Probabilistic Operator Algebras Seminar

(UCSD)

Using a new, non-K-theoretic approach involving embedding spaces in II1 factors with plenty of freely independent Haar unitaries, we prove that the reduced free group C*-algebras for different numbers of free generators are pairwise non-isomorphic. This recovers the seminal result of Pimsner and Voiculescu with a short new proof.

This talk is based on joint work with Srivatsav Kunnawalkam Elayavalli.

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(University of Maryland)

A group G is said to be a matricial field (MF) if it admits a “strongly converging” sequence of approximate homomorphisms into matrices, i.e. norms of polynomials converge to the corresponding value in the left regular representation. G is then said to be purely MF (PMF) if this sequence of maps into matrices can be chosen as actual homomorphisms. G is further said to be purely finite field (PFF) if the image of each homomorphism is finite. These questions have phenomenal applications to the study of \mathrm{C}^* and von Neumann algebras, spectral geometry, random walks and random graphs, spectral gaps of hyperbolic manifolds, minimal surface theory and Yau’s conjectures and even applied mathematics including but not limited to signal processing. By developing a new operator algebraic approach to the MF problems, we are able to prove the following result bringing several new examples into the fold. Suppose G is a MF (resp., PMF, PFF ) group and H is separable (i.e. H = \bigcap_{i=1}^N H_i where H_i < G are finite index subgroups) and K is a residually finite MF (resp., PMF, PFF) group. If either G or K is exact, then the amalgamated free product G *_H (H \times K)
is MF (resp., PMF, PFF). Our work has several applications. Firstly, as a consequence of MF, the Brown-Douglas-Fillmore semigroups of many new reduced C*-algebras are not groups. Secondly, we obtain that arbitrary graph products of residually finite exact MF (resp.,PMF, PFF) groups are MF (resp., PMF, PFF), yielding a significant generalization of the breakthrough work of Magee-Thomas. Thirdly our work resolves the open problem of proving PFF for 3-manifold groups, more generally all RAAGs. Prior to our paper, PFF results remained unknown even in the simple subcase of free products. These results are of further significance since PFF is th property that is used in Antoine Song’s approach towards the existence of minimal surfaces in spheres of negative curvature.

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(UC Berkeley)

We establish the finite free analogue to the classical and free Stam inequality, proved via a novel connection between the Jacobian of the finite free convolution map and the score vector (the finite free analogue of the Hilbert transform). Time permitting, we will then discuss using the same technique to prove monotonicity of the finite free Fisher information; applications to finite free entropy monotonicity; and obtaining a result of Shlyakhtenko and Tao in free probability as a limiting instance.

Reference: arXiv:2602.15822; joint work with Jorge Garza-Vargas and Nikhil Srivastava.

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(University of Toulouse)

I will describe a Boolean information-theoretic framework paralleling the classical and free settings, by introducing Microstates and Non-Microstates Boolean Fisher informations and relating them to Boolean entropy through Bruijn-type identities. We established Boolean analogues of the main functional inequalities, including logarithmic Sobolev, HWI, HSI, Talagrand and WSH-type inequalities. It yields new quantitative Berry-Esseen bounds and entropic convergence rates in the Boolean CLT.

Reference: arXiv.2602.23527; joint work with Kewei Pan.

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(University of Copenhagen)

The Namioka-Phelps tensor product of two convex compact sets associates two new compact convex sets: the minimal and the maximal tensor product where the former is contained in the latter. They show that the two tensor products agree if one of the two convex compact sets is a Choquet simplex. It remains an open problem, known as Barker’s conjecture, if the converse also holds. Barker’s conjecture was recently verified by Aubrun-Lami-Palazuelos-Plavala in the finite dimensional case (and was verified by Namioka and Phelps when one of the two convex compact sets is the square).
We show that Barker’s conjecture holds when the compact convex sets are state spaces of \mathrm{C}^*– algebras, and we describe the two Namioka-Phelps tensor products. The minimal tensor product is precisely the set of entangled states in the (minimal) tensor product of the \mathrm{C}^*-algebras, and one can describe the maximal tensor product in terms of positive maps. We identify the trace space of
the tensor product of \mathrm{C}^*-algebras as the Namioka-Phelps tensor product of the trace spaces and use this to say when the trace simplex of a tensor product of \mathrm{C}^*-algebras is the Poulsen simplex.

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