A bi-unitary connection in subfactor theory producing a subfactor of finite depth gives a 4-tensor appearing in a recent work on 2-dimensional topological order and anyons. Physicists have a special finite dimensional projection called a "projector matrix product operator" in this setting. We prove that the range of this projection of length k is naturally identified with the k-th higher relative commutant of the subfactor arising from the bi-unitary connection. This gives a further connection between 2-dimensional topological order and subfactor theory.

We classify outer actions (or $\mathscr{G}$-kernels) of discrete amenable groupoids on injective factors. Our method based on unified approach for classification of discrete amenable groups actions, and cohomology reduction theorem of discrete amenable equivalence relations. We do not use Katayama-Takesaki type resolution group approach.

We give a necessary and sufficient condition for d-tuple of unitary operators in a finite von Neumann algebras to be free Haar unitaries in terms of the norm of an operator involving the d unitary operators. We will discuss some motivations, related results and some aspects of the proof.
This talk is based on the joint work with Leonard Cadilhac arXiv:2103.02944.

Freed and Hopkins conjectured that the deformation classes of non-topological invertible quantum field theories are classified by a generalized cohomology theory called the Anderson dual of bordism theories. Two of the main difficulty of this problem are the following. First, we do not have the axioms for QFT's. Second, the Anderson dual is defined in an abstract way. In this talk, I will explain our results (arXiv:2106.09270) to give a new approach to this conjecture, in particular to overcome the second difficulty above. We construct a new, physically motivated model for the Anderson duals. This model is constructed so that it abstracts a certain property of invertible QFT's which physicists believe to hold in general. I will start from basic motivations for the classification problem, report the progress of our work and explain future directions.
This is the joint work with Kazuya Yonekura (Tohoku, physics).

The notion of sandwiched Rényi divergences (a one-parameter extension of the relative entropy) has played a role in quantum information, which has recently been generalized to the von Neumann algebra setting by Berta--Scholz--Tomamichel and Jenčová. In this talk we discuss properties of sandwiched Rényi divergences and their relation with the strong converse exponent (showing up in the strong converse type of quantum hypothesis testing), extending Mosonyi and Ogawa's result in the finite-dimensional case to the von Neumann algebra case (also the C*-algebra case).
The talk is based on ongoing work with Milán Mosonyi.

I would like to talk about bundles of the Cuntz algebras and their relationship to Dadarlat-Pennig's cohomology groups. I will also explain a one to one correspondence between the sets of bundles whose fibers are the Cuntz algebra with n+1 generators and the tensor product of n by n matrix algebra and the infinite Cuntz algebra.