Wigner's unitary-antiunitary theorem states that every surjective isometry of the set of rank one projections on a Hilbert space is given by a unitary or antiunitary operator. In this talk, I will survey the study of generalizations and improvements of this theorem in various directions.

I will talk about the $q$-Gaussians and one of the free probabilistic tools, the so-called conjugate system, which is given by the adjoint of the non-commutative derivative. The goal is to see that there exists a conjugate system for the $q$-Gaussians for any -1<$q$<1, which is an improvement of the result by Y. Dabrowski. This talk is based on the joint work with R. Speicher.

In a similar spirit to the duality theorems by Takesaki, Takai, and Baaj-Skandalis, we would like to compare the actions on C*-algebras of the two constructions of locally compact quantum groups, called the bicrossed product and the double crossed product, respectively. We will explain the properties of this duality procedure in the case of quantum doubles and its consequence for equivariant Kasparov theories concerning the quantum analog of the Baum-Connes conjecture.

Kajiwara and Watatani's previous research shows us that a Kajiwara--Watatani algebra has a natural maximal commutative subalgebra. I will talk about whether the subalgebras are Cartan (in the meaning of Renault) or not.

There exist various kinds of non-linear positive maps on C*-algebras. For example, consider functional calculus by a fixed continuous (or Borel) positive function, determinant map, n-fold self-tensor products, n-th singular value of compact operators, non-linear integrals by non-additive measures. Except for the work of non-linear completely positive maps by Ando-Choi, Hiai-Nakamura and Arveson, we give a systematic study of non-linear positive maps on C*-algebras or the positive parts of C*-algebras. In particular we consider monotone positive maps, that is, positive maps preserving the order. They include linear positive maps, functional calculus by a fixed operator monotone function, non-linear 2-positive maps, continuous *-multiplicative maps and non-linear traces of Choquet type and Sugeno type. We characterize some classes of them by their intrinsic properties. We also study non-linear and non-commutative integration theory and certain relations with majorization theory. This is a joint work with M. Nagisa.

In this talk, we discuss some recent results on the classification of Cuntz--Pimsner algebras associated to a C*-correspondence over a commutative C*-algebra. In particular, we consider C*-correspondences arising from a vector bundle over a compact metric space X, for which a left action is defined through a homeomorphism of X. When the homeomorphism is minimal and X is finite dimensional, we showed that the associated Cuntz--Pimsner algebras are classifiable.
Given a line bundle, we consider the so-called "orbit-breaking subalgebras", which can be seen as Cuntz--Pimsner algebras associated to non-finitely generated C*-correspondences. In this case, by showing that these subalgebras are centrally large, we will prove that they are classifiable. Lastly, we discuss applications to the study of relevant class of C*-algebras, e.g., quantum Heisenberg manifolds.
This talk is based on joint work with Archey, Forough, Georgescu, Jeong, K. Strung, M. G. Viola, arXiv:2202.10311.

pdf

Recently, P.-E. Caprace, A. Le Boudec, and N. Matte Bon proved that the Neretin group, which is the totally disconnected locally compact group consisting of almost automorphisms of the rooted tree, is not of type I and conjectured that a distinguished open subgroup of this group is not of type I either. In this talk, I will show that the group von Neumann algebra of this open subgroup is of type II and answer their question.

Von Neumann equivalence is an equivalence relation on the class of discrete countable groups that is coarser than Measure Equivalence and W*-equivalence. In this talk, I will explain that group exactness is a von Neumann equivalence invariant. The talk is based on arXiv:2202.13872.

α-induction is known to produce modular invariants from modules of chiral conformal field theories. In this talk, I will introduce recent developments in the equivariantization of this theory.

大昔(1998年)に聞いた無限対称群の表現論に関する講演中に感じたことに由来する試みを話す．

The notion of quantum graphs is a non-commutative analogue of classical graphs and recently developed in the interactions between theories of operator algebras, quantum information, non-commutative geometry, quantum groups, etc. It is well-known that the spectrum of the adjacency matrix can characterize some properties of a (regular) classical graph, for example, connectedness, bipartiteness, and expanders. It is natural to expect that quantum graphs have similar spectral characterizations. Indeed, Ganesan (2021) shows that such a spectral approach works for the chromatic numbers of quantum graphs. Similarly to the classical case, the degree of a regular quantum graph is shown to be the spectral radius of the adjacency matrix. Thus it makes sense to consider the behavior of the spectrum in $[-d,d]$ for $d$-regular undirected quantum graphs. We introduce bipartiteness and connectedness for quantum graphs in terms of graph homomorphisms, and we give their spectral characterizations for regular quantum graphs.