Abstracts

We study matrices whose entries are free or exchangeable noncommutative elements in some tracial W*-probability space. We provide quantitative estimates of their convergence to some operator-valued semicircular elements. Many random block matrices fit well in the framework of matrices with exchangeable entries. For instance, we obtain explicit rates of convergence for the limiting spectral distribution of self-adjoint Kronecker, Wigner and Wishart random matrices in independent or correlated blocks. Our approach relies on a noncommutative extension of the Lindeberg method and operator-valued Gaussian interpolation techniques.

Joint work with Guillaume Cébron.

We explore the interplay between free probability theory and other noncommutative probabilistic frameworks through the lens of noncommutative central limit theorems. We show how the original argument of Speicher can be adapted or generalized to produce a variety of known (and some new) noncommutative Gaussian laws, which have interesting connections to special functions, combinatorics, and mathematical physics. Parts of this talk are based on recent joint work with W. Ejsmont.

We adapt the theory of Loewner chains to non-commutative functions in the operator-valued upper half-plane over a $C^*$-algebra $\mathcal{B}$.  We define an $\mathcal{A}$-valued Loewner chain as a subordination chain $(F_t)_{t \geq 0}$ of self-maps of the $\mathcal{A}$-valued upper half-plane, such that each $F_t$ is the reciprocal Cauchy transform of some $\mathcal{A}$-valued non-commutative law.  Our first main result is that normalized Loewner chains which are Lipschitz with respect to $t$ correspond precisely to solutions of Loewner's evolution equation with respect to some vector field.  This is a direct generalization of a theorem of Bauer in the scalar-valued setting.  To achieve the bijection, we must interpret the differentiation with respect to time in a certain distributional sense, which is suitable for Lipschitz functions taking values in arbitrary Banach spaces.  We also describe the relationship between Loewner chains and monotone independence in the operator-valued setting as Schleissinger has done in the scalar-valued case.

In the theory of free probability, an operator $a$ is called R-diagonal if its $*$-distribution coincides with the $*$-distribution of a product of the form $u\cdot p$, where the sets $\{u,u^*\}$ and $\{p,p^*\}$ are freely independent and $u$ is a unitary distributed according to the Haar measure on the circle. It is due to this free factorization property that the class of R-diagonal operators constitutes a particularly well-behaved class of non-normal operators and their distributions yield answers to maximization problems involving free entropy. Bi-free probability theory was originated by Voiculescu as an extension of the free setting and involves the simultaneous study of left and right action of algebras on reduced free product spaces. In this talk, by utilising the combinatorial description of bi-free probability developed by Charlesworth, Nelson and Skoufranis, we will present the bi-free analogue of R-diagonal operators, namely bi-R-diagonal pairs of operators, and discuss a number of their properties that are of interest within the bi-free framework.

The eigenvalue distributions of various random matrix models show a deterministic behavior when their size tends to infinity. Free probability theory teaches us that these limits can often be described as the spectral distributions of certain noncommutative random variables and provides a powerful machinery to deal with them.

Indeed, analytic tools of operator-valued free probability theory were in recent years combined successfully with algebraic linearization techniques in order to compute – at least numerically – the limit of eigenvalue distributions for noncommutative polynomials and rational functions in tuples of asymptotically free random matrices.

With this approach, however, it remains as a challening problem to detect regularity properties of those limiting distributions, such as absence of atoms, Hölder continuity of their cumulative distribution functions, and absolute continuity. These questions are of particular interest, if one leaves the regime of asymptotic freeness and deals with more general distributions in the realm of Voiculescu's free analogues of entropy and Fisher information.

In my talk, which is based on joint work with Marwa Banna, Roland Speicher, and Sheng Yin, I will give a survey on those developments.

In 2012-2013, in a joint work with J. A. Mingo, we showed that unitarily invariant random matrices are asymptotically free from their transposes. The talk will present some recent developments of this result, with emphasis on GUE and Wishart ensembles.

