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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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Cited by 11 (6 self)
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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].
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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Cited by 4 (4 self)
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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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Cited by 1 (1 self)
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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.
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.
OPTIMAL ULTRACONTRACTIVITY FOR RDIAGONAL DILATION SEMIGROUPS
, 708
"... ABSTRACT. This paper contains sharp estimates for the smalltime behaviour of a natural class of oneparameter semigroups in free probability theory. We prove that the free OrnsteinUhlenbeck semigroup Ut, when restricted to the free SegalBargmann (holomorphic) space H0 introduced in [K] and [Bi1], ..."
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ABSTRACT. This paper contains sharp estimates for the smalltime behaviour of a natural class of oneparameter semigroups in free probability theory. We prove that the free OrnsteinUhlenbeck semigroup Ut, when restricted to the free SegalBargmann (holomorphic) space H0 introduced in [K] and [Bi1], is ultracontractive with optimal bound ‖Ut: H 2 0 → H ∞ 0 ‖ ∼ t −1. This was shown, as an upper bound, in [KS]; the lower bound is our main theorem here. These results are extended to a large class of noncommutative holomorphic spaces generated by Rdiagonal operators in a W ∗probability space. A surprising corollary is the fact that these holomorphic spaces (including H0) are not complex interpolation scale (even in the finiterank setting), contra to their commutative analogues. 1.
Submitted to the Annales de l’Institut Henri Poincaré Probabilités et Statistiques Twoparameter Noncommutative Central Limit Theorem ∗
"... Abstract. In 1992, Speicher showed the fundamental fact that the probability measures playing the role of the classical Gaussian in the various noncommutative probability theories (viz. fermionic probability, Voiculescu’s free probability, and qdeformed probability of Bo˙zejko and Speicher) all ar ..."
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Abstract. In 1992, Speicher showed the fundamental fact that the probability measures playing the role of the classical Gaussian in the various noncommutative probability theories (viz. fermionic probability, Voiculescu’s free probability, and qdeformed probability of Bo˙zejko and Speicher) all arise as the limits in a generalized Central Limit Theorem. The latter concerns sequences of noncommutative random variables (elements of a ∗algebra equipped with a state) drawn from an ensemble of pairwise commuting or anticommuting elements, with the respective limiting distributions determined by the average value of the commutation coefficients. In this paper, we derive a more general form of the Central Limit Theorem in which the pairwise commutation coefficients are arbitrary real numbers. The classical Gaussian statistics now undergo a secondparameter refinement as a result of controlling for the first and the second moments of the commutation coefficients. An application yields the random matrix models for the (q, t)Gaussian statistics, which were recently shown to have rich connections to operator algebras, special functions, orthogonal polynomials, mathematical physics, and random matrix theory.