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118
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 153 (21 self)
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Abstract. We prove that any isomorphism θ: M0 ≃ M of group measure space II1
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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Operator Algebras and Conformal Field Theory
 COMMUNICATIONS MATHEMATICAL PHYSICS
, 1993
"... We define and study twodimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite typ ..."
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Cited by 99 (2 self)
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We define and study twodimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite type III 1 factor. The conformal net determined by the algebras of local observables is proven to satisfy Haag duality. The representation of the Moebius group (and presumably of the entire Virasoro algebra) on the vacuum sector of a conformal field theory is uniquely determined by the TomitaTakesaki modular operators associated with its vacuum state and its conformal net. We then develop the theory of Moebius covariant representations of a conformal net, using methods of Doplicher, Haag and Roberts. We apply our results to the representation theory of loop groups. Our analysis is motivated by the desire to find a "backgroundindependent" formulation of conformal field theories.
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 78 (36 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∥
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 68 (9 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.
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.
A short survey of noncommutative geometry
 J. Math. Physics
, 2000
"... We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the RiemannHilbert problem. We discuss at length two issues. The first is the relevance of the paradigm of geom ..."
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Cited by 50 (4 self)
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We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the RiemannHilbert problem. We discuss at length two issues. The first is the relevance of the paradigm of geometric space, based on spectral considerations, which is central in the theory. As a simple illustration of the spectral formulation of geometry in the ordinary commutative case, we give a polynomial equation for geometries on the four dimensional sphere with fixed volume. The equation involves an idempotent e, playing the role of the instanton, and the Dirac operator D. It expresses the gamma five matrix as the pairing between the operator theoretic chern characters of e and D. It is of degree five in the idempotent and four in the Dirac operator which only appears through its commutant with the idempotent. It determines both the sphere and all its metrics with fixed volume form. We also show using the noncommutative analogue of the Polyakov action, how to obtain the noncommutative metric (in spectral form) on the noncommutative tori from the formal naive metric. We conclude on some questions related to string theory. I
Topological sectors and a dichotomy in conformal field theory
 Commun. Math. Phys
"... Let A be a local conformal net of factors on S 1 with the split property. We provide a topological construction of soliton representations of the nfold tensor product A ⊗ · · · ⊗ A, that restrict to true representations of the cyclic orbifold (A ⊗ · · · ⊗ A) Zn. We prove a quantum index the ..."
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Cited by 48 (23 self)
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Let A be a local conformal net of factors on S 1 with the split property. We provide a topological construction of soliton representations of the nfold tensor product A ⊗ · · · ⊗ A, that restrict to true representations of the cyclic orbifold (A ⊗ · · · ⊗ A) Zn. We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then A is not completely rational iff the symmetrized tensor product (A ⊗ A) flip has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of A have a conjugate sector then either A is completely rational or A has uncountably many different irreducible sectors. Thus A is rational iff A is completely rational. In particular, if the µindex of A is finite then A turns out to be strongly additive. By [31], if A is rational then the tensor category of representations of A is automatically modular, namely the braiding symmetry is nondegenerate. In interesting cases, we compute the fusion rules of the topological solitons and show that they determine all twisted sectors of the cyclic orbifold.