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.

The notion of the Haagerup approximation property, originally introduced for von Neumann algebras equipped with a faithful normal tracial state, is generalised to arbitrary von Neumann algebras. We discuss two equivalent characterisations, one in term of the standard form and the other in term of the approximating maps with respect to a fixed faithful normal semifinite weight. Several stability properties, in particular regarding the crossed product construction are established and certain examples are introduced. (C) 2014 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

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 study a faithful product type action of G_q that is the q-deformation<br />
of a connected semisimple compact Lie group G, and prove that such an action is<br />
induced from a minimal action of the maximal torus T of G_q. This enables us to<br />
classify product type actions of SU_q(2) up to conjugacy. We also compute the<br />
intrinsic group of G_{q,\Omega}, the 2-cocycle deformation of G_q that is<br />
naturally associated with the quantum flag manifold T\backslash G_q.

The paper is concerned with the extension of the classical study of probability measures on a compact group which are square roots of the Haar measure, due to Diaconis and Shahshahani, to the context of compact quantum groups. We provide a simple characterisation for compact quantum groups which admit no non-trivial square roots of the Haar state in terms of their corepresentation theory. In particular it is shown that such compact quantum groups are necessarily of Kac type and their subalgebras generated by the coefficients of a fixed two-dimensional irreducible corepresentation are isomorphic (as finite quantum groups) to the algebra of functions on the group of unit quaternions. An example of a quantum group whose Haar state admits no nontrivial square root and which is neither commutative nor cocommutative is given. (C) 2012 Elsevier B.V. All rights reserved.

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.

We establish a Galois correspondence for a minimal action of a compact quantum group G on a von Neumann factor M. This extends the result of Izumi, Longo and Popa who treated the case of a Kac algebra. Namely, there exists a one-to-one correspondence between the lattice of left coideals of G and that of intermediate subfactors of M(G) subset of M.

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 introduce the notions of approximate innerness and central triviality for endomorphisms on separable von Neumann factors, and we characterize them for hyperfinite factors by Connes-Takesaki modules of endomorphisms and modular endomorphisms which are introduced by Izumi. Our result is a generalization of the corresponding result obtained by Kawahigashi-Sutherland-Takesaki in automorphism case. (C) 2008 Elsevier Inc. All rights reserved.

We classify a certain class of minimal actions of a compact Kac algebra with<br />
amenable dual on injective factors of type III. Our main technical tools are<br />
the structural analysis of type III factors and the theory of canonical<br />
extension of endomorphisms introduced by Izumi.

Let G be a co- amenable compact quantum group. We show that a right coideal of G is of quotient type if and only if it is the range of a conditional expectation preserving the Haar state and is globally invariant under the left action of the dual discrete quantum group. We apply this result to the theory of Poisson boundaries introduced by Izumi for discrete quantum groups and generalize a work of Izumi- Neshveyev- Tuset on SUq ( N) for co- amenable compact quantum groups with the commutative fusion rules. More precisely, we prove that the Poisson integral is an isomorphism between the Poisson boundary and the right coideal of quotient type by a maximal quantum subgroup of Kac type. In particular, the Poisson boundary and the quantum flag manifold are isomorphic for any q- deformed classical compact Lie group.

We show the uniqueness of minimal actions of a compact Kac algebra with amenable dual on the AFD factor of type II1. This particularly implies the uniqueness of minimal actions of a compact group. Our main tools are a Rohlin type theorem, the 2-cohomology vanishing theorem, and the Evans-Kishimoto type intertwining argument.

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.