Authentic Publications (Published Papers & Preprints)

• [46] A remark on non-commutative $L^p$-spaces (with Shinya Kato)
  Studia Math., to appear.
  [arXiv:2307.01790] abstract
  Abstract: We explicitly describe the Haagerup and the Kosaki non-commutative $L^p$-spaces associated with a tensor product von Neumann algebra $M_1\bar{\otimes}M_2$ in terms of those associated with $M_i$ and usual tensor products of unbounded operators. The descriptions are then shown to be useful in the quantum information theory based on operator algebras.
  Comments: The motivation of this work came from Hiai's online lectures entitled "Quantum Analysis and Quantum Information" in Feb.--Mar., 2022 via Nagoya University. We gave the first rigorous proof of the additivity of the sandwich Renyi divergence in the full generality, and also formulate the $\alpha$-$z$ Renyi divergence in the general von Neumann algebra setup.


• [45] Lebesgue decomposition for positive operators revisited (with Yoshiki Aibara)
  C. R. Math. Rep. Acad. Sci. Canada, Vol.45 (2023), 37-55.
  [arXiv:2305.15085] abstract
  Abstract: We explain how Pusz--Woronowicz's notion of their functional calculus fits the theory of Lebesgue decomposition for positive operators on Hilbert spaces initially developed by Ando. In this way, we reconstruct the essential and fundamental part of the theory.
  Comments: I've been interested in this topic since my graduate school days. Because(!) my thesis advisor was (is) a specialist on the topic.


• [44] Spherical representations of $C^*$-flows III: Weight-extended branching graphs
  J. Aust. Math. Soc., to appear.
  [arXiv:2302.11113] abstract
  Abstract: We apply Takesaki's and Connes's ideas on structure analysis for type III factors to the study of links (a short term of Markov kernels) appearing in asymptotic representation theory.
  Comment: This was written as a suppliement to [40,43]. The purpose is to connect general links to the notion of dimension groups (or $K_0$-groups), and I used a standard trick in operator algebras.


• [43] Spherical representations of $C^*$-flows II: Representation system and Quantum group setup
  SIGMA, Vol.18 (2022), 050, 43 pages.
  [arXiv:2201.10931] abstract
  Abstract: This paper is a sequel to our previous study of spherical representations in the operator algebra setup. We first introduce possible analogs of dimension groups in the present context by utilizing the notion of operator systems and their relatives. We then apply our study to inductive limits of compact quantum groups, and establish an analogue of Olshanski's notion of spherical unitary representations of infinite-dimensional Gelfand pairs of the form $G < G\times G$ (via the diagonal embedding) in the quantum group setup. This, in particular, justifies Ryosuke Sato's approach to asymptotic representation theory for quantum groups.
  Comment: My research in this direction is still in progress gradually like the other direction [36,39]. Surprisingly, this paper of mine appeared just after a paper of my colleagues on the journal site.


• [42] Pusz--Woronowicz functional calculus and extended operator convex perspectives (with Fumio Hiai and Shuhei Wada)
  Integral Equation Operator Theory, Vol.94 (2022).
  [arXiv:2105.09549] abstract
  Abstract: In this article, we first study, in the framework of operator theory, Pusz and Woronowicz's functional calculus for pairs of bounded positive operators on Hilbert spaces associated with a homogeneous two-variable function on $[0,\infty)^2$. Our construction has special features that functions on $[0,\infty)^2$ are assumed only locally bounded from below and that the functional calculus is allowed to take extended semibounded self-adjoint operators. To analyze convexity properties of the functional calculus, we extend the notion of operator convexity for real functions to that for functions with values in $(-\infty,\infty]$. Based on the first part, we generalize the concept of operator convex perspectives to pairs of (not necessarily invertible) bounded positive operators associated with any operator convex function on $(0,\infty)$. We then develop theory of such operator convex perspectives, regarded as an operator convex counterpart of Kubo and Ando's theory of operator means. Among other results, integral expressions and axiomatization are discussed for our operator perspectives.
  Comment: The co-authors invited me to develop my naive idea, casually commented in [41, Remark 10 and subsection 4.2], in their on-going (at that time) joint project to establish operator convex perspectives in full generality, i.e. without assuming the invertibility of given operators, though I initially had no plan to study it further. As a bonus, I could learn many techniques related to operator convexity, operator means, quantum information, etc., throughout this collaboration. This is the longest paper on this publication list. We hope that this paper serves as a reference to a class of binary operations for two positive Hilbert space operators including operator means, connections and perspectives.


