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63
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
qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation ..."
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Cited by 64 (2 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
Operator algebras and conformal field theory  III. Fusion of positive energy representations of LSU(N) using bounded operators
, 1998
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Nets of subfactors
, 1994
"... A subtheory of a quantum field theory specifies von Neumann subalgebras A(O) (the ‘observables’ in the spacetime region O) of the von Neumann algebras B(O) (the ‘fields ’ localized in O). Every local algebra being a (type III1) factor, the inclusion A(O) ⊂ B(O) is a subfactor. The assignment of th ..."
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Cited by 51 (4 self)
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A subtheory of a quantum field theory specifies von Neumann subalgebras A(O) (the ‘observables’ in the spacetime region O) of the von Neumann algebras B(O) (the ‘fields ’ localized in O). Every local algebra being a (type III1) factor, the inclusion A(O) ⊂ B(O) is a subfactor. The assignment of these local subfactors to the spacetime regions is called a ‘net of subfactors’. The theory of subfactors is applied to such nets. In order to characterize the ‘relative position ’ of the subtheory, and in particular to control the restriction and induction of superselection sectors, the canonical endomorphism is studied. The crucial observation is this: the canonical endomorphism of a local subfactor extends to an endomorphism of the field net, which in turn restricts to a localized endomorphism of the observable net. The method allows to characterize, and reconstruct, local extensions B of a given theory A in terms of the observables. Various nontrivial examples are given. Several results go beyond the quantum field theoretical application.
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
Stochastic Schrödinger equations
 J. Phys. A: Math. Gen
"... A derivation of stochastic Schrödinger equations is given using quantum filtering theory. We study an open system in contact with its environment, the electromagnetic field. Continuous observation of the field yields information on the system: it is possible to keep track in real time of the best es ..."
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Cited by 27 (6 self)
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A derivation of stochastic Schrödinger equations is given using quantum filtering theory. We study an open system in contact with its environment, the electromagnetic field. Continuous observation of the field yields information on the system: it is possible to keep track in real time of the best estimate of the system’s quantum state given the observations made. This estimate satisfies a stochastic Schrödinger equation, which can be derived from the quantum stochastic differential equation for the interaction picture evolution of system and field together. Throughout the paper we focus on the basic example of resonance fluorescence. 1
An introduction to quantum filtering
, 2006
"... Abstract. This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation ..."
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Cited by 24 (13 self)
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Abstract. This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation
Algebraic orbifold conformal field theories
 Proceedings of National Academy of Sci. USA 97
, 2000
"... Abstract. We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categ ..."
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Cited by 22 (5 self)
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Abstract. We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also show that the irreducible representations of orbifolds of rank one lattice vertex operator algebras give rise to unitary modular categories and determine the corresponding modular matrices, which has been conjectured for some time. Cosets and orbifolds are two methods of producing new two dimensional conformal field theories from given ones (cf. [MS]). In [X2, 3,4, 5], unitary coset conformal field theories are formulated in the algebraic quantum field theory framework and such a formulation is used to solve many questions beyond the reach of other approaches.
An analogue of the KacWakimoto formula and black hole conditional entropy
 Commun. Math. Phys
"... Abstract. A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by Kac ..."
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Cited by 21 (10 self)
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Abstract. A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by KacWakimoto within the context of representations of certain affine Lie algebras. Our formula is model independent and its version in general Quantum Field Theory applies to black hole thermodynamics. The relative free energy between two thermal equilibrium states associated with a black hole turns out to be proportional to the variation of the conditional entropy in different superselection sectors, where the conditional entropy is defined as the ConnesStœrmer entropy associated with the DHR localized endomorphism representing the sector. The constant of proportionality is half of the Hawking temperature. As a consequence the relative free energy is quantized proportionally to the logarithm of a rational number, in particular it is equal to a linear function the logarithm of an integer once the initial state or the final state is taken fixed.