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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 expec ..."
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Cited by 120 (7 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 o ..."
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Cited by 88 (36 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
Positive representations of general commutation relations allowing wick ordering
 FUNCT ANAL
, 1995
"... We consider the problem of representing in Hilbert space commutation relations of the form aia ∗ j = δij1 + ∑ kℓ T kℓ ij a ∗ ℓ ak, where the T kℓ ij are essentially arbitrary scalar coefficients. Examples comprise the qcanonical commutation relations introduced by Greenberg, Bozejko, and Speicher, ..."
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Cited by 35 (7 self)
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We consider the problem of representing in Hilbert space commutation relations of the form aia ∗ j = δij1 + ∑ kℓ T kℓ ij a ∗ ℓ ak, where the T kℓ ij are essentially arbitrary scalar coefficients. Examples comprise the qcanonical commutation relations introduced by Greenberg, Bozejko, and Speicher, and the twisted canonical (anti)commutation relations studied by Pusz and Woronowicz, as well as the quantum group SνU(2). Using these relations, any polynomial in the generators ai and their adjoints can uniquely be written in “Wick ordered form ” in which all starred generators are to the left of all unstarred ones. In this general framework we define the Fock representation, as well as coherent representations. We develop criteria for the natural scalar product in the associated representation spaces to be positive definite, and for the relations to have representations by bounded operators in a Hilbert space. We characterize the relations between the generators ai (not involving a ∗ i) which are compatible with the basic relations. The relations may also be interpreted as defining a noncommutative differential calculus. For generic coefficients T kℓ ij, however, all differential forms of degree 2 and higher vanish. We exhibit conditions for this not to be the case, and relate them to the ideal structure of the Wick algebra, and conditions of positivity. We show that the differential calculus is compatible with the involution iff the coefficients T define a representation of the braid group. This condition is also shown to imply improved bounds for the positivity of the Fock representation. Finally, we study the KMS states of the group of gauge transformations defined by aj ↦ → exp(it)aj.
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 33 (12 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.
Noncommutative Symmetric Functions III: Deformations Of Cauchy And Convolution Algebras
"... This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple qanalogue of the shuffle product, which has unexpec ..."
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Cited by 23 (8 self)
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This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple qanalogue of the shuffle product, which has unexpected connections with quantum groups, hyperplane arrangements, and certain questions in theoretical physics (the quon algebra).
Some estimates for nonmicrostates free entropy dimension, with applications to qsemicircular families
 Int. Math. Res. Notices
"... Abstract. We give an general estimate for the nonmicrostates free entropy dimension δ ∗ (X1,..., Xn). If X1,..., Xn generate a diffuse von Neumann algebra, we prove that δ ∗ (X1,..., Xn) ≥ 1. In the case that X1,...,Xn are qsemicircular variables as introduced by Bozejko and Speicher and q 2 n &l ..."
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Cited by 23 (5 self)
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Abstract. We give an general estimate for the nonmicrostates free entropy dimension δ ∗ (X1,..., Xn). If X1,..., Xn generate a diffuse von Neumann algebra, we prove that δ ∗ (X1,..., Xn) ≥ 1. In the case that X1,...,Xn are qsemicircular variables as introduced by Bozejko and Speicher and q 2 n < 1, we show that δ ∗ (X1,...,Xn)> 1. We also show that for q  < √ 2−1, the von Neumann algebras generated by a finite family of qGaussian random variables satisfy a condition of Ozawa and are therefore solid: the relative commutant of any diffuse subalgebra must be hyperfinite. In particular, when these algebras are factors, they are prime and do not have property Γ. 1. Introduction. In [2], Bozejko and Speicher introduced a deformation of a free semicircular family of Voiculescu [12, 13], parameterized by a number q ∈ [−1, 1]. Their qsemicircular family X1,...,Xn is represented on a deformed Fock space and generates a finite von Neumann algebra, which is nonhyperfinite for n ≥ 2 and q ∈ (−1, 1) [7]. For q = 0, X1,...,Xn are
Appell polynomials and their relatives
 Int. Math. Res. Not
, 2004
"... ABSTRACT. This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can be considered as natural analogs of the Appell polynomials among polynomials in noncommuting va ..."
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Cited by 20 (1 self)
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ABSTRACT. This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can be considered as natural analogs of the Appell polynomials among polynomials in noncommuting variables. They also fit well into the framework of free probability. For the free Appell polynomials, a number of combinatorial and “diagram ” formulas are proven, such as the formulas for their linearization coefficients. An explicit formula for their generating function is obtained. These polynomials are also martingales for free Lévy processes. For more general free Sheffer families, a necessary condition for pseudoorthogonality is given. Another family investigated are the KailathSegall polynomials. These are multivariate polynomials, which share with the Appell polynomials nice combinatorial properties, but are always orthogonal. Their origins lie in the Fock space representations, or in the theory of multiple stochastic integrals. Diagram formulas are proven for these polynomials as well, even in the qdeformed case. 1.
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 19 (2 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.
A qdeformation of the Gauss distribution
 J. Math. Phys
, 2000
"... The qdeformed commutation relation aa #  qa # a = 11 for the harmonic oscillator is considered with q # [1, 1]. An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of a + a # ..."
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Cited by 19 (2 self)
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The qdeformed commutation relation aa #  qa # a = 11 for the harmonic oscillator is considered with q # [1, 1]. An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of a + a # in the vacuum state is explicitly calculated. This distribution is to be regarded as the natural q deformation of the Gaussian. 1995 PACS numbers: 02.50.Cw, 05.40.+j, 03.65.Db, 42.50.Lc 1991 MSC numbers: 81S25, 33D90, 81Q10 1 1
qLévy processes
 J. Reine Angew. Math
, 2004
"... ABSTRACT. We continue the investigation of the Lévy processes on a qdeformed full Fock space started in [1]. First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a chaotic representation property in terms of multiple integrals ..."
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Cited by 17 (1 self)
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ABSTRACT. We continue the investigation of the Lévy processes on a qdeformed full Fock space started in [1]. First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a chaotic representation property in terms of multiple integrals with respect to diagonal measures, in the style of Nualart and Schoutens. We define stochastic integration with respect to these processes, and calculate their combinatorial stochastic measures. Finally, we show that they generate infinite von Neumann algebras. 1.