S ( σ ) , S s ( σ ) and the fundamental group of σ , F ( σ ) , in the case G has infinite normal subgroups with the relative property (T) (e.g., when G itself has the property (T) of Kazhdan) and σ is an action of G on the hyperfinite II 1 factor by Connes–Stormer Bernoulli shifts of weights { t i } i . Thus, S s ( σ ) and F ( σ ) coincide with the multiplicative subgroup S of R + * generated by the ratios { t i / t j } i , j , while S ( σ ) = Z + * if S = { 1 } (i.e. when all weights are equal), and S ( σ ) = R + * otherwise. In fact, we calculate all the “1-cohomology picture” of σ t , t > 0 , and classify the actions ( σ , G ) in terms of their weights { t i } i . In particular, we show that any 1-cocycle for ( σ , G ) vanishes, modulo scalars, and that two such actions are cocycle conjugate iff they are conjugate. Also, any cocycle action obtained by reducing a Bernoulli action of a group G as above on N = ⊗ ¯ g ∈ G ( M n × n ( C ) , tr ) g to the algebra pNp, for p a projection in N, p ≠ 0 , 1 , cannot be perturbed to a genuine action.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/51b8ba88ee922449b57da5cdf6a3e6a7094fc366

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

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

In Mathematical Physics

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

In Statistical and Nonlinear Physics

1. Adrian Ioana
2. J. Peterson
3. Cyril Houdayer
4. I. Chifan
5. A. Furman
6. S. Vaes
7. Ben Hayes
8. Andrés Navas

Citations in the corpus

• A random matrix approach to the Peterson-Thom conjecture (2020, background): “ments in deformation/rigidity theory go beyond free group factors and apply to von Neumann algebras associated to groups which have a combination of approximation properties and nontrivial cohomology [61, 60, 62, 63, 56, 54, 14], as well as crossed product algebras associated to actions of such grou”
• Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I (2003, result): “Instead, Theorem 0.1 is in line with the rigidity results established over the last 25 years in von Neumann algebra theory ([C1,2], [CJ2], [Po8], [CoHa], [GoNe], [ Po1 –4]) and orbit equivalence ergodic theory ([Zi], [Moo], [GeGo], [Po8], [Ga1,2], [Fu1–3], [MoSh], [Hj], [PoSa], [Po6]).”
• Borel Equivalence Relations (2010, background): “Moreoever the discussion of Borel equivalence relations is organized around the Borel reducibility order, ≤B, rather than notions such as orbit equivalence, as found in say [20], [18], [21], [53], [28], or notions of isomorphism, as discussed in [9].”
• Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups (2005, methodology): “In Section 4 we recall from ([P1-4]) the notion of s-malleable action and the proof that (generalized) Bernoulli actions have this property, then use a deformation/rigidity argument from ([ P1 ,2]) to show that if a group contains a lar ge rigid part then any s-malleable -action satisfies the criter”
• On Ozawa's Property for Free Group Factors (2006, methodology): “was resolved in ([ P2 ,3]) by the discovery of “symmetric paths”, which allows patching intertwiners in a way that makes each incremental intertwiner have range matching the”
• Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups (2006, background): “nn algebra L(Γ) has countable outer automorphism group and countable fundamental group. Without being exhaustive, we cite the following von Neumann rigidity results obtained during the last 25 years: [3, 4, 6, 7, 8, 22, 37, 38, 40]. The first von Neumann strong rigidity theorem, deducing conjugacy of”
• L2-rigidity in von Neumann algebras (2009, background): “This type of rigidity phenomenon (and it’s relative version later introduced by Popa [P4]) has since led to the solution of many old problems in von Neumann algebras and orbit equivalence ergodic theory ([C2], [IPP], [ P3 ], [P4], [P5]).”
• Amalgamated free products of weakly rigid factors and calculation of their symmetry groups (2005, methodology): “lows a rigid inclusion. We’ll now use the deformation 2.2.2 to exploit the relative rigidity of subalgebras Q⊂ M1 ∗B M2 ⊂ M˜1 ∗B M˜2. This “deformation/rigidity” argument is inspired from (4.3-4.8 in [P3] and Sec. 4 in [P6I]) 3.3. Proposition. Let (M1,τ1),(M2,τ2) be finite von Neumann algebras with a”
• On Popa's Cocycle Superrigidity Theorem (2006, result): “In [22] Sorin Popa singles out a very interesting class of groups Ufin.”
• Cocycle Superrigidity for Profinite Actions of Property (T) Groups (2008, background): “te how to construct non orbit equivalent profinite actions for the groups SL n(Z)⋉Zn (n≥2). To do this, we first introduce a new isomorphism invariant for measurable equivalence relations. Note that in [P5], S. Popa studied a related version of this invariant for actions of countable groups on the hyp”
• Bass-Serre rigidity results in von Neumann algebras (2008, methodology): “ors; Intertwining techniques; Spectral gap; Amalgamated free products. 1 2 IONUT CHIFAN AND CYRIL HOUDAYER von Neumann algebra B. Proofs are based on Popa’s deformation/rigidity argument. We refer to [17, 18, 21, 23, 24, 25, 26, 27, 38] for some applications of this theory. For the purpose of this p”
• Rigidity results for wreath product II1 factors (2006, methodology): “Proposition 3.4. in [ Po5 ] then implies that there exists a finite dimensional”
• GROUP ACTIONS ON 1-MANIFOLDS: A LIST OF VERY CONCRETE OPEN QUESTIONS (2017, background): “Despite of its apparent simplicity, this shift action is fundamental in several contexts, and has attracted the attention of many people over the last decades [12, 25, 66, 118].”
• Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa) (2006, background): “⊂ G and an AdG-invariant character β ∈ CharK. Hence, K = {e}, p = 1 and we are done. 10. OUTER CONJUGACY OF W-RIGID GROUP ACTIONS ON THE HYPERFINITE II 1 FACTOR We discuss some of the results of Popa [52] on (cocycle) actions of w-rigid groups on the hyperfinite II 1 factor. As explained in the intro”