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
August 31: Therese Basa Landry: Compact quantum metric spaces from free probability
(UC Santa Barbara)
We initiate the study of quantum metric space structures for operator algebras arising from free probability, namely q-Gaussians and free Gibbs laws. Length functions on groups have been well-studied as a means of producing spectral triples which give rise to compact quantum metric spaces. For our first main result, we define compact quantum metric spaces for q-Gaussians using length-like functions by the same mathod as developed by Ozawa and Rieffel for spectral triples coming
from length functions and applied by others to hyperbolic groups, quantum groups of rapid decay, free products, and free graph algebras. Motivated by free transport results for free Gibbs laws, we describe a universal way of defining Lip-norms in terms of a generating set, which behaves well under changes of coordinates. As our construction can be applied to any given self-adjoint tuple (x_1,\dots,x_d) generating a unital C*-algebra A, our methods may be of wider interest for constructing
quantum metric spaces. This is joint work with David Jekel.
September 14: Tim Austin: Entropy and large deviations for random unitary representations
(University of Warwick)
This talk will be an introduction to “almost periodic entropy”. This quantity is defined for positive definite functions on a countable group, or more generally for positive functionals on a separable C*-algebra. It is an analog of Lewis Bowen’s “sofic entropy” from ergodic theory. This analogy extends to many of its properties but some important differences also emerge. I will not assume any prior knowledge about sofic entropy. After setting up the basic definition, I will focus on the special case of finitely generated free groups, about which the most is known. For free groups, results include a large deviations principle in a fairly strong topology for uniformly random representations. This, in turn, offers a new proof of the Collins-Male theorem on strong convergence of independent tuples of random unitary matrices, and a large deviations principle for operator norms to accompany that theorem.
September 21: Paola Zurlo: Fermionic optimal transport
(Universita degli Studi Aldo Moro, Bari)
We consider optimal transport between quantum dynamical systems on \mathbb{Z}_2– graded von Neumann algebras. In the usual, non-graded setting, Wasserstein distances between quantum systems can be defined in terms of transport plans on tensor products involving the commutant of one of the algebras. We briefly recall this construction and describe its extension to the graded setting, where the usual commutant and tensor product are replaced by the twisted commutant and the Fermi tensor product. The main point of the construction is a correspondence between fermionic and usual transport plans. More precisely, using cyclic representations associated with transport plans and the Klein isomorphisms induced by the gradings, we establish a one-to-one correspondence between fermionic transport plans and graded transport plans in the usual tensor product. This correspondence allows the Wasserstein distances in the fermionic setting to be related to the corresponding distances in the non-graded setting, and in particular yields their metric properties.