Schedule and abstracts

Vaughan Jones and I showed in the late 1990's that whenever a subfactor has an intermediate subfactor, the fundamental subfactor symmetries are captured by what we called the ``Fuss-Catalan algebras''. If the subfactor has two intermediate subfactors in general position, i.e. they form a quadrilateral of subfactors, describing the quantum symmetries is more complicated and not much progress had been made since the work of Grossman, Izumi and Jones some 20 years ago.

I will discuss recent work with Junhwi Lim in which we determine the quantum symmetries of a quadrilateral of subfactors when the four algebras form a co-commuting square and satisfy some additional conditions. The subfactor symmetries we find are related to partition algebras and Bell numbers.

The graphical operation of insplitting is key to understanding conjugacy of shifts of finite type (SFTs) in both one and two dimensions. In this paper, we consider two approaches to studying 2-dimensional SFTs: textile systems and rank-2 graphs. Nasu's textile systems describe all two-sided 2D SFTs up to conjugacy, whereas the 2-graphs (higher-rank graphs of rank 2) introduced by Kumjian and Pask yield associated C*-algebras. Both models have a naturally associated notion of insplitting (introduced for textile systems by Johnson and Madden and for 2-graphs by Eckhardt, et al).  We show that these notions do not coincide, raising the question of whether insplitting a 2-graph induces a conjugacy of the associated one-sided 2-dimensional SFTs. This talk focuses on the relationship between 2-graphs and textile systems, and how their notions of insplitting diverge.

A Hadamard matrix is a square matrix of complex numbers whose entries have modulus 1, and whose columns are pairwise orthogonal.  Hadamard matrices appear all over mathematics, and are well known in the field of operator algebras as basic building blocks in subfactor theory.  In this talk I will introduce various notions of notions of quantum symmetries and quantum equivalences of Hadamard matrices using ideas from quantum group theory. These symmetries, in the case of finite order (i.e., Butson-type) Hadamard matrices, turn out to have interesting applications in quantum information theory.  Time permitting, I will also discuss some conjectural connections to Hadamard subfactors. This is joint work with Daniel Gromada, Roberto Hernandez-Palomares, and Nicky Priebe.

A landmark result in C*-algebra theory is the classification of unital separable simple nuclear Z-stable C*-algebras satisfying the universal coefficient theorem (UCT) in terms of their K-theory and traces. I will discuss this result with a focus on the role of UCT. In the infinite setting, without the UCT, two unital Kirchberg algebras are isomorphic if and only they are KK-equivalent in a unit-preserving way. I’ll discuss some results along these lines in the finite case.

Classification theorems for simple nuclear C*-algebras classify unital *-homomorphisms up to approximate unitary equivalence: a sequence of unitaries conjugates one map ever closer to the other. Strengthening the sequence to a continuous path of unitaries starting at the identity (called strong asymptotic unitary equivalence) meets an obstruction that ordinary KK-theory cannot see. The type of path produced by absorption arguments starts at some unitary w, and the class of w in K1 can prevent deforming the starting point to 1. We introduce a unital variant of Kasparov's theory, KKu(A,B), that captures this obstruction. It fits in an exact sequence between K1(B) and KK(A,B), enjoys the expected formal properties, and yields existence and uniqueness theorems for unitaly absorbing maps. This is joint work with Jamie Gabe and Christopher Schafhauser.

We will present new examples of irreducible, hyperfinite subfactors with trivial standard invariant and interesting Jones indices. These are obtained by constructing new finite dimensional commuting squares. We will use two graph planar algebra embedding theorems and the classification of small index subfactors to show that our commuting square subfactors cannot have finite depth. We also present one-parameter families of commuting squares that, by a classification result of Kawahigashi, will also yield irreducible infinite depth subfactors. This is joint work with Dietmar Bisch.

Strong ergodicity of actions is very similar to fullness of von Neumann algebras. Exploiting this relation, we will show that if a locally compact second countable group G admits a strongly ergodic translation action by a countable subgroup, then G is almost unimodular (the image of G under the modular homomorphism is countable). Moreover for almost unimodular groups, strong ergodicity of a left action is equivalent to strong ergodicity of this action on the unimodular part of G. Time permitting, we will talk about rigidity or constructing non-injective non-full von Neumann algebras using translation actions on non almost unimodular groups.

I will discuss new work joint with Gao, Manzoor, Patchell where we found a new C*-algebraic approach to establish strong convergence of unitary respresentations for a wide class of countable groups. I will first give an overview of this program and discuss why it is of importance to a wide array of areas.

