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65
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 70 (8 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 52 (8 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.
Lectures on graded differential algebras and noncommutative geometry
, 1999
"... These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments. ..."
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Cited by 48 (5 self)
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These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.
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 33 (10 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.
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 30 (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 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 28 (8 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.
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 26 (7 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
Nonsingular transformations and spectral analysis of measures
 Bull. Soc. Math. France
, 1991
"... RÉSUMÉ. — Ce travail approfondit les interactions qui existent entre l’analyse harmonique des mesures et l’étude spectrale des systèmes dynamiques non singuliers. Il est centre ́ sur l’étude de sousgroupes remarquables du cercle, groupes de valeurs propres, groupes de quasiinvariance des mesu ..."
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Cited by 23 (0 self)
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RÉSUMÉ. — Ce travail approfondit les interactions qui existent entre l’analyse harmonique des mesures et l’étude spectrale des systèmes dynamiques non singuliers. Il est centre ́ sur l’étude de sousgroupes remarquables du cercle, groupes de valeurs propres, groupes de quasiinvariance des mesures..., dont les exemples les plus naturels sont définis par des conditions diophantiennes. La conjonction des points de vue permet d’obtenir nombre de résultats nouveaux dans les deux théories, y compris dans des problèmes classique d’analyse de Fourier. ABSTRACT. — This work explores in depth the interactions existing between harmonic analysis of measures and spectral theory of nonsingular dynamical systems. It focuses on the study of some classes of remarkable subgroups of the circle: eigenvalue groups, groups of quasiinvariance of measures..., the most natural examples of which are defined by diophantine conditions. The conjonction of these points of view leads to many new results in both theories, including some classical problems in Fourier analysis. 1.
Extension of the structure theorem of Borchers and its application to halfsided modular inclusions" manuscript, preliminary version
, 1995
"... Abstract. A result of H.W. Wiesbrock is extended from the case of a common cyclic and separating vector for the halfsided modular inclusion N ⊂ M of von Neumann algebras to the case of a common faithful normal semifinite weight and at the same time a gap in Wiesbrock’s proof is filled in. 1. ..."
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Cited by 21 (0 self)
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Abstract. A result of H.W. Wiesbrock is extended from the case of a common cyclic and separating vector for the halfsided modular inclusion N ⊂ M of von Neumann algebras to the case of a common faithful normal semifinite weight and at the same time a gap in Wiesbrock’s proof is filled in. 1.
Type III Factors Arising from CuntzKrieger Algebras
 Proc. Amer. Math. Soc
"... Abstract. We determine the types of factors arising from GNSrepresentations of quasifree KMS states on CuntzKrieger algebras. Applying our result to the CuntzKrieger algebras arising from the boundary actions of some amalgamated free product groups, we also determine the types of the harmonic m ..."
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Cited by 21 (2 self)
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Abstract. We determine the types of factors arising from GNSrepresentations of quasifree KMS states on CuntzKrieger algebras. Applying our result to the CuntzKrieger algebras arising from the boundary actions of some amalgamated free product groups, we also determine the types of the harmonic measures on the boundaries. 1.