Copenhagen
Groups and Operator Algebras Seminar

Abstract:  Voiculescu's freeness emerges when computing the asymptotic spectra of polynomials on random matrices with eigenspaces in generic positions, where they are randomly rotated using a uniform unitary random matrix.

We elaborate on this fundamental result by proposing a random matrix model, which we name the Vortex model, where U_N has the law of a uniform unitary random matrix but is conditioned to leave invariant a deterministic vector. In the high-dimensional regime, this model exhibits asymptotic conditional freeness.

If time permits, we will introduce cyclic-conditional freeness, which unifies three independences: infinitesimal freeness, cyclic monotone independence, and cyclic Boolean independence. Infinitesimal distributions in the Vortex model can be computed thanks to this new independence.

Abstract:  In this talk, I will introduce noncommutative convexity and present a brief overview of noncommutative Choquet theory, which provides a structural foundation for the study of noncommutative convex sets. Since the category of compact noncommutative convex sets is dual to the category of operator systems, which are unital self-adjoint subspaces of operators, noncommutative Choquet theory can potentially shed new light on problems of an operator-algebraic nature. I will discuss two recent examples of this. The first relates to the notion of self testing in quantum information theory, and the second provides a new characterization of operator systems satisfying an important approximation-theoretic property called hyperrigidity. This talk will feature joint work with Ken Davidson and Eli Shamovich. 

Abstract:  The study of locally compact groups is often partitioned into the unimodular and non-unimodular cases; that is, when the modular function of the group is trivial or not. But the modular function offers another natural dichotomy: it is either an open map onto the positive real numbers, or its kernel is open. In this talk I will discuss an operator algebraic perspective on this dichotomy and how it can be used to assign a Murray–von Neumann dimension to representations of groups in the latter class. This is based on joint work with Aldo Garcia Guinto.

Abstract:  In recent joint work with Corentin Le Bars, we establish the C*-simplicity of any discrete subgroup acting properly, cocompactly, and by automorphisms on a locally finite, thick affine building. While some of the techniques employed are standard, the proof crucially relies on the rich geometry of both affine and spherical buildings, as well as certain generalizations of some results for locally finite trees.

Abstract:  In his quest in constructing conformal field theories from subfactors Vaughan Jones found an unexpected connection with Richard Thompson's group T. This led to Jones' technology: a powerful method for constructing actions of certain groups. I will introduce the great lines of this fascinating connection. I will then explain how this approach permits to promote representations of certain basic algebras to more complicated ones like the Cuntz algebra. Thanks to some rigidity phenomena this will provide moduli spaces of irreducible classes of representations of the Cuntz algebra. These are joint works with Vaughan Jones and with Dilshan Wijesena.

Abstract:  TBA

Abstract:  TBA