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35
Cocycle and orbit equivalence superrigidity for malleable actions of wrigid groups
"... Abstract. We prove that if a countable discrete group Γ is wrigid, i.e. it contains an infinite normal subgroup H with the relative property (T) (e.g. Γ = SL(2, Z) ⋉ Z 2, or Γ = H × H ′ with H an infinite Kazhdan group and H ′ arbitrary), and V is a closed subgroup of the group of unitaries of a f ..."
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Cited by 34 (7 self)
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Abstract. We prove that if a countable discrete group Γ is wrigid, i.e. it contains an infinite normal subgroup H with the relative property (T) (e.g. Γ = SL(2, Z) ⋉ Z 2, or Γ = H × H ′ with H an infinite Kazhdan group and H ′ arbitrary), and V is a closed subgroup of the group of unitaries of a finite separable von Neumann algebra (e.g. V countable discrete, or separable compact), then any Vvalued measurable cocycle for a measure preserving action Γ � X of Γ on a probability space (X, µ) which is weak mixing on H and smalleable (e.g. the Bernoulli action Γ � [0,1] Γ) is cohomologous to a group morphism of Γ into V. We use the case V discrete of this result to prove that if in addition Γ has no nontrivial finite normal subgroups then any orbit equivalence between Γ � X and a free ergodic measure preserving action of a countable group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ ≃ Λ. There has recently been increasing interest in the study of measure preserving actions of groups on (nonatomic) probability spaces up to orbit equivalence (OE), i.e. up to isomorphisms of probability spaces taking the orbits of one action onto the orbits of
Free Quasifree States
, 1997
"... To a real Hilbert space and a oneparameter group of orthogonal transformations we associate a C∗algebra which admits a free quasifree state. This construction is a freeprobability analog of the construction of quasifree states on the CAR and CCR algebras. We show that under certain conditions, o ..."
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Cited by 33 (7 self)
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To a real Hilbert space and a oneparameter group of orthogonal transformations we associate a C∗algebra which admits a free quasifree state. This construction is a freeprobability analog of the construction of quasifree states on the CAR and CCR algebras. We show that under certain conditions, our C∗algebras are simple, and the free quasifree states are unique. The corresponding von Neumann algebras obtained via the GNS construction are free analogs of the ArakiWoods factors. Such von Neumann algebras can be decomposed into free products of other von Neumann algebras. For nontrivial oneparameter groups, these von Neumann algebras are type III factors. In the case the oneparameter group is nontrivial and almostperiodic, we show that Connes’ Sd invariant completely classifies these algebras.
KMS states and complex multiplication
 the proceedings of the Abel Symposium
, 2005
"... The following problem in operator algebra has been open for several years. Problem 1.1. For some number field K (other than Q) exhibit an explicit quantum statistical mechanical system (A, σt) with the following properties: (1) The partition function Z(β) is the Dedekind zeta function of K. ..."
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Cited by 23 (6 self)
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The following problem in operator algebra has been open for several years. Problem 1.1. For some number field K (other than Q) exhibit an explicit quantum statistical mechanical system (A, σt) with the following properties: (1) The partition function Z(β) is the Dedekind zeta function of K.
Representation theory and numerical AFinvariants  The representations and centralizers of certain states on O_d
, 1999
"... ..."
Amalgamated free product over Cartan subalgebra
 Pacific J. Math. 191
, 1999
"... Let A ⊇ D ⊆ B be two von Neumann algebras together with a common Cartan subalgebra. Then the amalgamated free product M = A ∗D B with respect to the unique conditional expectations from A, B onto D can be considered. In our previous paper [U1], the questions of its factoriality and type classificati ..."
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Cited by 11 (3 self)
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Let A ⊇ D ⊆ B be two von Neumann algebras together with a common Cartan subalgebra. Then the amalgamated free product M = A ∗D B with respect to the unique conditional expectations from A, B onto D can be considered. In our previous paper [U1], the questions of its factoriality and type classification
The role of type III factors in quantum field theory
 Reports in Mathematical Physics 55
, 2005
"... One of von Neumann’s motivations for developing the theory of operator algebras and his and Murray’s 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possi ..."