• [41] Pusz--Woronowicz's functional calculus revisited (with Kanae Hatano)
  Acta Sci. Math. (Szeged), Vol.87 (2021), 485-503.
  [arXiv:2012.13072] abstract
  Abstract: This note is a complement to Pusz--Woronowicz's works on functional calculus for two positive forms from the viewpoint of operator theory. Based on an elementary, self-contained and purely Hilbert space operator explanation of their functional calculus, we show that any operator connection type operations (including any operator perspectives) are captured by their functional calculus.
  Comment: By accident, we realized that some inetersting old works due to Pusz--Woronowicz have been unnoticed in some recent developments, and thus dicided to make it clear with a (hopefully easy-to-read) explanation in the framework of Hilbert space bounded operators.


• [40] Spherical representations of $C^*$-flows I
  Muenster J. Math., Vol.16 (2023),201--263.
  [arXiv:2010.15324] abstract
  Abstract: We propose an abstract framework of a kind of representation theory for $C^*$-flows, i.e., $C^*$-algebras equipped with one-parameter automorphism groups, as a proper generalization of Olshanski's formalism of unitary representation theory for infinite-dimensional groups such as the infinite-dimensional unitary group $\mathrm{U}(\infty)$. The present framework, in particular, clarifies some overlaps and/or similarities between a certain unitary representation theory of infinite-dimensional groups and existing works in operator algebras, and captures arbitrary projective chains arising from links.
  Comment: I've wanted to understand Olshanski's spherical representation theory for infinite-dimensional groups in the framework of operator algebras for a long time (actually since I attended a seminar at UCB in fall 1998). This is a very first abstract part of my attempt. The published version contains an appendix, where a generalization of Stratila--Voiculescu's result is given.


• [39] Matrix liberation process II: Relation to orbital free entropy
  Canad. J. Math., Vol.73 (2021), Issue 2, 493-541.
  [arXiv:1905.08013] abstract
  Abstract: We investigate the concept of orbital free entropy from the viewpoint of matrix liberation process. We will show that many basic questions around the definition of orbital free entropy are reduced to the question of full large deviation principle for the matrix liberation process. We will also obtain a large deviation upper bound for a certain family of random matrices that is an essential ingredient to define the orbital free entropy. The resulting rate function is made up into a new approach to free mutual information.
  Comment: The second output of my attempt to develop a new kind of random matrix models motivated from Voiculescu's liberation theory. My attempt is still growing gradually.


• [38] On Arveson's boundary theorem (with Kei Hasegawa)
  Math. Proc. Royal Irish Acad., Vol.119A (2019), No.1, 1-5.
  [arXiv:1810.10689] abstract
  Abstract: This short note aims to give an insight to Arveson's boundary theorem by means of non-commutative Poisson boundaries and to obtain new results from our insight.
  Comment1: Two different notions of boundaries are connected mathematically. This is the shortest paper on this publication list.
  Comment2: (Dec. 14, 2019) Following Yasuhiko Sato's comments, I confirmed that Theorem 1 and their corollaries hold with the same proof (with the help of Choi's "original" matrix trick) even for unital 2-positive maps. On the other hand, they do not hold in general for unital positive maps, because the transposition map on the $2\times2$ complex matrices is a counter-example.


