For each $k\geq 0$ we consider the Toeplitz algebra $\mathcal{T}_\lambda(M_k)$ of the monoid $M_k$ generated by the first $k+1$ generators in the infinite presentation of the Thompson group $F$. In this talk we will discuss some structural properties of $\mathcal{T}_\lambda(M_k)$ and its boundary quotient, generating sets and presentation. This talk is based on joint work with A. an Huef, M. Laca, B. Nucinkis and I. Raeburn.

We will introduce three open problems related to normal weights posed by Haagerup 50 years ago in his master’s thesis. Among these, we especially discuss how we could solve the last problem and its meaning. This will provide a complete set of the positive Hahn-Banach separation theorems on operator algebras and shed light on why the weak∗ topologies on the duals of C*-algebras are relatively difficult to work with. (arXiv:2501.16832.)

We introduce a two-parameter deformation of the classical Poisson distribution from the point of view of noncommutative probability theory, by defining a $(q, t)$-Poisson type operator (random variable) on the $(q, t)$-Fock space. We also derive a combinatorial moment formula of the $(q, t)$-Poisson type operator and the (q, t)-Poisson distribution. This is accomplished by means of a card arrangement technique, which encodes set partitions statistics of the restricted crossings and nestings in graphical way, and the obtained expression naturally exhibits a duality between these statistics.
This talk is based on the series of joint works with Nobuhiro Asai at Aichi Univ. Edu.

The area law is a fundamental property of quantum many-body systems, stating that the entanglement entropy of a region typically scales with its boundary rather than its volume. It plays a central role in both quantum statistical mechanics and quantum computation.
For one-dimensional quantum spin systems, Hastings (2007) provided the first rigorous proof that gapped ground states satisfy an area law. Later, Arad et al. (2013) introduced a different approach based on the AGSP formalism. Until very recently, however, all rigorous results were restricted to finite systems. In 2024, new developments finally established an area law in the infinite-volume setting, which is essential for the framework of quantum statistical mechanics.
In this talk, I will review the background and significance of the area law, then explain the conceptual differences between finite and infinite systems, with particular emphasis on the definition of ground states. I will then present recent progress on proving a one-dimensional area law for non-degenerate gapped ground states in the infinite setting, using operator-algebraic methods.

Given an amenable second countable Hausdorff locally compact étale groupoid $\mathcal G$, it is known that every primitive ideal of the corresponding C*-algebra is induced from an isotropy group. For certain classes of groupoids, we refine this result by describing the Jacobson topology on such induced ideals. Our results give, in principle, a complete description of the ideal structure of a large class of C*-algebras including, for example, all higher rank graph C*-algebras as well as transformation group C*-algebras defined by actions of groups of local polynomial growth and by amenable actions of lattices of semisimple Lie groups of real rank one. (Joint work with Johannes Christensen.)

In the 1990s, Izumi introduced a general method for constructing a certain type of unitary fusion category from a corresponding endomorphism of a Cuntz C*-algebra. This construction gives a very explicit model for the categories in question, and allows for concrete computations which are normally very difficult, including of the structure of the Drinfeld center. The two main families of such categories which have been studied so far are the Haagerup-Izumi categories and the near-group categories. In this talk I will describe work-in-progress with Masaki Izumi on constructing examples of "mixed" quadratic categories, which combine features of Haagerup-Izumi and near-group categories.

The notion of qausi-product actions of a compact group on a C*-algebra was introduced by Bratteli et al. in their attempt to seek an equivariant analogue of Glimm's characterization of non-type I C*-algebras. We show that a faithful minimal action of a second countable compact group on a separable C*-algebra is quasi-product whenever its fixed point algebra is simple. This was previously known only for compact abelian groups and for profinite groups. Our proof relies on a subfactor technique applied to finite index inclusions of simple C*-algebras in the purely infinite case, and also uses ergodic actions of compact groups in the general case. As an application, we show that if moreover the fixed point algebra is a Kirchberg algebra, such an action is always isometrically shift-absorbing, and hence is classifiable by the equivariant KK-theory due to a recent result of Gabe-Szabó.

A G-kernel is a group homomorphism from a (discrete) group G to Out(A), the outer automorphism group of a C*-algebra A. There are cohomological obstructions to lifting such a G-kernel to a group action. In the setting of von Neumann algebras, G-kernels on the hyperfinite II_1-factor have been completely understood via deep results of Connes, Jones and Ocneanu. In the talk I will explain how G-kernels on C*-algebras and their lifting obstructions can be interpreted in terms cohomology with coefficients in crossed modules. G-kernels, group actions and cocycle actions then give rise to induced maps on classifying spaces. For strongly self-absorbing C*-algebras these classifying spaces turn out to be infinite loop spaces creating a bridge to stable homotopy theory.
The talk is based on joint work with S. Giron Pacheco and M. Izumi, and with my PhD student V. Bianchi.

An action of a compact group G on a von Neumann algebra M is called prime if the fixed point algebra M^G is a factor. Then M admits the Izumi--Longo--Popa type decomposition. I will present an outline of the proof and its application when G=SU(2). This is a joint-work with Y. Arano.

Richard Thompson’s groups F, T, V are one of the most remarkable discrete infinite groups for their several unusual properties. On the other hand, the celebrated Cuntz algebra O has many fascinating properties and it is known that V embeds inside O. However, classifying the representations of the Thompson groups have proven to be very difficult, while for O its spectrum of irreducible classes of representations has no obvious topology aside for the coarse one.

The notion of KMS-bundles for C*-dynamical systems associated with general topological groups was introduced by Bratteli, Elliott, and Kishimoto. Building on the classical work of Sewell and Araki on KMS-states, we establish the properness of KMS-bundles for general C*-dynamical systems. Moreover, we show that the properness of the bundle, together with the weak Rohlin property for the underlying dynamical system, plays a fundamental role in the correspondence between KMS-weights and certain eigenvalue-type traces.

We study inclusions of von Neumann algebras that admit operator valued weights. We show that every positive element whose image under the operator valued weight is finite satisfies the weak Dixmier property with respect to the inclusion. This generalizes Marrakchi's result for conditional expectations and has several applications to type III factors within the framework of Popa's deformation/rigidity theory. We generalize Popa's intertwining techniques and Ozawa's relative solidity theorem, and construct new examples of prime type III factors.