Treeings are free generating systems of measured groupoids. We construct a treeing from a probability-measure-preserving action of certain HNN extensions (including Baumslag-Solitar groups) and discuss its application.

A countable group is called permutation stable if every asymptotic homomorphism into finite symmetric groups can be approximated by actual homomorphisms. For example, all finitely generated abelian groups are permutation stable. It is not known whether all finitely generated metabelian groups have this property. This question is related to the study of invariant random subgroups. I will talk about permutation stability of a basic example of metabelian groups.

A powerful tool in the mathematical analysis of infinite quantum systems are certain functional inequalities involving orthonormal families of functions. Most relevant for this talk are inequalities which provide control on the density of an infinite system of fermions. These estimates can be recast in terms of certain Schatten space estimates and, on several levels, orthogonality features heavily in the analysis. In the course of introducing these topics, I also hope to mention some open problems related to orthogonality and Schatten spaces.

We overview how we associate the Kasparov category for tensor category actions on C*-algebras. We also see some examples of tensor categories and see how the Kasparov category looks like.

Using non-linear Choquet integrals on $N={1,2,...}$, we define non-linear Choquet traces on the algebra of compact operators. The monotone measure on $N$ can be represented as the positive increasing sequence $\alpha$ ($0=\alpha(0)\le \alpha(1)\le \cdots$). We investigte conditions for $\alpha$ that the non-linear Choquet trace associated with $\alpha$ becomes subadditive on $K(H)$ or quasi-norm on some subset of $K(H)$.

In this talk, we will explore the connection between the theory of quasi-free states of gauge-invariant CAR algebras and determinantal point processes (DPPs) on discrete spaces. Specifically, we will investigate DPPs arising from orthogonal polynomials. Based on the theory of orthogonal polynomials, we will construct a one-parameter group of Bogoliubov automorphisms on a GICAR algebra. Moreover, this construction provides an operator-algebraic understanding of stochastic dynamics on the DPPs.

As an application of the classification theorem of amenable C*-algebras, we construct an endomorphism of the Jiang-Su algebra $\mathcal{Z}$ which does not admit a conditional expectation. This answers a question in the testamentary homework by E. Kirchberg. It is shown that any unital separable simple nuclear $\mathcal{Z}$-absorbing C*-algebra is non-transportable in the Cuntz algebra $\mathcal{O}_2$.

We introduce the class of freely quasi-infinitely divisible distributions to investigate decomposition of random variables with respect to the free additive convolution. This class includes the class of freely infinitely divisible distributions. Some distributional properties and examples will be presented. Lastly, we provide an answer to a question posed by Marek Bożejko.

The study of C*-algebras associated to étale groupoids, known to be groupoid C*-algebras, was initiated by Renault in 1980. It is a natural task to characterize the property of groupoid C*-algebras in terms of étale groupoids. In this talk, we investigate *-homomorphisms between groupoid C*-algebras. First, we prove that *-homomorphisms between groupoid C*-algebras can be described by closed invariant subsets, groupoid homomorphisms and cocycles under some assumptions. Then we obtain the structure theorem for the group of automorphisms which globally fix Cartan subalgebras. As a corollary, we show that group actions on groupoid C*-algebras factor through the abelianizations of the acting groups if the actions pointwisely fix the Cartan subalgebras.

Cocycles on groups are interesting objects, which (sometimes) reveal some approximation properties of groups and correspond to Lévy processes on groups. In this talk, we introduce cocycles on quantizations of groups and consider some examples.

Spherical functions are basic tools in describing representations of free groups. We shall here review them as analytic functionals on certain polynomial hypergroups with the associated spectral measures worked out explicitly. As an interesting example, Haagerup's positive definite functions are interpreted in this framework and investigated in terms of continued fraction expansions of their Stieltjes transforms.

We study free actions of discrete Kac algebras on type III factors whose canonical extension are modular endomorphisms in the sense of Izumi. If a Kac algebra has such actions, then it turns out this is a 2-cocycle twisting of a dual of compact group. We also discuss crossed product algebras. This talk is based on the joint work with R. Tomatsu (Mem. AMS. Vol. 245, no. 1160).