Berkeley Operator Algebras Seminar

Title: What is the set of Jones indices of irreducible, hyperfinite subfactors?

Abstract: Since Vaughan Jones introduced the theory of subfactors in 1983, it has been an open problem to determine the set of Jones indices of irreducible, hyperfinite subfactors. Jones' rigidity theorem establishes all indices between 1 and 4, but above 4, there are many open questions. Caceres and I have recently shown that all indices of finite depth subfactors between 4 and 5 are also realized by new hyperfinite subfactors with Temperley-Lieb-Jones standard invariant. They are non-amenable, but have certain nice asymptotic commutativity properties. Our work leads to a conjecture and some results regarding Jones' index problem. The construction involves new families of commuting squares, a graph planar algebra embedding theorem, and a few tricks that allow us to avoid solving large systems of linear equations to compute invariants of our subfactors. I will explain (somne of) these results, and I will try to make the talk accessible to non-experts in subfactors.

Title: A class of II$_1$ factors with no non-trivial crossed product decompositions. 

Abstract: In this talk, will present the first examples of separable II$_1$ factors $M$ that admit no non-trivial crossed product decompositions: $M\not\cong B\rtimes_\sigma G$, for any trace preserving action of an infinite countable group $G$ on a tracial von Neumann algebra $(B,\tau)$. Our approach relies on constructing separable II$_1$ factors $M$ with the strong rigidity property that every embedding of $M$ into its tensor product square $M\overline{\otimes}M$ arises from the canonical embeddings $x\mapsto x\otimes 1$ and $x\mapsto 1\otimes x$. This is joint work with Adriana Fernández Quero and Hui Tan.

Aareyan: Title: Non co-sofic IRS.
Abstract: The landmark quantum complexity result MIP$^*$=RE was used to prove the existence of a non Connes embeddable tracial von Neumann algebra. Recently, similar ideas were used to give a negative solution to the Aldous-Lyons conjecture: there is a non co-sofic IRS on any non-abelian free group. We define a notion of hyperlinearity for an IRS and show that there is a non co-hyperlinear IRS on any non-abelian free group. As a corollary, we prove that there is a relation whose von Neumann algebra is not Connes embeddable. We do this by significantly simplifying the reduction of Aldous-Lyons to non-local games, removing the need for subgroup tests entirely.

Lizzy: Title: Non sofic group.
Abstract: I will present the construction of a non sofic group as described by Francesco Fournier Facio, combining Proposition 2.3 of the OpenAI announcement with tools from small cancellation theory and the works of Kun and Kun-Thom on expanders and property T groups.

Title:

The Lima Bean Law

Abstract:

R-Diagonal elements are the most rotationally-invariant operators in free probability: a is R-diagonal if a has the same law as ua (and au) for any Haar unitary u freely independent from a.  Examples include circular elements and Haar unitaries.  They are the large-N limits of the most general kind of matrix with totally random eigenvectors; i.e. the (*-distribution) model is an ensemble of the form UAV where A is any matrix and U,V are Haar-distributed random matrices with {A,U,V} independent.  One of the biggest modern theorems in random matrix theory, the Single Ring Theorem [Guionnet--Krishnapur--Zeitouni, 2011], proved that the density of eigenvalues of such a random matrix model converges to the Brown measure of the corresponding R-diagonal.

In this talk I will discuss my recent work on ``random walk'' models of the form (1+a_1)...(1+a_k) for freely independent R-diagonal elements a_j.  We prove convergence of the density of eigenvalues for all these models as well.  This talk will focus on the Brown measure of such shifted products, which we are able to describe quite precisely thanks to a surprising result (which I will discuss) on operator-valued freeness vs. scalar-valued freeness in the context of a symmetry we call "circulant invariance".

Our main result is that, under diffusion rescaling, the Brown measure of any such k-step random walk superconverges to the large-N limit law of the standard Brownian motion on GL(N,C), yielding a very strong kind of Berger-Donsker scaling limit law (very strong because it holds even when the covariance of the terms is "wrong").  We call this scaling limit theorem The Lima Bean Law, for reasons that will become clear from pictures.

This is joint work with Driver, Hall, Ho, Nemish, Nikitopoulos, and Parraud.

Title: Strong convergence of unitary representations

Abstract: