Dr. John D. Williams Humboldt Fellow für August 2014 - Juli 2016 Universität des Saarlandes Fachrichtung Mathematik Postfach 151150 66041 Saarbrücken Zimmer 304 (Gebäude E 2 4) E-mail:   williams [at] math.uni-sb.de Curriculum Vitae

Research

My main research areas are analysis and probability. I focus on questions in non-commutative probability, random matrices, operator theory and operator algebras.

Publications

• On the Hausdorff Continuity of Free Levy Processes and Free Convolution Semigroups [pdf]
  Accepted for Publication, Journal of Mathematical Analysis and Applications.

  Let \mu denote a Borel probability measure and let {\mut}t>1 denote the free additive convolution semigroup of Nica and Speicher. We show that the support of these measures varies continuously in the Hausdorff metric for t>1. We utilize complex analytic methods and, in particular, a characterization of the absolutely continuous portion of these supports due to Huang.

• B-Valued Free Convolution for Unbounded Operators [pdf]
  Accepted for Publication, Indiana University Journal of Mathematics.

  Consider the $\mathcal{B}$-valued probability space $(\mathcal{A}, E, \mathcal{B})$, where $\mathcal{A}$ is a tracial von Neumann algebra. We extend the theory of operator valued free probability to the algebra of affiliated operators $\tilde{\mathcal{A}}$. For a random variable $X \in \tilde{\mathcal{A}}^{sa}$ we study the Cauchy transform $G_{X}$ and show that the operator algebra $(\mathcal{B} \cup \{X\})''$ can be recovered from this function. In the case where $\mathcal{B}$ is finite dimensional, we show that, when $X, Y \in \tilde{\mathcal{A}}^{sa}$ are assumed to be $\mathcal{B}$-free, the $\mathcal{R}$-transforms are defined on universal subsets of the resolvent and satisfy $$ \mathcal{R}_{X} + \mathcal{R}_{Y} = \mathcal{R}_{X + Y}. $$ Examples indicating a failure of the theory for infinite dimensional $\mathcal{B}$ are provided. Lastly, we show that the class of functions that arise as the Cauchy transform of affiliated operators is, in a natural way, the closure of the set of Cauchy transforms of bounded operators.

• Operator-Valued Jacobi Parameters and Examples of Operator-Valued Distributions [pdf]
  Submitted for Publication.

  We consider B-valued distributions arising from sequences of Jacobi parameters. The class of Meixner distributions are realized through these constructions and certain examples of their free convolution are calculated. Moreover, the authors develop a convenient combinatorial method for calculating the joint distributions of B-free Jacobi distributed random variables utilizing two-color non-crossing pairings. This precipitates several new examples of the convolution operation in this operator-valued setting and allows the construction of a counterexample wherein a free convolution of Jacobi distributed random variables is not itself Jacobi distributed . Additionally, the authors obtain a counting algorithm for certain distinguished subsets of the family of two-color non-crossing pairings using only free probabilistic techniques. Joint with Michael Anshelevich.

• Operator-Valued Monotone Convolution Semigroups [pdf]
  Accepted for publication, Documenta Mathematica.

  In a classic result, Bercovici and Pata showed that a natural bijection between the classical, free and Boolean innitely divisible measures held at the level of limit theorems of triangular arrays. This was later extended to include monotone convolution by Anshelevich and Williams. In recent years, operator-valued versions of free, Boolean and monotone probability have also been developed. In a recent paper, Belinschi, Popa and Vinnikov showed that this bijection holds for the operator-valued versions of free and Boolean probability. In this paper, the authors show that this too extends to operator-valued versions of monotone probability theory. To prove this result, the authors develop the theory of composition semigroups of non-commutative functions and largely recapture Berkson and Porta's classical results on composition semigroups of complex functions in this more general setting. Moreover, Williams' result on the classification of Cauchy transforms for non-commutative functions is extended to the Cauchy transforms associated to more general completely positive maps. Joint with Michael Anshelevich.