• [37] Free products in AQFT (with Roberto Longo and Yoh Tanimoto)
  Ann. Inst. Fourier (Grenoble), 69 (2019) no. 3, 1229-1258.
  [arXiv:1706.06070] abstract
  Abstract: We apply the free product construction to various local algebras in algebraic quantum field theory. If we take the free product of infinitely many identical half-sided modular inclusions with ergodic canonical endomorphism, we obtain a half-sided modular inclusion with ergodic canonical endomorphism and trivial relative commutant. On the other hand, if we take Moebius covariant nets with trace class property, we are able to construct an inclusion of free product von Neumann algebras with large relative commutant, by considering either a finite family of identical inclusions or an infinite family of inequivalent inclusions. In two dimensional spacetime, we construct Borchers triples with trivial relative commutant by taking free products of infinitely many, identical Borchers triples. Free products of finitely many Borchers triples are possibly associated with Haag-Kastler net having S-matrix which is nontrivial and non asymptotically complete, yet the nontriviality of double cone algebras remains open.
  Comment: I've been interested in the concept of half-sided modular inclusions since I knew it when I was a grad. I deeply thank Roberto and Yoh for giving me an amazing opportunity to work about it.


• [36] Matrix liberation process I: Large deviation upper bound and almost sure convergence
  J Theor Probab, Vol.32, No.2 (2019), 806-847.
  [arXiv:1610.04101] abstract
  Abstract: We introduce the concept of matrix liberation process, a random matrix counterpart of the liberation process in free probability, and prove a large deviation upper bound for its empirical distribution and several properties on its rate function. As a simple consequence we obtain the almost sure convergence of the empirical distribution of the matrix liberation process to that of the corresponding liberation process as continuous processes in the large $N$ limit.
  Comment: This is the first output of my attempt to develop a new kind of random matrix models motivated from Voiculescu's liberation theory.


• [35] A remark on orbital free entropy
  Arch. Math., Vol.108, No.6 (2017), 629-638.
  [arXiv:1610.04118] abstract
  Abstract: A lower estimate of the orbital free entropy $\chi_\mathrm{orb}$ under unitary conjugation is proved, and it together with Voiculescu's observation shows that the conjectural exact formula relating $\chi_\mathrm{orb}$ to the free entropy $\chi$ breaks in general in contrast to the case when given random multi-variables are all hyperfinite.
  Comment: I made this observation few years ago (probably in 2014), and wrote down this short note in relation with [36].


• [34] A free product pair rigidity result in von Neumann algebras
  J. Noncommut. Geom., Vol.13, No.2 (2019), 587-607.
  [arXiv:1610.01842] abstract
  Abstract: We prove that the free product pair of any finitely many copies of the unique amenable type III$_1$ factor endowed with weakly mixing states remembers the number of free components and the given states.
  Comment: This is the first result of such a complete restoration.


• [33] Rigidity of free product von Neumann algebras (with Cyril Houdayer)
  Compositio Math., Vol.152, No.12 (2016), 2461-2492.
  [arXiv:1507.02157] abstract
  Abstract: Let $I$ be any nonempty set and $(M_i, \varphi_i)_{i \in I}$ any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class $\mathcal C_\mathrm{anti-free}$ of (possibly type ${\rm III}$) von Neumann algebras including all nonprime factors, all nonfull factors and all factors possessing a Cartan subalgebra. For the free product $(M, \varphi) = \ast_{i \in I} (M_i, \varphi_i)$, we show that the free product von Neumann algebra $M$ retains the cardinality $|I|$ and each nonamenable factor $M_i$ up to stably inner conjugacy, after permutation of the indices. Our main theorem unifies all previous Kurosh-type rigidity results for free product type ${\rm II_1}$ factors and is new for free product type ${\rm III}$ factors. It moreover provides new rigidity phenomena for type ${\rm III}$ factors.
  Comment1: This is a sequel of [32], but the subject matters are different. We have developed quite useful tools and techniques for not necessarily tracial (amalgamated) free products in [32,33], and this paper is indeed a consequence from those.
  Comment2: We could clarify the solidity for free product von Neumann algebras completely, but couldn't do the strong solidity. Yusuke Isono finally resolved the strong solidity for free product von Neumann algebras completely, by which we should think that the general structural analysis of free product von Neumann algebras becomes complete. See arXiv:1902.01049. It's a very nice work.


