Probabilistic Operator Algebras Seminar

(UCLA)

We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus g \geq 2 are free group factors on 2g-1 generators. The key technical ingredient involves a proof that the element w= ABA^{-1}B^{-1} of the free group F_2 = \langle A, B \rangle is freely complemented in the group factor: L(F_2) = \mathrm{W}*(w) * \mathrm{W}^*(v) for some Haar unitary v \in L(F_2) that is freely independent from w. Combined with previous results, we conclude that for an arbitrary finitely generated torsion-free non-elementary discrete subgroup \Gamma \subset PSL_2(\mathbb{R}), L(􀀀) is a free group factor settling a conjecture
of de la Harpe and Voiculescu. This result was obtained using OpenAI’s ChatGPT Pro 6.0

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(University of Virginia)

We give an explicit example of a \mathrm{W}^*-bundle M over a compact, metrizable space K, which has each fiber M_p a \mathrm{II}_1[.katex]-factor with separable predual, and which satisfies the following negation of the generator problem : given any finite family [katex]a_1, ..., a_n \in M of continuous sections, there is a p \in K (depending upon that family) so that a_{1,p}, ..., a_{n,p} do not generate M_p. as a von Neumann algebra. More generally, if N is a sub-bundle of M with the property that each fiber is hyperfinite, or has a Cartan, or is generated by two commuting diffuse subalgebras, or has diffuse central sequence algebra, or is generated by a single sequential commutation orbit, then given any finite family a_1, ..., a_n \in M of continuous sections, there is a p \in K (depending upon that family) so that a_{1,p}, ..., a_{n,p} together with N_p do not generate M_p. We discuss implications for the generator problem for von Neumann algebras: e.g. there is no "continuous" way to take countably many generators for a tracial von Neumann algebra with separable predual and produce a single generator, at least if such a procedure works for all von Neumann algebras simultaneously.

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(CIMAT Guanajato)

We study the asymptotic distribution of the Partial Transpose of Random Unitarily Invariant Random Matrix Ensembles from a new perspective. We introduce a left G-circulant decomposition for matrices indexed by an arbitrary finite group G, extending the diagonal decomposition associated with cyclic groups. We show that, when A \in M_{|G||(\mathcal{A}) is free from M_{|G|}(\mathbb{C}), the components arising from the left G-circulant decomposition of At form a free family with the components associated with inverse pairs forming R-diagonal pairs. We also describe the distributions of these components in terms of the distribution of A. Our results recover the cyclic case and show that different group structures of the same order may lead to different free decompositions of the same matrix. This is joint work with Julian Zazuela-Obeso.

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(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.

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