Results 1  10
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110
On a class of type II1 factors with Betti numbers invariants
, 2002
"... We prove that a type II1 factor M can have at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class HT of factors M having such Cartan subalgebras A ⊂ M, the Betti numbers of the standard equivalence ..."
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Cited by 175 (29 self)
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We prove that a type II1 factor M can have at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class HT of factors M having such Cartan subalgebras A ⊂ M, the Betti numbers of the standard equivalence relation associated with A ⊂ M ([G2]), are in fact isomorphism invariants for the factors M, β HT n (M), n ≥ 0. The class HT is closed under amplifications and tensor products, with the Betti numbers satisfying β HT n (Mt) = β HT n (M)/t, ∀t> 0, and a Künneth type formula. An example of a factor in the class HT is given by the group von Neumann factor M = L(Z2 ⋊ SL(2, Z)), for which β HT 1 (M) = β1(SL(2, Z)) = 1/12. Thus, Mt ̸ ≃ M, ∀t ̸ = 1, showing that the fundamental group of M is trivial. This solves a long standing problem of R.V. Kadison. Also, our results bring some insight into a recent problem of A. Connes and answer a number of open questions on von Neumann algebras.
Strong rigidity of II1 factors arising from malleable actions of weakly rigid groups, I
"... Abstract. We prove that any isomorphism θ: M0 ≃ M of group measure space II1 ..."
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Cited by 148 (22 self)
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Abstract. We prove that any isomorphism θ: M0 ≃ M of group measure space II1
On the superrigidity of malleable actions with spectral gap
 J. Amer. Math. Soc
"... Abstract. We prove that if a countable group Γ contains a nonamenable subgroup with centralizer infinite and “weakly normal ” in Γ (e.g. if Γ is nonamenable and has infinite center or is a product of infinite groups) then any measure preserving Γaction on a probability space which satisfies certa ..."
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Cited by 79 (7 self)
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Abstract. We prove that if a countable group Γ contains a nonamenable subgroup with centralizer infinite and “weakly normal ” in Γ (e.g. if Γ is nonamenable and has infinite center or is a product of infinite groups) then any measure preserving Γaction on a probability space which satisfies certain malleability, spectral gap and weak mixing conditions is cocycle superrigid. We also show that if Γ � X is an arbitrary free ergodic action of such a group Γ and Λ � Y = T Λ is a Bernoulli action of an arbitrary infinite conjugacy class group, then any isomorphism of the associated II1 factors L ∞ X ⋊Γ ≃ L ∞ Y ⋊Λ comes from a conjugacy of the actions. 1.
Deformation and rigidity for group actions and von Neumann algebras
, 2007
"... We present some recent rigidity results for von Neumann algebras (II1 factors) and equivalence relations arising from measure preserving actions of groups on probability spaces which satisfy a combination of deformation and rigidity properties. This includes strong rigidity results for factors wit ..."
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Cited by 64 (7 self)
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We present some recent rigidity results for von Neumann algebras (II1 factors) and equivalence relations arising from measure preserving actions of groups on probability spaces which satisfy a combination of deformation and rigidity properties. This includes strong rigidity results for factors with calculation of their fundamental group and cocycle superrigidity for actions with applications to orbit equivalence ergodic theory.
An uncountable family of non orbit equivalent actions of Fn
 J. Amer. Math. Soc
, 2005
"... Recall that two ergodic probability measure preserving (p.m.p.) actions σi for i =1, 2 of two countable groups Γi on probability measure standard Borel spaces (Xi,µi) areorbit equivalent (OE) if they define partitions of the spaces into orbits that are isomorphic, more precisely, if there exists a m ..."
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Cited by 53 (16 self)
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Recall that two ergodic probability measure preserving (p.m.p.) actions σi for i =1, 2 of two countable groups Γi on probability measure standard Borel spaces (Xi,µi) areorbit equivalent (OE) if they define partitions of the spaces into orbits that are isomorphic, more precisely, if there exists a measurable, almost everywhere defined isomorphism f: X1 → X2 such that f∗µ1 = µ2 and the Γ1orbit of µ1almost every x ∈ X1 is sent by f onto the Γ2orbit of f(x). The theory of orbit equivalence, although underlying the “group measure space construction ” of Murray and von Neumann [MvN36], was born with the work of H. Dye who proved, for example, the following striking result [Dy59]: Any two ergodic p.m.p. free actions of Γ1 � Z and Γ2 � � j∈N Z/2Z are orbit equivalent. Through a series of works, the class of groups Γ2 satisfying Dye’s theorem gradually increased until it achieved the necessary and sufficient condition: Γ2 is infinite amenable [OW80]. In particular, all infinite amenable groups produce one and only one ergodic p.m.p. free action up to orbit equivalence (see also [CFW81] for a more
Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups
, 2006
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A new approach to induction and imprimitivity results
 J. Funct. Anal
"... In the framework of locally compact quantum groups, we provide an induction procedure for corepresentations as well as coactions on C ∗algebras. We prove imprimitivity theorems that unify the existing theorems for actions and coactions of groups. We essentially use von Neumann algebraic techniques. ..."
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Cited by 33 (0 self)
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In the framework of locally compact quantum groups, we provide an induction procedure for corepresentations as well as coactions on C ∗algebras. We prove imprimitivity theorems that unify the existing theorems for actions and coactions of groups. We essentially use von Neumann algebraic techniques. 1