Researcher Database

Reiji Tomatsu
Faculty of Science Mathematics Mathematics
Associate Professor

Researcher Profile and Settings

Affiliation

  • Faculty of Science Mathematics Mathematics

Job Title

    Associate Professor

E-mail

  • tomatsu@math.sci.hokudai.ac.jp

URL

Research Interests

  • flow   von Neumann algebra   Haagerup property   

Research Areas

  • Mathematics / Global analysis
  • Mathematics / Basic analysis

Research Activities

MISC

  • Reiji Tomatsu  2017/05   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    We study a relationship between the ultraproduct of a crossed product von Neumann algebra and the crossed product of an ultraproduct von Neumann algebra. As an application, the continuous core of an ultraproduct von Neumann algebra is described.
  • Rui Okayasu, Narutaka Ozawa, Reiji Tomatsu  2015/01   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    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 is recently generalized for 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 partly 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.
  • Reiji Tomatsu, Yoshimichi Ueda  2014/12   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    We prove that, for any type III$_1$ free product factor, its continuous core is full if and only if its $\tau$-invariant is the usual topology on the real line. This trivially implies, as a particular case, the same result for free Araki--Woods factors. Moreover, our method shows the same result for full (generalized) Bernoulli crossed product factors of type III$_1$.
  • Martijn Caspers, Rui Okayasu, Adam Skalski, Reiji Tomatsu  2014/04   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    The notion of the Haagerup approximation property, originally introduced for von Neumann algebras equipped with a faithful normal tracial state, is generalized to arbitrary von Neumann algebras. We discuss two equivalent characterisations, one in terms of the standard form and the other in terms 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.
  • Rui Okayasu, Reiji Tomatsu  2014/03   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    We introduce the notion of the $\alpha$-Haagerup approximation property for $\alpha\in[0,1/2]$ using a one-parameter family of positive cones studied by Araki and show that the $\alpha$-Haagerup approximation property actually does not depend on a choice of $\alpha$. This gives us a direct proof of the fact that two characterizations of the Haagerup approximation property are equivalent, one in terms of the standard form and the other in terms of completely positive maps. We also discuss the $L^p$-Haagerup approximation property for a non-commutative $L^p$-spaces associated with a von Neumann algebra ($1
  • Rui Okayasu, Reiji Tomatsu  2013/12   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    We attempt presenting a notion of the Haagerup approximation property for an arbitrary von Neumann algebra by using its standard form. We also prove the expected heredity results for this property.
  • Toshihiko Masuda, Reiji Tomatsu  2013/06   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    We will study two kinds of actions of a discrete amenable Kac algebra. The first one is an action whose modular part is normal. We will construct a new invariant which generalizes a characteristic invariant for a discrete group action, and we will present a complete classification. The second is a centrally free action. By constructing a Rohlin tower in an asymptotic centralizer, we will show that the Connes-Takesaki module is a complete invariant.
  • Reiji Tomatsu  2013/02   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    We will study a faithful product type action of G_q that is the q-deformation of a connected semisimple compact Lie group G, and prove that such an action is induced from a minimal action of the maximal torus T of G_q. This enables us to classify product type actions of SU_q(2) up to conjugacy. We also compute the intrinsic group of G_{q,\Omega}, the 2-cocycle deformation of G_q that is naturally associated with the quantum flag manifold T\backslash G_q.
  • Toshihiko Masuda, Reiji Tomatsu  2012/06   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    We will introduce the Rohlin property for flows on von Neumann algebras and classify them up to strong cocycle conjugacy. This result provides alternative approaches to some preceding results such as Kawahigashi's classification of flows on the injective type II$_1$ factor, the classification of injective type III factors due to Connes, Krieger and Haagerup and the non-fullness of type III$_0$ factors. Several concrete examples are also studied.
  • Uwe Franz, Adam Skalski, Reiji Tomatsu  Journal of Pure and Applied Algebra, Volume 216, Issue 10, 2012, Pages 2079-2093  2010/12   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    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.
  • Uwe Franz, Adam Skalski, Reiji Tomatsu  Journal of Noncommutative Geometry, Volume 7, Issue 1, 2013, pp. 221-254  2009/03   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    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 which do arise as Haar states on quantum subgroups. We also show that all idempotent states on the quantum groups U_q(2), SU_q(2), and SO_q(3) (q in (-1,0) \cup (0,1]) arise in this manner and list the idempotent states on the compact quantum semigroups U_0(2), SU_0(2), and SO_0(3). In the Appendix we provide a short new proof of coamenability of the deformations of classical compact Lie groups based on their representation theory.
  • Toshihiko Masuda, Reiji Tomatsu  2008/06   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    We classify a certain class of minimal actions of a compact Kac algebra with amenable dual on injective factors of type III. Our main technical tools are the structural analysis of type III factors and the theory of canonical extension of endomorphisms introduced by Izumi.
  • Toshihiko Masuda, Reiji Tomatsu  2008/02   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    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.
  • Reiji Tomatsu  2008/01   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    We establish a Galois correspondence for a minimal action of a compact quantum group ${\mathbb G}$ 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 ${\mathbb G}$ and that of intermediate subfactors of $M^{\mathbb G}\subset M$.
  • Reiji Tomatsu  2006/11   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    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 theory of Poisson boundaries introduced by Izumi for discrete quantum groups and generalize a work of Izumi-Neshveyev-Tuset on $SU_q(N)$ 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 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  2006/04   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    We show the uniqueness of minimal actions of a compact Kac algebra with amenable dual on the AFD factor of type II$_1$. 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   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    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 $q\in [-1,1]\setminus \{0\}$. 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.
  • Reiji Tomatsu  Journal of the Mathematical Society of Japan 58 (2006), 949-964  2003/02   [Not refereed] [Not invited]  Institution technical report and pre-print, etc.  
    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.

Educational Activities

Teaching Experience

  • Basic Analysis A
    開講年度 : 2017
    課程区分 : 学士課程
    開講学部 : 理学部
    キーワード : 初等関数,指数関数, 対数関数, 三角関数,整級数,複素積分,原始関数,不定積分,正則関数, コーシーの積分定理, コーシーの積分公式, 調和関数 テーラーの定理,留数定理
  • Exercises on Basic Analysis A
    開講年度 : 2017
    課程区分 : 学士課程
    開講学部 : 理学部
    キーワード : 初等関数,指数関数, 対数関数, 3角関数,複素積分 正則関数, コーシーの積分定理, コーシーの積分公式, 調和関数 テーラーの定理,留数定理,
  • Calculus I
    開講年度 : 2017
    課程区分 : 学士課程
    開講学部 : 全学教育
    キーワード : 数列, 収束, 関数, 極限, 微分, 偏微分, テイラ-の定理
  • Calculus II
    開講年度 : 2017
    課程区分 : 学士課程
    開講学部 : 全学教育
    キーワード : 原始関数, 積分, 重積分, リ-マン和, 変数変換


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