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UNIQUE ERGODICITY OF FREE SHIFTS AND SOME OTHER AUTOMORPHISMS OF C ∗ –ALGEBRAS
, 2006
"... Abstract. A notion of unique ergodicity relative to the fixed–point subalgebra is defined for automorphisms of unital C ∗ –algebras. It is proved that the free shift on any reduced amalgamated free product C ∗ –algebra is uniquely ergodic relative to its fixed–point subalgebra, as are autormorphisms ..."
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Abstract. A notion of unique ergodicity relative to the fixed–point subalgebra is defined for automorphisms of unital C ∗ –algebras. It is proved that the free shift on any reduced amalgamated free product C ∗ –algebra is uniquely ergodic relative to its fixed–point subalgebra, as are autormorphisms of reduced group C ∗ –algebras arising from certain automorphisms of groups. A generalization of Haagerup’s inequality, yielding bounds on the norms of certain elements in reduced amalgamated free product C ∗ –algebras is proved. 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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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.
RESOLVENTS OF RDIAGONAL OPERATORS
"... ABSTRACT. We consider the resolvent (λ − a) −1 of any Rdiagonal operator a in a II1factor. Our main theorem (Theorem 1.1) gives a universal asymptotic formula for the norm of such a resolvent. En route to its proof, we calculate the Rtransform of the operator λ − c  2 where c is Voiculescu’s ci ..."
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Cited by 2 (2 self)
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ABSTRACT. We consider the resolvent (λ − a) −1 of any Rdiagonal operator a in a II1factor. Our main theorem (Theorem 1.1) gives a universal asymptotic formula for the norm of such a resolvent. En route to its proof, we calculate the Rtransform of the operator λ − c  2 where c is Voiculescu’s circular operator, and give an asymptotic formula for the negative moments of λ − a  2 for any Rdiagonal a. We use a mixture of complex analytic and combinatorial techniques, each giving finer information where the other can give only coarse detail. In particular, we introduce partition structure diagrams in Section 4, a new combinatorial structure arising in free probability. 1.
ENUMERATION OF NONCROSSING PAIRINGS ON BIT STRINGS
"... ABSTRACT. A noncrossing pairing on a bitstring matches 1s and 0s in a manner such that the pairing diagram is nonintersecting. By considering such pairings on arbitrary bitstrings 1 n1 0 m1 ..."
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ABSTRACT. A noncrossing pairing on a bitstring matches 1s and 0s in a manner such that the pairing diagram is nonintersecting. By considering such pairings on arbitrary bitstrings 1 n1 0 m1
THE STRONG ASYMPTOTIC FREENESS OF HAAR AND DETERMINISTIC MATRICES
, 2011
"... Abstract. In this paper, we are interested in sequences of qtuple of N ×N random matrices having a strong limiting distribution (i.e. given any noncommutative polynomial in the matrices and their conjugate transpose, its normalized trace and its norm converge). We start with such a sequence having ..."
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Abstract. In this paper, we are interested in sequences of qtuple of N ×N random matrices having a strong limiting distribution (i.e. given any noncommutative polynomial in the matrices and their conjugate transpose, its normalized trace and its norm converge). We start with such a sequence having this property, and we show that this property pertains if the qtuple is enlarged with independent unitary Haar distributed random matrices. Besides, the limit of norms and traces in noncommutative polynomials in the enlarged family can be computed with reduced free product construction. This extends results of one author (C. M.) and of Haagerup and Thorbjørnsen. We also show that aptuple of independent orthogonal and symplectic Haar matrices have a strong limiting distribution, extending a recent result of Schultz.
Theory (60), and Combinatorics (05). Specifically: I work in random matrices and
"... free probability theory; functional inequalities such as logarithmic Sobolev inequalities and Haagerup inequalities in spaces of regular functions; stochastic analysis; and enumerative combinatorics of set partitions. I have three published papers, three submitted for publication, and many other pro ..."
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free probability theory; functional inequalities such as logarithmic Sobolev inequalities and Haagerup inequalities in spaces of regular functions; stochastic analysis; and enumerative combinatorics of set partitions. I have three published papers, three submitted for publication, and many other projects in progress. I was awarded an NSF grant DMS0701162 (20072010). I have coauthors and collaborators at institutions in the U.S., Canada, the U.K., France, and Poland. I have had a measure of experience supervising research: in 2007, I ran a very successful Research Experience for Undergraduates project, which has produced two papers whose results are described below; in addition, I am currently supervising a PhD student at MIT, whose work is in random matrix theory and combinatorics. 1. OVERVIEW A unifying theme to most of my work is Gaussian structures. Recall that the standard normal law (Gaussian distribution) is given by γ(dx) = (2π) −1/2 exp(−x 2 /2)dx. By Gaussian structures I mean mathematical objects that owe their properties to some (apparent or hidden) connection with the measure γ. Most of my work has to do with either L pspaces of Gaussian random variables or Gaussiantype random matrices. For example, the following kinds of objects interest me:
kdivisible noncrossing partitions ∗
"... We derive a formula for the moments and the free cumulants of the multiplication of k free random variables in terms of kequal and kdivisible noncrossing partitions. This leads to a new simple proof for the bounds of the rightedge of the support of the free multiplicative convolution µ ⊠k, given ..."
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We derive a formula for the moments and the free cumulants of the multiplication of k free random variables in terms of kequal and kdivisible noncrossing partitions. This leads to a new simple proof for the bounds of the rightedge of the support of the free multiplicative convolution µ ⊠k, given by Kargin in [5], which show that the support grows at most linearly with k. Moreover, this combinatorial approach generalize the results of Kargin since we do not require the convolved measures to be identical. We also give further applications, such as a new proof of the limit theorem of Sakuma and Yoshida [11]. Keywords: Free probability; Free multiplicative convolution; Noncrossing partitions.
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.
A NONCROSSING BASIS FOR NONCOMMUTATIVE INVARIANTS OF SL(2,C)
, 905
"... Abstract. Noncommutative invariant theory is a generalization of the classical invariant theory of the action of SL(2,C) on binary forms. The dimensions of the spaces of invariant noncommutative polynomials coincide with the numbers of certain noncrossing partitions. We give an elementary combinator ..."
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Abstract. Noncommutative invariant theory is a generalization of the classical invariant theory of the action of SL(2,C) on binary forms. The dimensions of the spaces of invariant noncommutative polynomials coincide with the numbers of certain noncrossing partitions. We give an elementary combinatorial explanation of this fact by constructing a noncrossing basis of the homogeneous components. Using the theory free stochastic measures this provides a combinatorial proof of the MolienWeyl formula in this setting. Invariant theory has played a major role in 19th century mathematics. It has seen a revival in the last decades and one of the recent generalizations is noncommutative invariant theory. The study of noncommutative invariants of SL(n,C) has been initiated by Almkvist, Dicks, Formanek and Kharchenko [6, 5, 2], see [1] for a survey. An approach using Young tableaux was realized by Teranishi [16] and the symbolic method was adapted from the classical to the noncommutative setting by Tambour [15]. The latter provides the ground on which we establish a natural basis of the noncommutative invariants which is in bijection with certain noncrossing partitions. It arose after computer experiments and subsequent consulting of Sloane’s database [13]. This bijection is applied to provide a