• [32] Asymptotic structure of free product von Neumann algebras (with Cyril Houdayer)
  Math. Proc. Cambridge Philos. Soc., Vol.161, No.3 (2016), 489-516.
  [arXiv:1503.02460] abstract
  Abstract: Let $(M, \varphi) = (M_1, \varphi_1) \ast (M_2, \varphi_2)$ be the free product of any $\sigma$-finite von Neumann algebras endowed with any faithful normal states. We show that whenever $Q \subset M$ is a von Neumann subalgebra with separable predual such that both $Q$ and $Q \cap M_1$ are the ranges of faithful normal conditional expectations and such that both the intersection $Q \cap M_1$ and the central sequence algebra $Q' \cap M^\omega$ are diffuse (e.g. $Q$ is amenable), then $Q$ must sit inside $M_1$. This result generalizes the previous results of the first named author in 2014 and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion $M_1 \subset M$ in arbitrary free product von Neumann algebras.
  


• [31] A characterization of the fullness of continuous cores of type III$_1$ free product factors (with Reiji Tomatsu)
  Kyoto J. Math., Vol.56, No.3 (2016), 599-610.
  [arXiv:1412.2418] abstract
  Abstract: 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$_1$.
  Comment1: This note leads to the end of the project, which began in [22], to compute all the basic invariants for free product von Neumann algebras. I deeply thank Reiji for this collaboration. The basic invariants mean: central decomposition, Murray--von Neumann--Connes's type, size of the central sequences -- fullness, Connes's two invariants for full type III$_1$ factors.
  Comment2: Amine Marrakchi generalized this result to arbitrary full type III$_1$ factors. See arXiv:1605.09613. It's a striking work.


• [30] Absence of Cartan subalgebras in continuous cores of free product von Neumann algebras
  Proc. Japan Acad. Ser. A., Vol.90, No.10 (2014), 151-155.
  [arXiv:1401.6489] abstract
  Abstract: We show that the continuous core of any type III free product factor has no Cartan subalgebra. This is a complement to previous works due to Houdayer--Ricard and Boutonnet--Houdayer--Raum.
  Comment: This note is equipped with full proofs.


• [29] Orbital free pressure and its Legendre transform (with Fumio Hiai)
  Comm. Math. Phys., Vol.334, No.1 (2015), 275-300.
  [arXiv:1310.3877] abstract
  Abstract: Orbital counterparts of the free pressure and its Legendre transform (or $\eta$-entropy) are introduced and studied in comparison with other entropy quantities in free probability theory and in relation to random multi-matrix models.


• [28] Remarks on free mutual information and orbital free entropy (with Masaki Izumi)
  Nagoya Math. J., 220 (2015), 45-66.
  [arXiv:1306.5372] abstract
  Abstract: The present notes provide a proof of $i^*(\mathbb{C}P+\mathbb{C}(I-P)\,;\mathbb{C}Q+\mathbb{C}(I-Q)) = -\chi_\mathrm{orb}(P,Q)$ for any pair of projections $P,Q$ with $\tau(P)=\tau(Q)=1/2$. The proof includes new extra observations, such as a subordination result in terms of Loewner equations. A study of the general case is also given.
  Comment: Only the remaining problem to prove $i^* = - \chi_\mathrm{orb}$ in the case of two projections had been to show that $H(t,\zeta)$ is of $H^{3/2}$-class in $\zeta$ for any $t > 0$. See [28, Lemma 4.4] for more details. This was confirmed in a stronger form by Hamdi in his two papers [Nagoya Math. J., to appear; Complex Anal. Theory (2018)] using only complex analysis.


• [27] Orbital free entropy, revisited
  Indiana Univ. Math. J., Vol.63, No.2 (2014), 551-577.
  [arXiv:1210.6421] abstract
  Abstract: We give another definition of orbital free entropy introduced by Hiai, Miyamoto and us, which does not need the hyperfiniteness assumption for each given random multi-variable. The present definition is somehow related to one of its several recent approaches due to Biane and Dabrowski, but can be shown to agree with the original definition completely and is much closer to the original approach.
  