• Analytic Function Theory for Operator-Valued Free Probability [pdf]
  Accepted for publication. Crelle's Journal.

  It is a classical result in complex analysis that the class of functions that arise as the Cauchy transform of probability measures may be characterized entirely in terms of their analytic and asymptotic properties. Such transforms are a main object of study in non-commutative probability theory as the function theory encodes information on the probability measures and the various convolution operations. In extending this theory to operator-valued free probability theory, the analogue of the Cauchy transform is a non-commutative function with domain equal to the non-commutative upper-half plane. In this paper, we prove an analogous characterization of the Cauchy transforms, again, entirely in terms of their analytic and asymptotic behavior. We further characterize those functions which arise as the Voiculescu transform of $\boxplus$-infinitely divisible $\mathcal{B}$-valued distributions. As an immediate consequence, this theorem, combined with an existing result of Popa and Vinnikov, may be used to produce analogues of the Nevanlinna representation for non-commutative functions with the appropriate asymptotics. In addition to this, we may define semigroups of completely positive maps associated to infinitely divisible distributions.

• Quantum symmetric states on free product C*-algebras. [pdf]
  Accepted for publication, Transactions of the AMS.

  We introduce quantum symmetric states on universal unital free product C*-algebras of the form B=*_1^\infty A for an arbitrary unital C^*--algebra A, as a generalization of the notion of quantum exchangeable random variables. Extending and building on the proof of the noncommutative de Finetti theorem of K\"ostler and Speicher, we prove a de Finetti type theorem that characterizes quantum symmetric states in terms of amalgamated free products over the tail algebra, and we provide a convenient description of the set of all quantum symmetric states on B in terms of von Neumann algebras generated by homomorphic images of A and the tail algebra. This description allows a characterization of the extreme quantum symmetric states, and we provide some examples of these. Similar results are proved for the subset of tracial quantum symmetric states. The central quantum symmetric states are those for which the tail algebra is in the center of the von Neumann algebra, and we show that the central quantum symmetric states form a Choquet simplex whose extreme points are the free product states, while the tracial central quantum symmetric states form a Choquet simplex whose extreme points are the free product traces. Joint with Ken Dykema and Claus Koestler.

• Limit theorems for monotonic convolution and the Chernoff product formula. [pdf]
  International Mathematics Research Notices. IMRN (2014), no. 11, 2990 - 3021. MR 3214313.

  Bercovici and Pata showed that the correspondence between classically, freely, and Boolean infinitely divisible distributions holds on the level of limit theorems. We extend this correspondence also to distributions infinitely divisible with respect to the additive monotone convolution. Because of non-commutativity of this convolution, we use a new technique based on the Chernoff product formula. We also study this correspondence for multiplicative monotone convolution, where the Bercovici-Pata bijection no longer holds. Joint with Michael Anshelevich.

• An Analogue of Hincin's Characterization of Infinite Divisibility for Operator-Valued Free Probability. [pdf]
  Journal of Functional Analysis, 2014, Vol. 267(1), pp. 1-14.

  In this paper, we utilize the Steinitz Lemma to provide a unified approach to Hincin type infinite divisibility theorems for various probabilistic categories. New results are obtained for operator valued free probability theory.

• A Khintchine Decomposition for Free Probability. [pdf]
  Annals of Probability. Volume 50(5), 2012, 2236--2263.

  In this paper, we show that an arbitrary probability measure that is supported on the real line may be decomposed into the additive sum of an infinitely divisible divisor and infinitely many "prime" divisors. These theorems are in direct analogy with Khintchine's result in classical probability theory. We also prove a number of compactness theorems, in particular, showing that the set of all divisors of a measure is weakly compact up to translation. Analagous results are proven in the multiplicative cases.

• Uniform Convergence and the Free Central Limit Theorem. [pdf]
  Complex Analysis and Operator Theory (23 July 2010), pp. 1-9-9.

  In this paper, we prove a number of superconvergence results for unbounded random variables with regard to the central limit theorem.

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Aktualisiert: 16. September 2015   John Williams

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