I'll give an account of our recent classification result of outer actions of poly-Z groups $G$ on Kirchberg algebras $A$ in terms of the associated classifying maps. Roughly speaking such actions are completely classified by the associated continuous fields of $A$ over the classifying space $BG$ (caution: this is not a correct statement in the unital case). In particular, we determined the number of cocycle conjugacy classes of outer actions of $Z^m$ on the Cuntz algebras.
This is joint work with Hiroki Matui.

We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsions. This formulates the Baum-Connes assembly map for general discrete quantum groups possibly with torsion. As an application, we show that the group C*-algebra of a discrete quantum group in a certain class satisfies the UCT.

非従順群の単純環への従順作用は、これまで長らく見落とされていた対象である。実際フォンノイマン環の世界では、そのような作用は許容されないことが40年以上前にDelarocheにより示されている。しかし近年の私の発見により、C*環の世界では、接合積(の包絡環)やCuntz-Pimsner環の無限テンソル積作用として、自然に現れることが明らかになってきた。これはC*環の世界でしか観測できない真に新しい数学的現象である。これらの作用が共通して持つ「従順性」が中心化列により捉えられることを見た後、従順群についてG. Szaboが示していた定理が完全群に拡張されることを見ていく。私はこれを非可換・連続版のConnes-Feldmann-Weiss定理への第一歩であるとみなしている。

We study Polish groups consisting of unitary operators on separable Hilbert spaces. In particular, we study some boundedness properties of those groups.
We also answer a question by Pestov, showing that the p-unitary group associated with a properly infinite semifinite von Neumann algebra does not have property (FH).
This talk is based on joint works by Yasumichi Matsuzawa.

The classical Loewner's theorem states that operator monotone functions on real intervals are described by holomorphic functions on the upper half-plane. We characterize local order isomorphisms on operator domains by biholomorphic automorphisms of the generalized upper half-plane, which is the collection of all operators with positive invertible imaginary part. We describe such maps in an explicit manner, and examine properties of maximal local order isomorphisms. Moreover, in the finite-dimensional case, we explain that every order embedding of a matrix domain is a homeomorphic order isomorphism onto another matrix domain.
This is joint work with Peter Semrl (Ljubljana).

We will explain our recent results on gapped ground states of 1-dim fermion systems using elementary methods of operator algebras.

I will talk about recent progress on the classification problem of symmetry protected topological (SPT) phases in one-dimensional quantum systems.

I’ll talk about the matrix liberation process and the orbital free entropy. However, my attempt is still in progress toward the conjectural identification between the orbital free entropy and Voiculescu’s mutual free information as well as a better understanding of free independence.