About
The Probabilistic Operator Algebras Seminar (POAS) is an online seminar organized by Dan-Virgil Voiculescu that features talks on recent advances in free probability, operator algebras, random matrices, and related topics. It meets usually weekly from 9:00-10:30am Pacific time. This site is managed by David Jekel. Please contact [email protected] for more information or to be added to the email list.
Upcoming Talks
December 1: Ben Hayes: Selfless inclusions of C*-algebras
(University of Virginia)
I will talk about joint work with Kunnawalkam Elayavalli, Robert, and Patchell. In it, we introduce and study a natural notion of selflessness for inclusions of C*-probability spaces, which in particular necessitates all intermediate C*-algebras to be selfless in the sense of Robert. We identify natural sources of selfless inclusions in the realms of \mathcal{Z}-stable and free product C*-algebras. As an application of this, we prove selflessness for a new family of C*-probability spaces outside the regime of free products and group C*-algebras. These include the reduced free unitary compact quantum groups.
January 12 : Marek Bożejko: A Poisson-Type Operator Deformed by Generalized Fibonacci Numbers and Its Combinatorial Moment Formula
(University of Wrocław)
We introduce a two-parameter deformation of the classical Poisson distribution from the viewpoint of noncommutative probability theory, by defining a (q,t)-Poisson type operator (random variable) on the (q,t)-Fock space. From the analogous viewpoint of the classical Poisson limit theorem in probability theory, we are naturally led to a family of orthogonal polynomials, which we call the (q,t)-Charlier polynomials. These generalize the q-Charlier polynomials of Saitoh Yoshida and reflect deeper combinatorial symmetries through the additional deformation parameter t .A central feature of this paper is the derivation of a combinatorial moment formula of the (q,t)-Poisson type operator and the (q,t)-Poisson distribution. This is accomplished by means of a card arrangement technique, which encodes set partitions together with crossing and nesting statistics. The resulting expression naturally exhibits a duality between these statistics, arising from a structure rooted in generalized Fibonacci numbers. Our approach provides a concrete framework where methods in combinatorics and theory of orthogonal polynomials are used to investigate the probabilistic properties arising from the (q,t)-deformation.
January 26: Dimitri Shlyakhtenko: Strong 1-boundedness via non-microstates free entropy techniques
(UCLA)
In this joint work with Ben Major, we show that for a tracial von Neumann algebra M, a variety of decomposition properties (such as having a Cartan subalgebra, a non-trivial tensor product decomposition, or property Gamma) imply the existence of an amenable subalgebra N (in M, or possibly in its ultrapower) and a generating set X for M, such that the algebra generated by the
semicircular perturbation of the set X actually contains M itself. Using properties of Voiculescu’s non-microstates free entropy and a recent result of Jekel and Pi, this gives a short proof of strong 1-boundedness (in the sense of Jung) for such algebras.
February 2: David Jekel and Brent Nelson: Strong convergence of general Gaussian matrices to operator-valued semicirculars
(University of Copenhagen, Michigan State University)
In joint work with Yoonkyeong Lee and Jennifer Pi, we establish a framework for weak and strong convergence of matrix models to operator-valued semicircular systems parametrized by operator-valued covariance matrices \eta = (\eta_{i,j})_{i,j \in I}. Non-commutative polynomials are replaced by covariance polynomials that can involve iterated applications of \eta_{i,j}, leading to the notion of covariance laws. We give sufficient conditions for weak and strong convergence of general Gaussian random matrices and deterministic matrices to a B-valued semicircular family and generators of the base algebra B. In particular, we obtain operator-valued strong convergence for continuously weighted Gaussian Wigner matrices, such as Gaussian band matrices with a continuous cutoff, and we construct natural strongly convergent matrix models for interpolated free group factors.
February 9: Martin Peev: Renormalising Singular PDEs Driven by q-Gausssian Noises
(Imperial College London)
When attempting to construct QFTs that include Fermions using the methods of Stochastic Quantisation, one is naturally forced to consider noncommutative stochastic PDEs. I shall show how to formulate SPDEs driven by noncommutative noises in terms of algebra-valued singular PDEs. Furthermore, I will describe how one can renormalise singular products appearing in such equations, using the example of free probability and q-Gaussian noises. Finally, I will show how one can derive operator insertion estimates, which are crucial in solving singular SPDEs, such as the \Phi_2^4-equation, by changing the topology on the algebra of q-Gaussian. This talk is based on joint work with Ajay Chandra and Martin Hairer.