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
November 9: Luis Daniel Abreu: A computable wandering and tracelike vector for modular orbits in the Bergman space
(University of Vienna)
We construct a function \Phi such that the orbit under the representation of \Gamma = PSL(2;mathbb{Z}) is an orthonormal basis for the Bergman space with weight q = 12. Moreover, we show that \Phi is effectively computable as a holomorphic function on the upper half-plane (in the precise sense of computable analysis), by providing an effective procedure. This constructs a wandering and tracelike vector for PSL(2;mathbb{Z}), whose abstract existence was proved by Sir Vaughan Jones in his last paper, where the corresponding construction was left as a problem. The construction of the function, which can also be represented as an orthonormal wavelet in the half-line, reproduces the method used in the signal approximation practice: the off-diagonal decay of the Gramian operator allows to control its restriction to nested finite subsets of PSL(2;mathbb{Z}), where it becomes a finite matrix and can be inverted, leading to a convergent sum where all objects and bounds are computable in the cyclotomic field generated by the 12th-root of unity over the rationals. Since modular actions on the Bergman kernel enter the construction, \Phi displays Bergman and modular reminiscences, despite appearing to be modular itself. As suggested by Jones, it provides a computable implementing vector for the abstract anti-isomorphism between the von Neumann algebra M_12(\Gamma) and its commutant. With minor adaptations, the construction works for any cocompact Fuchsian group, but the more involved proof of computability is only executed for PSL(2;mathbb{Z}).
November 16: Dima Shlyakhtenko: OpenAI’s proof of isomorphism of free group factors, and further developments
(UCLA)
We will present a version of OpenAI’s proof of free group factor isomorphism. A strengthening of the proof also shows that the reduced C*-algebras of free groups have strong reduced free products absorption properties. Moreover, C*-algebras generated by n=2, 3,… or countably many semicircular elements are all isomorphic, and the same is true for reduced free products of copies of the Jiang-Su algebra. A common ingredient is a certain automorphism approximation property; a relative version of that property appeared in our recent work on the identification of von Neumann algebras of Fuchsian groups.
November 23: Samuel Gurrola Viramontes: Real Infinitesimal Freeness of Wigner Matrices
(CIMAT Guanajato)
In this talk, within the framework of real infinitesimal free probability introduced by Cebron and Mingo, we will discuss the calculation of the real infinitesimal free cumulants of independent complex Wigner matrices. Our approach relies on establishing a combinatorial relation between annular noncrossing partitions and families of directed graphs. As a consequence, we show that independent complex Wigner matrices are asymptotically real infinitesimally free. In particular, under mild conditions, a complex Wigner matrix is asymptotically infinitesimally free from its transpose. This is based on joint work with James A. Mingo (arXiv:2609.24023).
November 30: Hari Bercovici: The hyperinvariant subspace problem
(Indiana University Bloomington)
Among the results announced by Open AI, there is a solution of the hyperinvariant subspace problem: there is an operator T on a Hilbert space that is not a scalar multiple of the identity, does have nontrivial invariant subspaces, but none of these subspaces is invariant under every operator in the commutant of T. In particular, the commutant (T)’ is a nonselfadjoint transitive algebra. The construction uses weighted shifts with few invariant subspaces, and a modicum of ergodic theory: a transformation known as an odometer. The resulting operator T is quasinilpotent. I will argue that the example is correct. Moreover, the example can be modified so that it yields a contraction with essential spectrum equal to the closed unit disk. Such contractions are known to have a large variety of invariant subspaces, and one could ask whether the techniques used to produce these subspaces can be improved to yield a hyperinvariant subspace.
December 7: David Jekel: On counterexamples to the unification of \chi and \chi^*
(University of Ottawa)
Open AI recently generated the first examples with -\infty < \chi < \chi^*. I will contextualize and give a detailed proof of similar example, and more explicitly show the separation between free Fisher information and the large-n limit of classical Fisher information. The upper bound on free Fisher information comes from a cumulant-bound for Fisher information that draws on higher-order conjugate variable/Stein kernel techniques from classical and free settings. The lower bound on classical Fisher information comes from volume bounds on the space of matrices admitting a noticeable block structure with respect to some projections reminiscent of the volume arguments in Voiculescu’s paper on absence of Cartan algebras and many subsequent works. I will derive the
upper bound using a sup-formula and showing the relationship with my study of model theory and Wasserstein distance in the context of random matrices.