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29
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].
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
Popa M.: Feynman Diagrams and Wick products associated with qFock space
 Proc. Natl. Acad. Sci. USA 100
, 2003
"... Abstract. It is shown that if one keeps track of crossings, Feynman diagrams can be used to compute qWick products and normal products in terms of each other. 1. ..."
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Cited by 11 (1 self)
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Abstract. It is shown that if one keeps track of crossings, Feynman diagrams can be used to compute qWick products and normal products in terms of each other. 1.
Factoriality of qgaussian von neumann algebras
 Comm. Math. Phys
"... We prove that the von Neumann algebras generated by n qGaussian elements, are factors for n � 2. 1 ..."
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Cited by 8 (1 self)
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We prove that the von Neumann algebras generated by n qGaussian elements, are factors for n � 2. 1
Rosenthal type inequalities for free chaos
, 2005
"... Let A denote the reduced amalgamated free product of a family ..."
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Cited by 8 (5 self)
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Let A denote the reduced amalgamated free product of a family
H∞ FUNCTIONAL CALCULUS AND SQUARE FUNCTIONS ON Noncommutative L^Pspaces
, 2006
"... In this work we investigate semigroups of operators acting on noncommutative L pspaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and H ∞ functional calculus. We discuss several examples of noncommutative diffusion semig ..."
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Cited by 8 (3 self)
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In this work we investigate semigroups of operators acting on noncommutative L pspaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and H ∞ functional calculus. We discuss several examples of noncommutative diffusion semigroups. This includes Schur multipliers, qOrnsteinUhlenbeck semigroups, and the noncommutative Poisson semigroup on free groups.
Standard dilations of qcommuting tuples
 Indian Statistical Institute, Bangalore preprint
, 2003
"... Here we study dilations of qcommuting tuples. In [BBD] the authors gave the correspondence between the two standard dilations of commuting tuples and here these results have been extended to qcommuting tuples. We are able to do this when qcoefficients ‘qij ’ are of modulus one. We introduce ‘maxi ..."
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Cited by 5 (1 self)
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Here we study dilations of qcommuting tuples. In [BBD] the authors gave the correspondence between the two standard dilations of commuting tuples and here these results have been extended to qcommuting tuples. We are able to do this when qcoefficients ‘qij ’ are of modulus one. We introduce ‘maximal qcommuting subspace ’ of a ntuple of operators and ‘standard qcommuting dilation’. Our main result is that the maximal qcommuting subspace of the standard noncommuting dilation of qcommuting tuple is the ‘standard qcommuting dilation’. We also introduce qcommuting Fock space as the maximal qcommuting subspace of full Fock space and give a formula for projection operator onto this space. This formula for projection helps us in working with the completely positive maps arising in our study. The first version of the Main Theorem (Theorem 19) of the paper for normal tuples using some tricky norm estimates and then use it to prove the general version of this theorem.
THE KERNEL OF FOCK REPRESENTATIONS OF WICK ALGEBRAS WITH BRAIDED OPERATOR OF COEFFICIENTS
, 2001
"... It is shown that the kernel of the Fock representation of a certain Wick algebra with braided operator of coefficients T, T   ≤ 1, coincides with the largest quadratic Wick ideal. Improved conditions on the operator T for the Fock inner product to be strictly positive are given. 1. Introduction ..."
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Cited by 3 (1 self)
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It is shown that the kernel of the Fock representation of a certain Wick algebra with braided operator of coefficients T, T   ≤ 1, coincides with the largest quadratic Wick ideal. Improved conditions on the operator T for the Fock inner product to be strictly positive are given. 1. Introduction. The problem of positivity of the Fock space inner product is central in the study of the Fock representation of Wick algebras (see [2], [3], [5], [6]). The paper [6] presents several conditions on the coefficients of the Wick algebra for the Fock inner product to be positive. If the operator of coefficients of
EXTENSIONS OF POSITIVE DEFINITE FUNCTIONS ON FREE GROUPS
, 2005
"... Abstract. An analogue of Krein’s extension theorem is proved for operatorvalued positive definite functions on free groups. The proof gives also the parametrization of all extensions by means of a generalized type of Szegö parameters. One singles out a distinguished completion, called central, which ..."
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Cited by 3 (1 self)
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Abstract. An analogue of Krein’s extension theorem is proved for operatorvalued positive definite functions on free groups. The proof gives also the parametrization of all extensions by means of a generalized type of Szegö parameters. One singles out a distinguished completion, called central, which is related to quasimultiplicative positive definite functions. An application is given to factorization of noncommutative polynomials. 1.
Derived noncommutative continuous Bernoulli shifts
 In preparation
"... Abstract: We introduce a noncommutative extension of TsirelsonVershik’s noises [TV98, Tsi04], called (noncommutative) continuous Bernoulli shifts. These shifts encode stochastic independence in terms of commuting squares, as they are familiar in subfactor theory [Pop83, GHJ89]. Such shifts are, i ..."
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Cited by 2 (1 self)
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Abstract: We introduce a noncommutative extension of TsirelsonVershik’s noises [TV98, Tsi04], called (noncommutative) continuous Bernoulli shifts. These shifts encode stochastic independence in terms of commuting squares, as they are familiar in subfactor theory [Pop83, GHJ89]. Such shifts are, in particular, capable of producing Arveson’s product system of type I and type II [Arv03]. We investigate the structure of these shifts and prove that the von Neumann algebra of a (scalarexpected) continuous Bernoulli shift is either finite or of type III. The role of (‘classical’) Gstationary flows for TsirelsonVershik’s noises is now played by cocycles of continuous Bernoulli shifts. We show that these cocycles provide an operator algebraic notion for Lévy processes. They lead, in particular, to units and ‘logarithms ’ of units in Arveson’s product systems [Kös04a]. Furthermore, we introduce (noncommutative) white noises, which are operator algebraic versions of Tsirelson’s ‘classical ’ noises. We give examples coming from probability, quantum probability and from Voiculescu’s theory of free probability [VDN92]. Our main result is a bijective correspondence between additive and unital shift cocycles. For the proof of the correspondence we develop tools which are of interest on their own: noncommutative extensions of stochastic Itô integration, stochastic logarithms and exponentials.