I studied the structure of symmetries of operator algebras. With Hiroki Matui, we studied discrete group actions on C*-algebras, and partially obtained a classification invariant by using topological properties of discrete groups and the automorphism groups of C*-algebras.
With Vaughan Jones, Scott Morrison, David Penneys, Emily Peters, Noah Snyder, we completely classified subfactors of index less than or equal to 5. With Pinhas Grossman and Noah Snyder, we gave detailed description of the structure of the Morita equivalence class of the fusion categories arising from the Asaeda-Haagerup subfactors.

I jointly work with A. Skalski and F. Uwe on a compact quantum group whose Haar state admits a non-trivial square root. Also, with Toshihiko Masuda(Kyushu university), I studied the action of the additive real group on a von Neumann algebra. We especially obtain a classification of Rohlin flows up to strong cocycle conjugacy.

We will study two kinds of actions of a discrete amenable Kac algebra. The<br />
first one is an action whose modular part is normal. We will construct a new<br />
invariant which generalizes a characteristic invariant for a discrete group<br />
action, and we will present a complete classification. The second is a<br />
centrally free action. By constructing a Rohlin tower in an asymptotic<br />
centralizer, we will show that the Connes-Takesaki module is a complete<br />
invariant.

We will introduce the Rohlin property for flows on von Neumann algebras and<br />
classify them up to strong cocycle conjugacy. This result provides alternative<br />
approaches to some preceding results such as Kawahigashi&#039;s classification of<br />
flows on the injective type II$_1$ factor, the classification of injective type<br />
III factors due to Connes, Krieger and Haagerup and the non-fullness of type<br />
III$_0$ factors. Several concrete examples are also studied.

Unlike for locally compact groups, idempotent states on locally compact<br />
quantum groups do not necessarily arise as Haar states of compact quantum<br />
subgroups. We give a simple characterisation of those idempotent states on<br />
compact quantum groups which do arise as Haar states on quantum subgroups. We<br />
also show that all idempotent states on the quantum groups U_q(2), SU_q(2), and<br />
SO_q(3) (q in (-1,0) \cup (0,1]) arise in this manner and list the idempotent<br />
states on the compact quantum semigroups U_0(2), SU_0(2), and SO_0(3). In the<br />
Appendix we provide a short new proof of coamenability of the deformations of<br />
classical compact Lie groups based on their representation theory.

We develop theory of multiplicity maps for compact quantum groups, as an<br />
application, we obtain a complete classification of right coideal<br />
$C^*$-algebras of $C(SU_q(2))$ for $q\in [-1,1]\setminus \{0\}$. They are<br />
labeled with Dynkin diagrams, but classification results for positive and<br />
negative cases of $q$ are different. Many of the coideals are quantum spheres<br />
or quotient spaces by quantum subgroups, but we do have other ones in our<br />
classification list.

Z.-J. Ruan has shown that several amenability conditions are all equivalent<br />
in the case of discrete Kac algebras. In this paper, we extend this work to the<br />
case of discrete quantum groups. That is, we show that a discrete quantum<br />
group, where we do not assume its unimodularity, has an invariant mean if and<br />
only if it is strongly Voiculescu amenable.