In 1992, Stein introduced a class of groups generalizing Higman-Thompson's groups. The group can be described as groups of piecewise linear maps on the reals with prescribed slopes and noncontinuous points. In this talk, we give a classification result for these groups. The proof uses cohomology of ample groupoids.

A random walk in a Gromov-hyperbolic space determines the hitting measure on the boundary if it has positive speed. This hitting measure is called the harmonic measure associated with the walk. I will talk about dimensional properties of the harmonic measures associated with random walks on random graphs in Gromov-hyperbolic spaces.

We first present the homotopy groups of the automorphism groups of Cuntz-Krieger algebras in terms of their underlying matrices. We also show that the homotopy groups are complete invariants of the isomorphism classes of the Cuntz-Krieger algebras. As a result, we see that the isomorphism classes of the Cuntz-Krieger algebras are completely characterized by only the group structure of their extension groups that are computable by hand. This talk is a joint work with Taro Sogabe and also the introduction to the following Sogabe's talk.

　I would like to talk about some K-theoretic invariants specific to the case of C*-algebras with finitely generated K-groups. We will focus on the invariant including the direct sum of strong and weak Ext group, the difference of the ranks of K0 and K1 groups, and I would like to introduce a hierarchy in the class of unital Kirchberg algebras with finitely generated K-groups to generalize our previous research classifying Cuntz--Krieger algebras via their homotopy groups of the automorphism groups. I will also explain that two groups built from the strong and weak extension groups completely remember three data: K0, K1-groups and the position of the unit. This is the joint work with Prof. Kengo Matsumoto.

The $\alpha$-$z$-Rényi divergences, extensively studied in these years, are a family of quantum extensions of the classical Rényi divergence. Among the family, the Petz type Rényi divergence (in the case $z=1$) and the sandwiched Rényi divergence (in the case $z=\alpha$) are of quite use, having the operational interpretation from quantum hypothesis testing. Recently, the extension of the $\alpha$-$z$-Rényi divergences to the general von Neumann algebra setting was proposed by Kato and Ueda and studied by Kato. In this talk, based on joint work with A. Jenčová, I will present further properties of this extension, focusing on variational expressions, the data processing inequality, the sufficiency theorem, and monotonicity properties in the parameters $\alpha,z$.

U. Haagerup proved the inequality for operator norms of homogeneous polynomials in generators of free groups with respect to the left regular representation, which was improved by T. Kemp and R. Speicher in the holomorphic setting. In this talk, I will explain another approach to their inequality in the case of the q-circular system. This talk is based on a joint project with T. Kemp.

We give a new complete description theorem of the intermediate operator algebras, which unifies the discrete Galois correspondence results and the crossed product splitting results, and involves $2$-cocycles. As an application, we obtain a Galois’s type result for Bisch—Haagerup type inclusions arising from isometrically shift-absorbing actions of compact-by-discrete groups. Based on my preprint arxiv:2406.00304.

Inclusions arising from actions of discrete (quantum) groups on factors were studied by Izumi-Longo-Popa and others. The correspondence between intermediate subfactors and subgroups is called the Galois correspondence. Analogues for actions on C*-algebras were also studied by Izumi, Cameron-Smith, Peligrad, and others. In this talk, I will discuss the Galois correspondence for quasi-product actions of compact groups on C*-algebras. The notion of a quasi-product action was introduced by Brattelli-Elliott-Kishimoto. Recently, Izumi proved that every minimal action with a simple fixed-point algebra is a quasi-product action.

The Razak-Jacelon algebra $\mathcal{W}$ is the simple separable nuclear monotracial $\mathcal{Z}$-stable C$^*$-algebra which is $KK$-equivalent to $\{0\}$. A finite group action $\gamma$ on $\mathcal{W}$ is said to be $\mathcal{W}$-absorbing if there exist a simple separable nuclear monotracial C$^*$-algebra $A$ and a finite group action $\alpha$ on $A$ such that $\gamma$ is cocycle conjugate to $\alpha\otimes \mathrm{id}_{\mathcal{W}}$ on $A\otimes \mathcal{W}$. In this talk, we completely classify outer $\mathcal{W}$-absorbing actions on $\mathcal{W}$ of finite groups up to conjugacy and cocycle conjugacy.

We characterize the simplicity of Pimsner algebras for non-proper C*-correspondences. Partly using this criterion, we provide several examples of actions of tensor categories on Kirchberg algebras.

In this talk, we give a mathematically rigorous formulation of a proposal by A. Kitaev, stating that the `space' of invertible gapped phases forms a generalized cohomology theory. We also present its applications to the study of symmetry-protected topological phases of quantum spin systems.

Property A is the amebable-type condition on metric spaces, which extends the group exactness. For an action of a discrete group on a set, we show that the Schreier graph has property A if and only if the permutation representation generates an exact C*-algebra.