• [26] On the geometry of von Neumann algebra preduals (with Miguel Martin)
  Positivity, Vol.18, No.3 (2014), 519-530.
  [arXiv:1209.3391] abstract
  Abstract: Let $M$ be a von Neumann algebra and let $M_\star$ be its (unique) predual. We study when for every $\varphi\in M_\star$ there exists $\psi\in M_\star$ solving the equation $\|\varphi \pm \psi\|=\|\varphi\|=\|\psi\|$. This is the case when $M$ does not contain type I nor type III$_1$ factors as direct summands and it is false at least for the unique hyperfinite type III$_1$ factor. We also characterize this property in terms of the existence of centrally symmetric curves in the unit sphere of $M_\star$ of length $4$. An approximate result valid for all diffuse von Neumann algebras allows to show that the equation has solution for every element in the ultraproduct of preduals of diffuse von Neumann algebras and, in particular, the dual von Neumann algebra of such ultraproduct is diffuse. This shows that the Daugavet property and the uniform Daugavet property are equivalent for preduals of von Neumann algebras.
  Comment: I've not met the coauthor in person.


• [25] Discrete cores of type III free product factors
  Amer. J. Math., Vol.138, No.2 (2016), 367-394.
  [arXiv:1207.6838] abstract
  
  Abstract: We give a general description of the discrete decompositions of type III factors arising as central summands of free product von Neumann algebras based on our previous works. This enables us to give several precise structural results on type III free product factors.
  Comment1: I'd wanted to reconstruct Dykema's works in the mid 90s since great while ago. Actually, this work completes the attempt with very strong (and probably optimal) results because mine deals with arbitrary almost periodic states. I was so busy when I wrote the first version, and hence it took a year to provide the final and submitted version.
  Comment2: Hartglass and Nelson [arXiv:1810.01924] succeeded in identifying a large class of free product von Neumann algebras (including free products of finite dimensional algebras) with free Araki-Woods factors with almost periodic states. That is quite a great progress in the direction. The remaining question is to remove assumptions on given states from their work (which Dykema's statement also requires but my result [25] doesn't). This kind of questions is beyond the general structural analysis of free product von Neumann algebras.


• [24] Some analysis on amalgamated free products of von Neumann algebras in non-tracial setup
  J. London Math. Soc., Vol.88, No.1 (2013), 25-48.
  [arXiv:1203.1806] abstract
  
  Abstract: Several techniques together with some partial answers are given to the questions of factoriality, type classification and fullness for amalgamated free product von Neumann algebras.
  Comment: Part of this work was done during my "second" stay in IHP, May 2011. One of the original motivations of this research is indeed a question about plain free product von Neumann algebras, but itself is later resolved by Reiji Tomatsu and myself in [31].


• [23] On type III$_1$ factors arising as free products
  Math. Res. Lett., Vol.18, No.5 (2011), 909-920.
  [arXiv:1101.4991] abstract
  
  Abstract: Type III$_1$ factors arising as (direct summands of) von Neumann algebraic free products are investigated. In particular we compute Connes' Sd- and $\tau$- invariants for those type III$_1$ factors without any extra assumption.
  Comment: This was done during the new year holidays of 2011 as a sequel to [22].


• [22] Factoriality, type classification and fullness for free product von Neumann algebras
  Adv. Math., Vol.228, No.5 (2011), 2647-2671.
  [arXiv:1011.5017] abstract
  
  Abstract: We give a complete answer to the questions of factoriality, type classification and fullness for arbitrary free product von Neumann algebras.
  Comment: I'd wanted to solve the questions since 1995. Thus this paper is a great favorite of mine. A crucial idea was gifted suddenly during the period (Apr.--Aug. 2010) of my heavy teaching duty. I realized that too much duty sometimes enables (or even forces) one to face the problem from quite a natural angle because of no time to spare.


• [21] On the predual of non-commutative $H^\infty$
  Bull. London Math. Soc., Vol.43, No.5 (2011), 886-896.
  [arXiv:1002.3672] abstract
  
  Abstract: The unique predual $M_\star/A_\perp$ of a non-commutative $H^\infty$-algebra $A = H^\infty(M,\tau)$ is investigated. In particular, we will prove the liftability property of weakly relatively compact subsets in $M_\star/A_\perp$ to $M_\star$.
  Comment: I spent a year on proving the main theorem of this paper. The situation around me became not good at that time (and had been so until 2017).


