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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 76 (16 self)
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Abstract. We prove that any isomorphism θ: M0 ≃ M of group measure space II1
Von Neumann Algebras
, 2009
"... The purpose of these notes is to provide a rapid introduction to von Neumann algebras which gets to the examples and active topics with a minimum of technical baggage. In this sense it is opposite in spirit from the treatises of ..."
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Cited by 70 (2 self)
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The purpose of these notes is to provide a rapid introduction to von Neumann algebras which gets to the examples and active topics with a minimum of technical baggage. In this sense it is opposite in spirit from the treatises of
The Conformal spin and statistics theorem
 Commun. Math. Phys
, 1996
"... During the recent years Conformal Quantum Field Theory has become a widely studied topic, especially on a low dimensional spacetime, because of physical motivations such as the desire of a better understanding of twodimensional critical phenomena, and also for its rich mathematical structure provi ..."
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Cited by 64 (23 self)
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During the recent years Conformal Quantum Field Theory has become a widely studied topic, especially on a low dimensional spacetime, because of physical motivations such as the desire of a better understanding of twodimensional critical phenomena, and also for its rich mathematical structure providing remarkable connections with different areas such as Hopf algebras, low dimensional topology, knot invariants, subfactors
Doob’s inequality for noncommutative martingales
 J. reine angew. Math
"... Abstract. Let 1 ≤ p < ∞ and (xn)n∈N be a sequence of positive elements in a noncommutative Lp space and (En)n∈N be an increasing sequence of conditional expectations, then En(xn) ∥ ≤ cp xn∥ ..."
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Cited by 46 (27 self)
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Abstract. Let 1 ≤ p < ∞ and (xn)n∈N be a sequence of positive elements in a noncommutative Lp space and (En)n∈N be an increasing sequence of conditional expectations, then En(xn) ∥ ≤ cp xn∥
Noncommutative Burkholder/Rosenthal inequalities
 Ann. Probab
, 2000
"... Abstract. We show norm estimates for the sum of independent random variables in noncommutative Lp spaces for 1 < p < ∞ following previous work by the authors. These estimates generalize Rosenthal’s inequalities in the commutative case. Among other applications, we derive a formula for pnorm of the ..."
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Cited by 46 (25 self)
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Abstract. We show norm estimates for the sum of independent random variables in noncommutative Lp spaces for 1 < p < ∞ following previous work by the authors. These estimates generalize Rosenthal’s inequalities in the commutative case. Among other applications, we derive a formula for pnorm of the eigenvalues for matrices with independent entries, and characterize those symmetric subspaces and unitary ideal spaces which can be realized as subspaces of noncommutative Lp for 2 < p < ∞. 0. Introduction and Notation Martingale inequalities have a long tradition in probability. The applications of the work of Burkholder and his collaborators [B73,?, BDG72, B71a, B71b, BGS71, BG70, B66] ranges from classical harmonic analysis to stochastical differential equations and the geometry of Banach spaces. When proving the estimates for the ‘little square function ’ Burkholder
Mathematical Theory of NonEquilibrium Quantum Statistical Mechanics
, 2002
"... We review and further develop a mathematical framework for nonequilibrium quantum statistical mechanics recently proposed in [JP4, JP5, JP6, Ru3, Ru4, Ru5, Ru6]. In the algebraic formalism of quantum statistical mechanics we introduce notions of nonequilibrium steady states, entropy production and ..."
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Cited by 33 (3 self)
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We review and further develop a mathematical framework for nonequilibrium quantum statistical mechanics recently proposed in [JP4, JP5, JP6, Ru3, Ru4, Ru5, Ru6]. In the algebraic formalism of quantum statistical mechanics we introduce notions of nonequilibrium steady states, entropy production and heat fluxes, and study their properties. Our basic paradigm is a model of a small (finite) quantum system coupled to several independent thermal reservoirs. We exhibit examples of such systems which have strictly positive entropy production.
Classification of local conformal nets. Case c < 1
"... We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of AD2nE6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We f ..."
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Cited by 27 (13 self)
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We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of AD2nE6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We first identify the nets generated by irreducible representations of the Virasoro algebra for c<1 with certain coset nets. Then, by using the classification of modular invariants for the minimal models by CappelliItzyksonZuber and the method of αinduction in subfactor theory, we classify all local irreducible extensions of the Virasoro nets for c<1 and infer our main classification result. As an application, we identify in our classification list certain concrete coset nets studied in the literature.
H.: Equilibrium statistical mechanics of Fermion lattice systems
"... We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the noncommutativity of local algebras for disjoint regions. Our major result is the equivalence of the KMS condition and the variational principle wit ..."
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Cited by 26 (7 self)
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We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the noncommutativity of local algebras for disjoint regions. Our major result is the equivalence of the KMS condition and the variational principle with a minimal assumption for the dynamics and without any explicit assumption on the potential. Its proof applies to spin lattice systems as well, yielding a vast improvement over known results. All formulations are in terms of a C ∗dynamical systems for the Fermion (CAR) algebra A with all or a part of the following assumptions: (I) The interaction is even, namely, the dynamics αt commutes with the evenoddness automorphism Θ. (Automatically satisfied when (IV) is assumed.) (II) The domain of the generator δα of αt contains the set A ◦ of all strictly local elements of A. (III) The set A ◦ is the core of δα.
Construction of Quantum Field Theories with Factorizing SMatrices
, 2007
"... A new approach to the construction of interacting quantum field theories on twodimensional Minkowski space is discussed. In this program, models are obtained from a prescribed factorizing Smatrix in two steps. At first, quantum fields which are localized in infinitely extended, wedgeshaped region ..."
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Cited by 21 (5 self)
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A new approach to the construction of interacting quantum field theories on twodimensional Minkowski space is discussed. In this program, models are obtained from a prescribed factorizing Smatrix in two steps. At first, quantum fields which are localized in infinitely extended, wedgeshaped regions of Minkowski space are constructed explicitly. In the second step, local observables are analyzed with operatoralgebraic techniques, in particular by using the modular nuclearity condition of Buchholz, d’Antoni and Longo. Besides a modelindependent result regarding the ReehSchlieder property of the vacuum in this framework, an infinite class of quantum field theoretic models with nontrivial interaction is constructed. This construction completes a program initiated by Schroer in a large family of theories, a particular example being the SinhGordon model. The crucial problem of establishing the existence of local observables in these models is solved by verifying the modular nuclearity condition, which here amounts to a condition on analytic properties of form factors of observables localized in wedge regions. It is shown that the constructed models solve the inverse scattering problem for the considered class of Smatrices. Moreover, a proof of asymptotic completeness is obtained by explicitly computing total sets of scattering states. The structure of these collision states is found to be in agreement with the heuristic formulae underlying the ZamolodchikovFaddeev algebra.