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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 expectation ..."
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Cited by 65 (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].
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 29 (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.
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 < ..."
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Cited by 13 (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
Coherent states of the qcanonical commutation relations
 455–471. OLA BRATTELI AND PALLE E.T. JORGENSEN
, 1994
"... Abstract. For the qdeformed canonical commutation relations a(f)a †(g) = (1 − q) 〈f, g〉1I + q a †(g)a(f) for f, g in some Hilbert space H we consider representations generated from a vector Ω satisfying a(f)Ω = 〈f, ϕ〉Ω, where ϕ ∈ H. We show that such a representation exists if and only if ‖ϕ ‖ ≤ ..."
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Cited by 9 (6 self)
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Abstract. For the qdeformed canonical commutation relations a(f)a †(g) = (1 − q) 〈f, g〉1I + q a †(g)a(f) for f, g in some Hilbert space H we consider representations generated from a vector Ω satisfying a(f)Ω = 〈f, ϕ〉Ω, where ϕ ∈ H. We show that such a representation exists if and only if ‖ϕ ‖ ≤ 1. Moreover, for ‖ϕ ‖ < 1 these representations are unitarily equivalent to the Fock representation (obtained for ϕ = 0). On the other hand representations obtained for different unit vectors ϕ are disjoint. We show that the universal C*algebra for the relations has a largest proper, closed, twosided ideal. The quotient by this ideal is a natural qanalogue of the Cuntz algebra (obtained for q = 0). We discuss the Conjecture that, for d < ∞, this analogue should, in fact, be equal to the Cuntz algebra itself. In the limiting cases q = ±1 we determine all irreducible representations of the relations, and characterize those which can be obtained via coherent states.
Hypercontractivity in noncommutative holomorphic spaces
 Commun. Math. Phys
, 2005
"... ABSTRACT. We prove an analog of Janson’s strong hypercontractivity inequality in a class of noncommutative “holomorphic ” algebras. Our setting is the qGaussian algebras Γq associated to the qFock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1, 1]. We construct subalgebras Hq ⊂ Γq, a qSeg ..."
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Cited by 8 (6 self)
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ABSTRACT. We prove an analog of Janson’s strong hypercontractivity inequality in a class of noncommutative “holomorphic ” algebras. Our setting is the qGaussian algebras Γq associated to the qFock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1, 1]. We construct subalgebras Hq ⊂ Γq, a qSegalBargmann transform, and prove Janson’s strong hypercontractivity L 2 (Hq) → L r (Hq) for r an even integer. 1.
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
The Free Quon Gas Suffers Gibbs' Paradox
, 1993
"... . We consider the Statistical Mechanics of systems of particles satisfying the qcommutation relations recently proposed by Greenberg and others. We show that although the commutation relations approach Bose (resp. Fermi) relations for q ! 1 (resp. q ! \Gamma1), the partition functions of free gases ..."
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Cited by 2 (1 self)
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. We consider the Statistical Mechanics of systems of particles satisfying the qcommutation relations recently proposed by Greenberg and others. We show that although the commutation relations approach Bose (resp. Fermi) relations for q ! 1 (resp. q ! \Gamma1), the partition functions of free gases are independent of q in the range \Gamma1 ! q ! 1. The partition functions exhibit Gibbs' Paradox in the same way as a classical gas without a correction factor 1=N ! for the statistical weight of the Nparticle phase space, i.e. the Statistical Mechanics does not describe a material for which entropy, free energy, and particle number are extensive thermodynamical quantities. PACS Classification: 12.90.+b, 05.30.d, 03.65.Fd 1 FB Physik, Universitat Osnabruck, Postfach 4469, D4500 Osnabruck, Germany. 2 Electronic mail: reinwer@dosuni1.rz.UniOsnabrueck.DE 1. Introduction In series of papers [Gr2,Gr1,Moh] O.W. Greenberg and collaborators have suggested a new way of interpolating betwee...
Stability of the C ∗algebra associated with the twisted CCR.
, 2001
"... Yurii Samoilenko The universal enveloping C ∗algebra Aµ of twisted canonical commutation relations is considered. It is shown that for any µ ∈ (−1, 1) the C ∗algebra Aµ is isomorphic to the C ∗algebra A0 generated by partial isometries ti, t ∗ i, i = 1,..., d satisfying the relations t ∗ i tj = δ ..."
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Yurii Samoilenko The universal enveloping C ∗algebra Aµ of twisted canonical commutation relations is considered. It is shown that for any µ ∈ (−1, 1) the C ∗algebra Aµ is isomorphic to the C ∗algebra A0 generated by partial isometries ti, t ∗ i, i = 1,..., d satisfying the relations t ∗ i tj = δij(1 − ∑ k<i tkt ∗ k), tjti = 0, i ̸ = j. It is proved that Fock representation of Aµ is faithful.
DETERMINANTS AND INVERSION OF GRAM MATRICES IN FOCK REPRESENTATION OF {qkl} CANONICAL COMMUTATION RELATIONS AND APPLICATIONS TO HYPERPLANE ARRANGEMENTS AND QUANTUM GROUPS. PROOF OF AN EXTENSION OF ZAGIER’S CONJECTURE
, 2003
"... 1 Multiparametric quon algebras, Focklike representations and determinants 1 1.1 qijcanonical commutation relations................... 1 1.2 The algebra f............................... 2 ..."
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1 Multiparametric quon algebras, Focklike representations and determinants 1 1.1 qijcanonical commutation relations................... 1 1.2 The algebra f............................... 2