Updated on 2022/05/26

TOMATSU, Reiji

Affiliation
Faculty of Education and Integrated Arts and Sciences, School of Education
Job title
Professor
Homepage URL

### Concurrent Post

• Faculty of Education and Integrated Arts and Sciences   Graduate School of Education

### Degree

• 東京大学   博士(数理科学)

### Research Experience

• 2015
-

Hokkaido University

### Research Areas

• Mathematical analysis

• Basic analysis

### Research Interests

• Rohlin性

• flow

• 量子群

• 作用

• 量子旗多様体

• 連結単純lie群

• von Neumann環

• Kac環

• 自己同型

• von Neumann algebra

• Haagerup property

• 無限テンソル積作用

• 誘導作用

### Papers

• Rohlin Flows on von Neumann Algebras

Toshihiko Masuda, Reiji Tomatsu

MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY   244 ( 1153 ) VII - +  2016.11  [Refereed]

View Summary

We will classify Rohlin flows on von Neumann algebras up to strong cocycle conjugacy. This result provides alternative approaches to some preceding results such as Kawahigashi's classification of flows on the injective type II1 factor, the classification of injective type III factors due to Connes, Krieger and Haagerup and the non-fullness of type III0 factors. Several concrete examples are also studied.

• Haagerup Approximation Property for Arbitrary von Neumann Algebras

Rui Okayasu, Reiji Tomatsu

PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES   51 ( 3 ) 567 - 603  2015.09  [Refereed]

View Summary

We attempt presenting a notion of the Haagerup approximation property for an arbitrary von Neumann algebra by using its standard form. We also prove the expected heredity results for this property.

• Idempotent states on compact quantum groups and their classification on U-q(2), SUq(2), and SOq(3)

Uwe Franz, Adam Skalski, Reiji Tomatsu

JOURNAL OF NONCOMMUTATIVE GEOMETRY   7 ( 1 ) 221 - 254  2013  [Refereed]

View Summary

Unlike for locally compact groups, idempotent states on locally compact quantum groups do not necessarily arise as Haar states of compact quantum subgroups. We give a simple characterisation of those idempotent states on compact quantum groups that do arise as Haar states on quantum subgroups. We also show that all idempotent states on the quantum groups U-q(2), SUq(2), and SOq(3) (q is an element of (-1, 0) boolean OR (0, 1]) arise in this manner and list the idempotent states on the compact quantum semigroups U-0(2), SU0(2), and SO0(3). In the Appendix we provide a short new proof of the coamenability of deformations of classical compact Lie groups based on their representation theory.

• Compact quantum ergodic systems

Reiji Tomatsu

JOURNAL OF FUNCTIONAL ANALYSIS   254 ( 1 ) 1 - 83  2008.01  [Refereed]

View Summary

We develop theory of multiplicity maps for compact quantum groups. As an application, we obtain a complete classification of right coideal C*-algebras of C(SU(q)(2)) for q epsilon [-1, 1) \ {10}. They are labeled with Dynkin diagrams, but classification results for positive and negative cases of q are different. Many of the coideals are quantum spheres or quotient spaces by quantum subgroups, but we do have other ones in our classification list. (c) 2007 Elsevier Inc. All rights reserved.

• Amenable discrete quantum groups

Reiji Tomatsu

JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   58 ( 4 ) 949 - 964  2006.10  [Refereed]

View Summary

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

• A paving theorem for amenable discrete Kac algebras

Reiji Tomatsu

INTERNATIONAL JOURNAL OF MATHEMATICS   17 ( 8 ) 905 - 919  2006.09  [Refereed]

View Summary

We extend the Ornstein-Weiss theorem on paving of amenable discrete groups to amenable discrete Kac algebras.

### Misc

• Reiji Tomatsu

2017.05

Internal/External technical report, pre-print, etc.

View Summary

We study a relationship between the ultraproduct of a crossed product von<br />
Neumann algebra and the crossed product of an ultraproduct von Neumann algebra.<br />
As an application, the continuous core of an ultraproduct von Neumann algebra<br />
is described.

• Rui Okayasu, Narutaka Ozawa, Reiji Tomatsu

Mathematica Scandinavica   121 ( 1 ) 75 - 91  2017

Internal/External technical report, pre-print, etc.

View Summary

The Haagerup approximation property (HAP) is defined for finite von Neumann algebras in such a way that the group von Neumann algebra of a discrete group has the HAP if and only if the group itself has the Haagerup property. The HAP has been studied extensively for finite von Neumann algebras and it was recently generalized to arbitrary von Neumann algebras by Caspers-Skalski and Okayasu-Tomatsu. One of the motivations behind the generalization is the fact that quantum group von Neumann algebras are often infinite even though the Haagerup property has been defined successfully for locally compact quantum groups by Daws-Fima-Skalski-White. In this paper, we fill this gap by proving that the von Neumann algebra of a locally compact quantum group with the Haagerup property has the HAP. This is new even for genuine locally compact groups.