• [20] On peak phenomena for non-commutative $H^\infty$
  Math. Ann., Vol.343, No.2 (2009), 421-429.
  [arXiv:0802.3449] abstract
  
  Abstract: A non-commutative extension of Amar and Lederer's peak set result is given. As its simple applications it is shown that any non-commutative $H^\infty$-algebra $H^\infty(M,\tau)$ has unique predual, and moreover some restriction in some of the results of Blecher and Labuschagne are removed, making them hold in full generality.
  Comment: This paper is a favorite of mine. The uniqueness of predual of non-commutative $H^\infty$ was obtained by my former student Shintaro Sewatari in his master thesis under some assumption as an application of Blecher--Labuschagne's work. After almost a decade, Blecher--Labuschagne [Trans. Amer. Math. Soc. 370 (2018), 8215--8236] succeeded in generalizing my result [20] to the general $\sigma$-finite case.


• [19] Orbital approach to microstate free entropy, (with Fumio Hiai and Takuho Miyamoto)
  Internat. J. Math., Vol.20, No.2 (2009), 227-273.
  [math.OA/0702745] abstract
  
  Abstract: Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy $\chi_\mathrm{orb}$ for non-commutative self-adjoint random variables (also for "hyperfinite random multi-variables"). Besides its basic properties the relation of $\chi_\mathrm{orb}$ with the usual free entropy $\chi$ is shown. Moreover, the dimension counterpart $\delta_\mathrm{0,orb}$ of $\chi_\mathrm{orb}$ is discussed, and we obtain the relation of $\delta_\mathrm{0,orb}$ with the original free entropy dimension $\delta_0$ with applications to $\delta_0$ itself.
  Comment: This is probably the best among the joint works with Fumio Hiai (on free probability).


• [18] Notes on microstate free entropy of projections, (with Fumio Hiai)
  Publ. RIMS, Vol.44, No.1 (2008), 49-89.
  [math.OA/0605633] abstract
  
  Abstract: We study the microstate free entropy $\chi_{\mathrm{proj}}(p_1,...,p_n)$ of projections, and establish its basic properties similar to the self-adjoint variable case. Our main contribution is to characterize the pair-block freeness of projections by the additivity of $\chi_\mathrm{proj}$ (Theorem 4.1), in the proof of which a transportation cost inequality plays an important role. We also briefly discuss the free pressure in relation to $\chi_\mathrm{proj}$.


• [17] Remarks on HNN extensions in operator algebras
  Illinois J. Math., Vol.52, No.3 (2008 in print; 2009 on web), 705-725.
  [math.OA/0601706 <-- old ver.] abstract
  
  Abstract: It is shown that any HNN extension is precisely a compression by a projection of a certain amalgamated free product in the framework of operator algebras. As its applications several questions for von Neumann algebras or $C^*$-algebras arising as HNN extensions are considered. Comment: An observation given in the paper shows that `graphs of operator algebras' are reduced to amalgamated free products in essence.
  


• [16] A log-Sobolev type inequality for free entropy of two projections, (with Fumio Hiai)
  Annales IHP Probab. Stat., Vol.45, No.1 (2009), 239-249.
  [math.OA/0601171] abstract
  
  Abstract: We prove an inequality between the free entropy and the mutual free Fisher information for two projections, regarded as a free analog of the logarithmic Sobolev inequality. The proof is based on the random matrix approximation procedure via the Grassmannian random matrix model of two projections.


• [15] Notes on treeability and costs for discrete groupoids in operator algebra framework
  Abel Symposia I, (2006), 259-279.
  [math.OA/0504262] abstract
  
  Abstract: We provide an operator algebraic interpretation of discrete measurable groupoids in the course of re-proving (and slightly generalizing) a result on treeability due to Adams and Spatzier. Then, we reconstruct Gaboriau's beautiful work on costs of equivalence relations in operator algebra framework, avoiding any measure theoretic argument, and clarify what kind of his results can or cannot be generalized to the non-principal groupoid case.
  Comment: This originally came from my intensive course at the university of Tokyo, Nov. 2004. One of my students made the consideration in the latter half of this paper complete, see arXiv:1502.01555.