Stochastic maps are a generalization of functions in that they assign to each point in the domain a probability measure on the codomain. In this talk we will discuss the category of stochastic maps. In particular, we will explore diagramatically formulating the notion of a disintegration of a positive measure and transfering this to the category of $C^\ast$-algebras utilizing the existence of a contravariant functor. Disintegrations in the non-commutative setting are related to reversible processes in quantum information theory and conditional probabilities in non-commutative probability.  This is joint work with Arthur Parzygnat (UConn).

In the context of non-commutative geometry, on the one hand free spheres (Wang), and more generally, partially commutative spheres (defined by Weber-Speicher) may be defined. In a similar way, related affine quantum isometry groups may be defined as well. On another hand, Goswami and Bhowmick have defined the notion of compact quantum group of Riemannian isometry, and have proved (with Banica) that, in the case of free spheres (in a suitable context), affin quantum isometry group and quantum group of Riemannian isometry match. The two purposes of this talk will be the following ones: firstly, introduce the definitions of quantum isometry groups and explain the context we are working with, and secondly, introduce my results about the matching between affine and Riemannian quantum isometry groups in the context of a slightly enlarged class of partially commutative spheres.

We prove  the existence and uniqueness of a discrete nonnegative harmonic function for a random walk satisfying finite range, centering and ellipticity conditions, killed when leaving a globally Lipschitz domain in $\mathbb{Z}^d$. Our method is based on a systematic use of comparison arguments and discrete potential-theoretical techniques.

This is a joint workwith Sami Mustapha.

The notions of free entropy developed by Voiculescu have had a profound impact on free probability and operator algebras.  In this talk, joint work with Ian Charlesworth will be discuss that generalizes the notion of non-microstate free entropy to the bi-free setting.  In particular, the notions of free derivations, conjugate variables, Fisher information, and entropy are extended to handle pairs of algebras from which many interesting results, complications, and questions arise.

The most general quantum measurements in quantum channels are described by quantum channels, and intrinsic uncertainties emerge from the state-channel interaction.   In this talk, we give uncertainty relations for arbitrary quantum channels in terms of generalized quasi-metric adjusted skew information which was defined by the author for arbitrary operators. We illustrate the uncertainty relations by explicit examples.

This talk is based on a recent joint-work with Tobias Mai and Roland Speicher. The free field is the skew field (aka division ring) which extends the noncommutative ring of polynomials in several formal variables with some universal property. In other words, it is the skew field of rational functions generated by non-commuting formal variables. In this talk, we address the question when a tuple of operators in a finite von Neumann algebra can generate the free field. It turns out that the quantity $\Delta$ introduced by A. Connes and D. Shlyakhtenko in their paper, $L^2$-homology for von Neumann algebras, gives a description for such operators. Actually, for a tuple of operators, the maximality of the associated $\Delta$ is equivalent to the realization of the free field by these operators.

On one side, $\Delta$ is related to many concepts, for examples, free entropy dimension and dual system, in free probability; and on the other side, the realization of free field leads to some Atiyah property. Therefore, a lot of interesting consequences can be extracted from this equivalence of the maximality of $\Delta$ and the realization of the free field.

In 1629 Albert Girard gave formulae for the power sums of several commuting variables in terms of the elementary symmetric functions; his result was subsequently often attributed to Newton.

Over a century later Edward Waring proved that an arbitrary symmetric polynomial in finitely many commuting variables could be expressed as a polynomial in the elementary symmetric functions of those variables.

In 1939 Margarete Wolf showed that there is no finite algebraic basis for the algebra of symmetric functions in $d \gt 1$ non-commuting variables, so there is no finite set of ‘elementary symmetric functions’ in the non-commutative case.

Nevertheless, Jim Agler, John McCarthy and I have proved analogues of Girard's and Waring's theorems for symmetric functions in two non-commuting variables. We find three free polynomials $f, g, h$ in two non-commuting indeterminates $x, y$ such that every symmetric polynomial in $x$ and $y$ can be written as a polynomial in $f, g, h$ and $1/g$.  In particular, power sums can be written explicitly in terms of $f$ ,$g$ and $h$. To do this we developed the notion of a non-commutative manifold.