• Rui Okayasu, Reiji Tomatsu

Journal of Operator Theory   75 ( 2 ) 259 - 288  2016

Internal/External technical report, pre-print, etc.

View Summary

We introduce the notion of the a-Haagerup approximation property (α-HAP) for α ∈ [0, 1/2] using a one-parameter family of positive cones studied by Araki and show that the a-HAP actually does not depend on the choice of α. This enables us to prove the fact that the Haagerup approximation properties introduced in two ways are actually equivalent, one in terms of the standard form and the other in terms of completely positive maps. We also discuss the Lp-Haagerup approximation property (Lp-HAP) for a noncommutative Lp-space associated with a von Neumann algebra for p ∈ (1,∞) and show the independence of the Lp-HAP on the choice of p.

INTERNATIONAL MATHEMATICS RESEARCH NOTICES   ( 19 ) 9857 - 9887  2015

Internal/External technical report, pre-print, etc.

View Summary

We introduce a natural generalization of the Haagerup property of a finite von Neumann algebra to an arbitrary von Neumann algebra (with a separable predual) equipped with a normal, semi-finite, faithful weight and prove that this property does not depend on the choice of the weight. In particular, this defines the Haagerup property as an intrinsic invariant of the von Neumann algebra. We also show that such a generalized Haagerup property is preserved under taking crossed products by actions of amenable locally compact groups. Our results are motivated by recent examples from the theory of discrete quantum groups, where the Haagerup property appears a priori only with respect to the Haar state.

• Reiji Tomatsu, Yoshimichi Ueda

2014.12

Internal/External technical report, pre-print, etc.

View Summary

We prove that, for any type III$_1$ free product factor, its continuous core<br />
is full if and only if its $\tau$-invariant is the usual topology on the real<br />
line. This trivially implies, as a particular case, the same result for free<br />
Araki--Woods factors. Moreover, our method shows the same result for full<br />
(generalized) Bernoulli crossed product factors of type III$_1$.

• Martijn Caspers, Rui Okayasu, Adam Skalski, Reiji Tomatsu

COMPTES RENDUS MATHEMATIQUE   352 ( 6 ) 507 - 510  2014.06

Internal/External technical report, pre-print, etc.

View Summary

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.

• Toshihiko Masuda, Reiji Tomatsu

2013.06

Internal/External technical report, pre-print, etc.

View Summary

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.

• Reiji Tomatsu

2013.02

Internal/External technical report, pre-print, etc.

View Summary

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.

• Uwe Franz, Adam Skalski, Reiji Tomatsu

JOURNAL OF PURE AND APPLIED ALGEBRA   216 ( 10 ) 2079 - 2093  2012.10

Internal/External technical report, pre-print, etc.

View Summary

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.

• Toshihiko Masuda, Reiji Tomatsu

2012.06

Internal/External technical report, pre-print, etc.

View Summary

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.

• 従順因子環への群・量子群作用について

戸松 玲治, 増田 俊彦

數學   64 ( 1 ) 24 - 46  2012.01

• Reiji Tomatsu

JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK   633   165 - 182  2009.08

Internal/External technical report, pre-print, etc.

View Summary

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.

• Uwe Franz, Adam Skalski, Reiji Tomatsu

Journal of Noncommutative Geometry, Volume 7, Issue 1, 2013, pp. 221-254    2009.03

Internal/External technical report, pre-print, etc.

View Summary

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.

• Toshihiko Masuda, Reiji Tomatsu

ADVANCES IN MATHEMATICS   220 ( 4 ) 1075 - 1134  2009.03

Internal/External technical report, pre-print, etc.

View Summary

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.

• Toshihiko Masuda, Reiji Tomatsu

2008.06

Internal/External technical report, pre-print, etc.

View Summary

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.

• Reiji Tomatsu

COMMUNICATIONS IN MATHEMATICAL PHYSICS   275 ( 1 ) 271 - 296  2007.10

Internal/External technical report, pre-print, etc.

View Summary

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.

• Toshihiko Masuda, Reiji Tomatsu

COMMUNICATIONS IN MATHEMATICAL PHYSICS   274 ( 2 ) 487 - 551  2007.09

Internal/External technical report, pre-print, etc.

View Summary

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.

• Reiji Tomatsu

2004.12

Internal/External technical report, pre-print, etc.

View Summary

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.

• Reiji Tomatsu

Journal of the Mathematical Society of Japan 58 (2006), 949-964    2003.02

Internal/External technical report, pre-print, etc.

View Summary

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.

### Syllabus

• School of Education

2022   full year

• School of Education

2022   fall semester

• School of Education

2022   full year

• School of Education

2022   fall semester

• School of Education

2022   full year

• School of Education

2022   fall semester

• School of Education

2022   full year

2022   fall semester

2022   spring semester

2022   fall semester

2022   spring semester

2022   fall semester

2022   spring semester

2022   fall semester

2022   spring semester

2022   fall semester