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Cited by 11 (0 self)
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One of von Neumann’s motivations for developing the theory of operator algebras and his and Murray’s 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e., factors of type I in the terminology of Murray and von Neumann, are perfectly adequate. In relativistic quantum field theory (RQFT), on the other hand, factors of type III occur naturally. The same holds true in quantum statistical mechanics of infinite systems. In this brief review some physical consequences of the type III property of the von Neumann algebras corresponding to localized observables in RQFT and their difference from the type I case will be discussed. The cumulative effort of many people over more than 30 years has established a remarkable uniqueness result: The local algebras in RQFT are generically isomorphic to the unique, hyperfinite type III1 factor in Connes ’ classification of 1973. Specific theories are characterized by the net structure of the collection of these isomorphic algebras for different spacetime regions, i.e., the way they are embedded into each other. John von Neumann was the father of the Hilbert space formulation of quantum mechanics [1] that has been the basis of almost all mathematically rigorous investigations of the theory to this day. We start by recalling the main concepts and explaining some notations.
Amenable discrete quantum groups
 J. Math. Soc. Japan
"... Abstract. Z.J. Ruan has shown that several amenability conditions are all equivalent in the case of discrete Kac algebras. In this paper we extend this work to the case of discrete quantum groups with a quite different method. That is, we show that a discrete quantum group, where we do not assume i ..."
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Cited by 9 (2 self)
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Abstract. Z.J. Ruan has shown that several amenability conditions are all equivalent in the case of discrete Kac algebras. In this paper we extend this work to the case of discrete quantum groups with a quite different method. That is, we show that a discrete quantum group, where we do not assume its unimodularity, has an invariant mean if and only if it satisfies a certain condition, which is called strongVoiculescu amenability in the case of Kac algebras. 1.
KMS states of quasifree dynamics on Pimsner algebras
 J. Funct. Anal
"... Abstract. A continuous oneparameter group of unitary isometries of a rightHilbert C*bimodule induces a quasifree dynamics on the CuntzPimsner C*algebra of the bimodule and on its Toeplitz extension. The restriction of such a dynamics to the algebra of coefficients of the bimodule is trivial, an ..."
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Cited by 8 (5 self)
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Abstract. A continuous oneparameter group of unitary isometries of a rightHilbert C*bimodule induces a quasifree dynamics on the CuntzPimsner C*algebra of the bimodule and on its Toeplitz extension. The restriction of such a dynamics to the algebra of coefficients of the bimodule is trivial, and the corresponding KMS states of the ToeplitzCuntzPimsner and CuntzPimsner C*algebras are characterized in terms of traces on the algebra of coefficients. This generalizes and sheds light onto various earlier results about KMS states of the gauge actions on Cuntz algebras, CuntzKrieger algebras, and crossed products by endomorphisms. We also obtain a more general characterization, in terms of KMS weights, for the case in which the inducing isometries are not unitary, and accordingly, the restriction of the quasifree dynamics to the algebra of coefficients is nontrivial.
Cocycle and orbit equivalence superrigidity for Bernoulli actions of Kazhdan groups. arXiv: math
, 2006
"... Abstract. We prove that if a countable discrete group Γ contains an infinite normal subgroup with the relative property (T) (e.g. Γ = SL(2, Z) ⋉ Z 2, or Γ = H × H ′ with H an infinite Kazhdan group and H ′ arbitrary) and V is a closed subgroup of the group of unitaries of a finite von Neumann algeb ..."
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Cited by 8 (1 self)
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Abstract. We prove that if a countable discrete group Γ contains an infinite normal subgroup with the relative property (T) (e.g. Γ = SL(2, Z) ⋉ Z 2, or Γ = H × H ′ with H an infinite Kazhdan group and H ′ arbitrary) and V is a closed subgroup of the group of unitaries of a finite von Neumann algebra (e.g. V countable discrete, or separable compact), then any Vvalued measurable cocycle for a Bernoulli Γaction is cohomologous to a group morphism of Γ into V. We use this result to prove that if in addition Γ has no nontrivial finite normal subgroups, then any orbit equivalence between a Bernoulli Γaction and a free ergodic measure preserving action of some group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ ≃ Λ. Bernoulli actions of Kazhdan groups were shown in ([P2,3]) to have sharp rigidity properties, to the extent that both the group and the action (up to conjugacy) can be completely recovered from the isomorphism class of the associated group von Neumann algebra. When applied to algebra isomorphisms coming from orbit equivalence (OE)
Compact abelian group actions on injective factors
 J. Funct. Anal
, 1992
"... Abstract. We classify compact abelian group actions on injective type III factors up to conjugacy, which completes the final step of classification of compact abelian group actions on injective factors. The purpose of this paper is to provide a classification, up to conjugacy, of actions of a (separ ..."
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Cited by 7 (4 self)
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Abstract. We classify compact abelian group actions on injective type III factors up to conjugacy, which completes the final step of classification of compact abelian group actions on injective factors. The purpose of this paper is to provide a classification, up to conjugacy, of actions of a (separable) compact abelian group on injective factors of type III (Theorem 3.1). Studying automorphism groups has been a powerful and fruitful approach to