Fusion categories encode new types of quantum symmetries. An object in a fusion category in the operator algebraic approach is typically a bimodule over II_1 factors or an endomorphism of a type III factor, but it is also known that it is represented with a bi-unitary connection in subfactor theory. We explore this aspect in terms of recent advances in the tensor network approach to 2-dimensional topological order.

In this talk we will discuss the explicit construction of 2D conformal nets extending the tensor product of chiral nets whose representation categories possess enough automorphisms under the assumption of braiding cancellation. Our construction for the local net of operator algebras for the full 2D CFT is made by introducing charged fields acting as shifts between different charged sectors.
For the example of the Heisenberg conformal net as chiral components, we define charged fields acting as (twisted) shifts between charged sectors. In this case, in order to recover locality for the 2D conformal net that we construct, we introduce a 2-cocycle on an even lattice $Q$ that encodes the interplay between the DHR categories of the left and right chiral component.
This talk is based on a joint project with L. Giorgetti, Y. Tanimoto, arXiv:2301.12310, arXiv:2506.01008.

We would like to explain a Pimsner construction that gives us an ergodic automorphism on a corner of Toeplitz--Pimsner algebra. We will explain the Pimsner construction using extension of C*-algebras and how the construction provides an ergodic automorphism where the important example is the automorphism shifting generators of infinite Cuntz algebra. If time permits, we will also explain some technical details and our future work.
This talk is a joint work with Prof. Kengo Matsumoto.

The homotopy groups of the automorphism group of C*-algebra play a crucial role in the classification of C*-algebra bundles. I will talk about a description of the homotopy groups of the equivariant automorphism group of a Kirchberg algebra with a compact group action, using equivariant KK-theory. This is an equivariant version of Dadarlat's result.

We consider the problem of computing Itzykson Zuber integrals for matrices with a tensor structure against a random rank one operator. The solution clarifies and generalizes previous results by Gurau, and establishes an unexpected link with Voiculescu’s S-transform. Actually, it allows to re-derive Voiculescu’s S-transform theory directly with matrix integrals, without the knowledge of asymptotic freezes.
Based on joint work with Nicolas Delporte, Manasa Nagatsu, Reiko Toriumi.

Free semicircular and circular random variables are fundamental objects in free probability theory. In this talk, we prove that every square-integrable noncommutative rational function in free semicircular variables is a bounded operator, and using the ideas behind this result, we compute the spectra of several classes of polynomials in free circular variables.

We establish a crossed product decomposition theorem for stabilized Cuntz--Pimsner algebras. This result extends Cuntz's classical decomposition for the Cuntz algebras and reveals an implicit symmetric structure within Cuntz--Pimsner algebras. In this talk, among other applications, I would like to explain how to apply this theorem to provide a short, alternative, and independent solution to the reduced Hao--Ng isomorphism problem. Unlike previous approaches, which heavily rely on non-self-adjoint operator algebras and their C*-envelopes, our proof builds solely on basic facts about C*-algebras.
Joint work with Miho Mukohara (Kyushu), arXiv:2605.21128v2.

A $G$-kernel is a group homomorphism from a group $G$ into the outer automorphism group $\mathrm{Out}(M)$ of an operator algebra $M$. The obstruction to lifting a $G$-kernel to a genuine action is measured by a cohomological invariant taking values in the third group cohomology $H^3(G;\mathbb{T})$. In this talk, I will discuss two aspects of this lifting problem for compact groups. First, extending Wassermann's construction, I will show that every class in $H^3(G;\mathbb{T})$ occurs as the obstruction of a $G$-kernel on a full factor. For compact, simple, simply-connected Lie groups, this yields concrete models for their string groups. Second, I will introduce the Rohlin property for compact group actions, and use it to prove a cohomology vanishing theorem on the hyperfinite II_1 factor.

We study the law of large numbers for non-additive functionals on operator algebras like operator norm or non-linear traces of the Choquet type. Main objects are a sequence of self-adjoint unitaries for Powers' binary shifts, a sequence of projections with generalized Jones relations and a sequence of unitaries in infinite dimensional irrational rotation algebras. This is a joint work with Masaru Nagisa.

Given a graph $C=(V, E)$ and a family of vertex groups $(G_v)$ indexed by the set of vertices $V$, their graph product group $G_C$ is the free product of $(G_v)$, subject to the condition that elements from adjacent vertex groups commute. The rigidity problem for this construction concerns how much information of the original graph or the vertex groups remains in the group von Neumann algebras of their graph product groups. In this talk, we review the recent progress on this topic and present a new approach using bi-exactness, the concept due to Ozawa.

This talk reports on new insights and a new construction for flows on classifiable C*-algebras satisfying the (rational) Rokhlin property. Other than the induced flow on the traces, a key invariant to understand flows of this type is given by Connes' rotation map, which provides a pairing between the $K_1$-group and the set of traces fixed by the flow. When the C*-algebra is unital and the flow has the Rokhlin property, Kishimoto observed that every fixed trace induces, through the rotation map, a real homomorphism on the $K_1$-group with dense range. The first insight of this talk is that this can be viewed as an instance of a more precise global relationship between the rotation map of a (rational) Rokhlin flow and the tracial pairing map within the Elliott invariant. This more general statement naturally extends to the nonunital case, such as stably projectionless C*-algebras, and can lead to different dynamical behavior compared to the unital case.
In the main part of the talk, we investigate the problem of finding (rational Rokhlin) flows on classifiable C*-algebras inducing some given data as its invariant. The main result asserts that under a set of conditions on the desired invariant data, one can indeed find such a flow. If the underlying C*-algebra in question is $\mathcal{Z}_0$-stable, one obtains a more satisfactory conclusion, allowing for arbitrary trace-scaling behavior in conjunction with arbitrary (compatible) pairings between $K_1$ and traces as the rotation map. This yields many examples of flows with genuinely new properties compared to the literature.
All of this is joint work with Johannes Christensen and Robert Neagu.

A rooted random graph is said to be sofic if it is a weak*-limit of finite graphs with uniform roots. Recently, the Aldous-Lyons conjecture, which states that every unimodular random graph is sofic, was refuted. However, we prove that every unimodular random planar graph is sofic. This work relies on the study of treeability of the equivalence relations generated by planar Borel graphs.