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13
Strong Haagerup inequalities for free Rdiagonal elements
 J. FUNCT. ANAL
, 2007
"... In this paper, we generalize Haagerup’s inequality [H] (on convolution norm in the free group) to a very general context of Rdiagonal elements in a tracial von Neumann algebra; moreover, we show that in this “holomorphic” setting, the inequality is greatly improved from its originial form. We give ..."
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In this paper, we generalize Haagerup’s inequality [H] (on convolution norm in the free group) to a very general context of Rdiagonal elements in a tracial von Neumann algebra; moreover, we show that in this “holomorphic” setting, the inequality is greatly improved from its originial form. We give combinatorial proofs of two important special cases of our main result, and then generalize these techniques. En route, we prove a number of moment and cumulant estimates for Rdiagonal elements that are of independent interest. Finally, we use our strong Haagerup inequality to prove a strong ultracontractivity theorem, generalizing and improving the one in [Bi2].
Free infinite divisibility for QGaussians
"... Abstract. We prove that the qGaussian distribution is freely infinitely divisible for all q ∈ [0, 1]. 1. ..."
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Abstract. We prove that the qGaussian distribution is freely infinitely divisible for all q ∈ [0, 1]. 1.
Rdiagonal dilation semigroups
"... ABSTRACT. This paper addresses extensions of the complex OrnsteinUhlenbeck semigroup to operator algebras in free probability theory. If a1,..., ak are ∗free Rdiagonal operators in a II1 factor, then Dt(ai1 · · · ain) = e −nt ai1 · · · ain defines a dilation semigroup on the nonselfadjoint ..."
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ABSTRACT. This paper addresses extensions of the complex OrnsteinUhlenbeck semigroup to operator algebras in free probability theory. If a1,..., ak are ∗free Rdiagonal operators in a II1 factor, then Dt(ai1 · · · ain) = e −nt ai1 · · · ain defines a dilation semigroup on the nonselfadjoint operator algebra generated by a1,..., ak. We show that Dt extends (in two different ways) to a semigroup of completely positive maps on the von Neumann algebra generated by a1,..., ak. Moreover, we show that Dt satisfies an optimal ultracontractive property: �Dt: L 2 → L ∞ � ∼ t −1 for small t> 0. 1.
T.: Hypercontractivity for logsubharmonic functions
"... Abstract. We prove strong hypercontractivity (SHC) inequalities for logarithmically subharmonic functions on R n and different classes of measures: Gaussian measures on R n, symmetric Bernoulli, symmetric uniform probability measures, as well as their convolutions µ1 ∗ µ2 if one of the convolved mea ..."
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Abstract. We prove strong hypercontractivity (SHC) inequalities for logarithmically subharmonic functions on R n and different classes of measures: Gaussian measures on R n, symmetric Bernoulli, symmetric uniform probability measures, as well as their convolutions µ1 ∗ µ2 if one of the convolved measures is compactly supported. A slightly weaker strong hypercontractivity property holds for any symmetric measure on R. For all 1–dimensional measures for which we know the (SHC) holds, we prove a Log–Sobolev inequality which is stronger than the classical one. This result is extended to all dimensions for compactlysupported measures. 1.
THE (q, t)GAUSSIAN PROCESS
, 2012
"... The (q, t)Fock space Fq,t(H), introduced in this paper, is a deformation of the qFock space of Bożejko and Speicher. The corresponding creation and annihilation operators now satisfy the commutation relation aq,t(f)aq,t(g) ∗ − q aq,t(g)∗aq,t(f) = 〈f, g〉H tN, with N denoting the usual number op ..."
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Cited by 3 (2 self)
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The (q, t)Fock space Fq,t(H), introduced in this paper, is a deformation of the qFock space of Bożejko and Speicher. The corresponding creation and annihilation operators now satisfy the commutation relation aq,t(f)aq,t(g) ∗ − q aq,t(g)∗aq,t(f) = 〈f, g〉H tN, with N denoting the usual number operator, and generate a Hilbert space representation of the ChakrabartiJagannathan deformed quantum oscillator algebra. The moments of the deformed field operator sq,t(h): = aq,t(h) + aq,t(h) ∗, the present analogue of the Gaussian random variable, are encoded by the joint statistics of crossings and nestings in pair partitions. The restriction of the vacuum expectation state to the (q, t)Gaussian algebra is not tracial for t 6 = 1. The q = 0 < t specialization yields a new singleparameter deformation of the full Boltzmann Fock space of free probability. This refinement is particularly natural as the probability measure associated with the deformed semicircular element turns out to be encoded via the RogersRamanujan continued fraction, the tAiry function, the tCatalan numbers of CarlitzRiordan, and the firstorder statistics of the reduced Wigner process.
STRONG HAAGERUP INEQUALITIES WITH OPERATOR COEFFICIENTS
, 903
"... Abstract. We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, Hd denotes the subspace of the von Neumann algebra of a free group FI spanned by the words of length d in the generators (but not their inverses), then we provide in this paper an explicit upper bound on ..."
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Abstract. We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, Hd denotes the subspace of the von Neumann algebra of a free group FI spanned by the words of length d in the generators (but not their inverses), then we provide in this paper an explicit upper bound on the norm on Mn(Hd), which improves and generalizes previous results by KempSpeicher (in the scalar case) and Buchholz and ParcetPisier (in the nonholomorphic setting). Namely the norm of an element of the form P i=(i1,...,i d) ai ⊗ λ(gi1... gi d) is less than 4 5 √ e(‖M0 ‖ 2 + · · · + ‖Md ‖ 2) 1/2, where M0,..., Md are d+1 different blockmatrices naturally constructed from the family (ai) i∈I d for each decomposition of I d ≃ I l ×I d−l with l = 0,..., d. It is also proved that the same inequality holds for the norms in the associated noncommutative Lp spaces when p is an even integer, p ≥ d and when the generators of the free group are more generally replaced by ∗free Rdiagonal operators. In particular it applies to the case of free circular operators. We also get inequalities for the nonholomorphic case, with a rate of growth of order d+1 as for the classical Haagerup inequality. The proof is of combinatorial nature and is based on the definition and study of a symmetrization process for partitions.