TOMATSU, Reiji

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

Concurrent Post 【 display / non-display 】

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

Degree 【 display / non-display 】

• 東京大学   博士(数理科学)

Research Experience 【 display / non-display 】

• 2015
-

Hokkaido University

Research Areas 【 display / non-display 】

• Mathematical analysis

• Basic analysis

• Rohlin性

• flow

• 量子群

• 作用

• 量子旗多様体

Papers 【 display / non-display 】

• Rohlin Flows on von Neumann Algebras

Toshihiko Masuda, Reiji Tomatsu

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

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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]

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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]

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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]

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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]

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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.

Misc 【 display / non-display 】

• Reiji Tomatsu

2017.05

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

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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.

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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.

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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.

• Martijn Caspers, Adam Skalski

INTERNATIONAL MATHEMATICS RESEARCH NOTICES   ( 19 ) 9857 - 9887  2015

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

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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.

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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$_1$.

Syllabus 【 display / non-display 】

• Graduate School of Education

2021   fall semester

• Graduate School of Education

2021   spring semester

• Graduate School of Education

2021   fall semester

• Graduate School of Education

2021   spring semester

• Graduate School of Education

2021   fall semester