I will discuss joint work with Jekel-Kunnawalkam Elayavalli where we show that finitely generated, Property (T) algebras are strongly 1-bounded, generalizing previous work of Voiculescu, Ge-Shen, Jung-Shlyakhtenko, Jung, and Shlyakhtenko. Prior knowledge of Property (T), strong 1-boundedness, or free entropy theory will not be assumed. I will largely stick to the case of von Neumann algebras of Property (T) groups.

I will discuss characterizations of property (T) for $\mathrm{II}_1$ factors by weak spectral gap in inclusions into tracial von Neumann algebras. I will explain how this is related to the non-weakly-mixing property of the bimodules containing almost central vectors, where we obtain a $\mathrm{II}_1$ factor version of a characterization of property (T) of Bekka and Valette.

We show that representations of the Thompson group F yield a large class of bilateral stationary noncommutative Markov processes. As a partial converse, bilateral stationary Markov processes in tensor dilation form (and in particular, in the commutative setting) are shown to yield representations of F. We point out analogous results between unilateral stationary Markov processes and representations of the Thompson monoid F+. This is joint work with Claus Koestler and Stephen J. Wills.

A commuting square can be characterized by an array of complex numbers called a biunitary. In this talk we will build a planar algebra associated to the commuting square and identify the biunitary as an element in this planar algebra. The biunitary can be used to embed a subfactor planar algebra into the commuting square planar algebra. We then construct finite dimensional representations of the fusion algebra of the commuting square subfactor. These representations can be used to prove that many subfactors from commuting squares are infinite depth.

We introduce a K-theoretic invariant for actions of unitary fusion categories on unital C*-algebras. We show that for inductive limits of finite dimensional actions of fusion categories on unital AF-algebras, this is a complete invariant. In particular, this gives a complete invariant for inductive limit actions of finite groups on AF-algebras. This is joint work with Roberto Hernández Palomares and Corey Jones (arXiv: 2207.11854).

Rieffel's quantum Gromov-Hausdorff distance is a complete metric on the class of order unit spaces equipped with L-seminorms (a noncommutative analogue to the Lipschitz seminorm), where distance zero provides, in part, an order unit isomorphism. Latrémolière's dual Gromov-Hausdorff propinquity is a complete metric on the class of unital C*-algebras equipped with Leibniz L-seminorms (where Leibniz adds a product rule-type property), where distance zero provides, in part, a *-isomorphism. We will show that if we define a version of the Gromov-Hausdorff propinquity for order unit spaces (and forget the Leibniz property), then we recover exactly the quantum Gromov-Hausdorff distance. As a consequence, we are able to place L-seminorms on the Bunce-Deddens algebras for which the standard circle algebras that form the Bunce-Deddens algebras converge in the quantum Gromov-Hausdorff distance by utilizing Latrémolière's completeness argument. We will also discuss more recent work where we define a version of the Gromov-Hausdorff propinquity on the class of unital C*-algebras equipped with strongly Leibniz L-seminorms (where strongly adds a quotient rule-type property) to provide new strongly Leibniz L-seminorms. (This talk includes joint work with Stephan Ramon Garcia, Elena Kim, Frédéric Latrémolière, and Timothy Rainone).

A quantum graph is a triple that consists of a finite-dimensional C*-algebra, a state, and a quantum adjacency matrix. Analogous to the Cuntz-Krieger algebra of a classical graph, the quantum Cuntz-Krieger (QCK) algebra of a quantum graph is generated by the operator coefficients of matrix partial isometries. In this talk, we discuss connections between a QCK algebra and a Cuntz-Pimsner algebra associated to a quantum graph correspondence, and in the complete quantum graph case, connections between the QCK algebra and a particular Exel crossed product. We end by discussing the challenges in defining the "infinite path space" for a quantum graph.

Consider $X$ an infinite compact metric space, $\mathscr{V}$ a locally trivial vector bundle over $X$ and $\alpha : X \to X$ a homeomorphism (often assumed minimal). We can construct a C*-correspondence $\mathcal{E}$ over $C(X)$ from the module of sections of $\mathscr{V}$, where we use the homeomorphism $\alpha$ to twist the left multiplication. As we shall see, many tractable and interesting C*-correspondences over $C(X)$ do in fact arise in this manner.
In this talk, I will discuss some of the structural properties of the resulting Cuntz-Pimsner algebra $\mathcal{O}(\mathcal{E})$. Under the additional assumption that $\mathscr{V}$ is a line bundle the Cuntz-Pimsner algebra is a generalized crossed product, suggesting additional means of investigation. For Cuntz-Pimsner algebras arising from line bundles we can identify a large subalgebra of $\mathcal{O}(\mathcal{E})$. I will describe this subalgebra and list some of the consequences of this result. This is based on joint work with Adamo, Archey, Forough, Jeong, Strung and Viola.

Relative Toeplitz algebras of directed graphs were introduced by Spielberg in 2002 to describe certain subalgebras corresponding to subgraphs. They can also be used to describe quotients of graph algebras corresponding to subgraphs. We use the latter relationship to answer a question posed in a recent paper regarding pushout diagrams of graphs that give rise to pullback diagrams of the respective graph C*-algebras. We introduce a new category of relative graphs to this end, and we prove our results using graph groupoids and their C*-algebras. This is joint work with Jack Spielberg.

Properly proximal groups were introduced recently by Boutonnet, Ioana, and Peterson, where they generalized several rigidity results to the setting of higher-rank groups. In this talk, I will how the notion of proper proximality fits in the setting of von Neumann algebras. I will also describe several applications, including that the group von Neumann algebra of a non-amenable inner-amenable group cannot embed into a free group factor, which solves a problem of Popa. This is joint work with Srivatsav Kunnawalkam Elayavalli and Jesse Peterson.

The dynamic asymptotic dimension (DAD) was first introduced by Guentner, Willett, and Yu to study the K-theory and nuclear dimension of crossed products (among other things). We will discuss the relationship between the asymptotic dimension of box spaces of groups and the DAD of group actions on profinite completions; as well as some dimension-theoretic properties of DAD. Together, these results allow us to calculate the dimension of box spaces of many elementary amenable groups, specifically the Baumslag-Solitar groups BS(1, n). Time allowing, we may discuss a similar relationship between DAD and warped cones.