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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.
Fidaleo F. Unique mixing of the shift on the C ∗ –algebras generated by the q–canonical commutation relations
 Houston J. Math
"... Abstract. The shift on the C ∗ –algebras generated by the Fock representation of the q–commutation relations has the strong ergodic property of unique mixing, when q  < 1. 1. introduction The q–commutation relations have been studied in the physics literature, see e.g. [8]. These are the relati ..."
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Cited by 1 (0 self)
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Abstract. The shift on the C ∗ –algebras generated by the Fock representation of the q–commutation relations has the strong ergodic property of unique mixing, when q  < 1. 1. introduction The q–commutation relations have been studied in the physics literature, see e.g. [8]. These are the relations aia + j − qa+ j ai = δij1, i, j ∈ Z where −1 ≤ q ≤ 1. This gives an interpolation between the canonical commutation relations (Bosons) when q = 1 and the canonical anticommutation relations (Fermions) when q = −1, while when q = 0 we have freeness (cf. [16]). In [3], (see also [9] and [7]) a Fock representation of these relations was found, giving annihilators ai and their adjoints, the creators a + i, acting on a Hilbert space with a vacuum vector Ω. The C ∗ –algebras and von Neumann algebras generated by sets of these operators or by their real parts ai+a + i have
The von Neumann algebra generated by tgaussians
, 2006
"... We study the tdeformation of gaussian von Neumann algebras. When the number of generators is fixed, it is proved that if t sufficiently close to 1, then these algebras do not depend on t. In the same way, the notion of conditionally free von Neumann algebras often coincides with freeness. 1 ..."
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Cited by 1 (0 self)
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We study the tdeformation of gaussian von Neumann algebras. When the number of generators is fixed, it is proved that if t sufficiently close to 1, then these algebras do not depend on t. In the same way, the notion of conditionally free von Neumann algebras often coincides with freeness. 1
U.F.R des Sciences et Techniques
, 2006
"... Asymptotic matricial models and QWEP property for qArakiWoods algebras ..."
BessisMoussaVillani conjecture and generalized Gaussian random variables
, 2007
"... In this paper we give the solution of BessisMoussaVillani conjecture (BMV) conjecture for the generalized Gaussian random variables G(f) = a(f) + a∗(f), where f is in the real Hilbert space H. The main examples of generalized Gaussian random variables are qGaussian random variables, (−1 ≤ q ≤ 1) ..."
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In this paper we give the solution of BessisMoussaVillani conjecture (BMV) conjecture for the generalized Gaussian random variables G(f) = a(f) + a∗(f), where f is in the real Hilbert space H. The main examples of generalized Gaussian random variables are qGaussian random variables, (−1 ≤ q ≤ 1), related to q −CCR relation and others commutation relations. We will prove that (BMV) conjecture is true for all operators A = G(f), B = G(g); i.e. we will show that the function F (x) = tr(exp(A+ ixB)) is positive definite function on the real line. The case q = 0,i.e. when G(f) are the free Gaussian (Wigner) random variables and the operators A and B are free with respect to the vacuum trace was proved by M.Fannes and D.Petz [23]. 1 Generalized Gaussian Random Variable. Generalized Gaussian random variables, G(f) were introduced in our paper with R.Speicher [16], where the main example was coming from the qCCR relation for q ∈ [−1, 1]: a(f)a∗(g) − qa∗(g)a(f) =< f, g> I,