Continuing off of Sri's talk, I will give a proof of strong convergence for a large class of groups. This proof will go through combinatorial group theory, Toeplitz algebras in operator algebras, and Agol and Haglund–Wise in geometric group theory.

We will discuss several new results that allow one to prove strong convergence by first showing some "ambient" strong convergence and then upgrading it. This includes settings such as Toeplitz algebras, amalgamated free products, operator-valued semicirculars, HNN extensions, and crossed products by exact groups. Most of the talk will be based on joint work with Srivatsav Kunnawalkam Elayavalli and the rest will be based on joint work with Srivatsav Kunnawalkam Elayavalli and Mahan Mj.

In joint work with Mícheál Ó Cobhthaigh, we develop a new strong convergence model for reduced graph products of C*-algebras and establish new MF permanence results as an application. In our model, taking a graph and associated graph product as input, a “doubled graph” is defined and graph product enlarged accordingly to accommodate the use of universality properties for Toeplitz algebras. We adapt previous strong convergence ideas of Magee–Thomas in the process of building our model and proving needed estimates and combinatorial identities.

I’ll discuss how K-homology can be realized entirely by ucp maps to matrix algebras for quasidiagonal C*-algebras.  I’ll then discuss applications to measuring the size of the space of approximate representations of groups modulo the space of actual representations.

The Fourier algebra A(G) of a locally compact group G is a Banach algebra that captures both the topology and the group structure of G through its unitary representation theory. An important structural property of Banach algebras is the notion of amenability. Various amenability properties of the Fourier algebra, and their connections to the underlying group structure, have been extensively studied over the past few decades. In this talk, we discuss a quantitative invariant for amenability known as the amenability constant. For Fourier algebras, explicit formulas for the amenability constant of A(G) are known only in limited settings. In particular, B.E. Johnson computed the finite-group case. We present new upper bounds for the amenability constant in the case of discrete groups, and present new classes of examples where the amenability constant can be explicitly computed. This talk is based on joint work with Yemon Choi.

During this talk, we will construct a functional calculus based on smooth functions for the Sobolev subalgebra of a tracial C*-probability space with the rapid decay property. Some consequences will be discussed. Joint work with Ben Hayes.

Tracial joint spectral measures were introduced last year by Otte Heinävaara. Given a pair of (hermitian) matrices, their TJSM is a measure on the plane that establishes a kind of ‘joint calculus’ for the pair, regardless of whether or not they commute. I will present examples of TJSM-s for pairs of infinite-rank operators in tracial von Neumann algebras and advertise open directions of inquiry based on these observations.

The crossed product is a fundamental construction of II1 factors, and a natural question is whether every separable II1 factor arises from a noncommutative dynamical system in this way. I will discuss joint work with Adriana Fernández Quero and Adrian Ioana in which we construct the first separable II1 factors that cannot be decomposed as $B\rtimes_{\sigma} G$ for any trace-preserving action of any countably infinite group G on any tracial von Neumann algebra B. Our approach relies on a rigidity phenomenon for embeddings. Specifically, for the constructed factors, all embeddings into their tensor products come from the canonical embeddings, which is incompatible with the comultiplication associated to an infinite crossed product decomposition.

Boundary properties such as biexactness and proper proximality play an important role in the rigidity theory of groups and von Neumann algebras. I will discuss an upgrading technique that obtains actual biexactness (proper proximality) from relative biexactness (proper proximality). First, if a separable von Neumann algebra M is biexact relative to an expected biexact and mixing subalgebras B, then M is biexact. Second, if a subalgebra A is properly proximal relative to a mixing subalgebra, then A decomposes into a properly proximal part and parts that are amenable relative to the mixing subalgebra. This generalizes the previous results of Ding and Kunnawalkam Elayavalli. As applications, we obtain new biexact and properly proximal results for amalgamated free product and graph product von Neumann algebras. This is based on joint works with Srivatsav Kunnawalkam Elayavalli and with Kai Toyosawa.

We describe an approach to the model theory of W*-probability spaces based on totally bounded elements. We introduce these elements and some key technical results about them. Joint work with Ando, Goldbring, Hart, and Sinclair.

The category of locally compact quantum groups has earned a reputation for the uncharacteristic alignment among their topological (C*-algebraic) and their measure theoretic (W*-algebraic) expressions, any of which retain the information of a locally compact group in the classical (or dual-to-classical) setting. In this talk, we define almost unimodular quantum groups, extending the notion due to Garcia Guinto and Nelson for classical groups, and continue the quantum group theme by demonstrating that the representation theory of such a quantum group (encoded by an associated universal object) retains features from the von Neumann algebra picture. This is based on joint work with Aldo Garcia Guinto and Brent Nelson.