• [14] Free transportation cost inequalities for non-commutative multi-variables, (with Fumio Hiai)
  Inf. Dim. Analysis and Quant. Prob., Vol. 9, No.3 (2006), 391-412.
  [math.OA/0501238] abstract
  
  Abstract: The free analogue of the transportation cost inequality so far obtained for measures is extended to the noncommutative setting. Our free transportation cost inequality is for tracial distributions of noncommutative self-adjoint (also unitary) multi-variables in the framework of tracial $C^*$-probability spaces, and it tells that the Wasserstein distance is dominated by the square root of the relative free entropy with respect to a potential of additive type (corresponding to the free case) with some convexity condition. The proof is based on random matrix approximation procedure.
  Comment: We felt difficult to publish this. That's my first experience of this kind.


• [13] A free logarithmic Sobolev inequality on the unit circle, (with Fumio Hiai and Denis Petz)
  Canad. Math. Bull., Vol. 49, No. 3 (2006), 389-406.
  abstract
  
  Abstract: Free analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.
  Comment: This paper is a companion to [11].


• [12] HNN extensions of von Neumann algebras
  J. Funct. Anal., Vol. 225, No. 2 (2005), 383-426.
  [math.OA/0312439] abstract
  
  Abstract: Reduced HNN extensions of von Neumann algebras (as well as $C^*$-algebras) will be introduced, and their modular theory, factoriality and ultraproducts will be discussed. In several concrete settings, detailed analysis on them will be also carried out.
  Comment: I learned from Yasuo Watatani that HNN extensions are important in group theory through casual conversations with him. Then I tried to lay groundwork for them in operator algebras. This is essentially my first output after arriving at Kyushu.


• [11] Free transportation cost inequalities via random matrix approximation, (with Fumio Hiai and Denis Petz)
  Probab. Th. Relat. Fields, Vol. 130, No. 2 (2004), 199-221.
  abstract
  
  Abstract: By means of random matrix approximation procedure, we re-prove Biane and Voiculescus free analog of Talagrands transportation cost inequality for measures on the real line in a more general setup. Furthermore, we prove the free transportation cost inequality for measures on the circle as well by extending the method to special unitary random matrices.
  Comment: I learned much on LDP and random matrices from this joint work and the next one [13]. My personal starting point was that I naively thought that there should be a relation between the co-authors's previous work on free relative entropy in the $1$-dimensional case and Biane--Voiculescu's free Wasserstein distance.


• [10] Free product actions and their applications
  Quantum Probability and White Noise Analysis, Vol. 16 (2003), 388-411.
  [math.OA/0211168] abstract
  
  Abstract: A quick review of a series of our recent works on free product actions, partly in collaborations with Dimitri Shlyakhtenko and with Fumio Hiai, is given. We also work out type III theoretic subfactor analysis on subfactors associated with free product actions of $\mathrm{SU}_q(n)$. This part can be read as a supplementary appendix to those works.
  


• [9] Automorphisms of free product type and their crossed-products, (with Fumio Hiai)
  J. Operator Theory, Vol. 50, No.1 (2003), 119--130.
  [math.OA/0211163] abstract
  
  Abstract: A continuous family of non-outer conjugate aperiodic automorphisms whose crossed-products are all isomorphic is given on every interpolated free group factor. An explicit "duality" relationship between compact co-commutative Kac algebra (minimal) free product actions and free shift actions is also discussed.
  Comment: This joint work was started around the end of 1999, and we completed writing in 2001 when I was at MSRI.


• [8] Irreducible subfactors of $L(\mathbb{F}_\infty)$ of index $\lambda > 4$ , (with Dimitri Shlyakhtenko)
  J. reine angew. Math. (Crelle's Journal) Vol. 548 (2002), 149 -- 166.
  [math.OA/0010202] abstract
  
  Abstract: By utilizing an irreducible inclusion of type III$_{q^2}$ factors coming from a free-product type action of the quantum group $\mathrm{SU}_q(n)$, we show that the free group factor $L(\mathbb{F}_\infty)$ possesses irreducible subfactors of arbitrary index $\lambda >4$. Combined with earlier results of Radulescu, this shows that $L(\mathbb{F}_\infty)$ has irreducible subfactors with any possible index value.
  Comment: This papar was written at MSRI in 2000.


• [7] Fullness, Connes' $\chi$-groups, and ultra-products of amalgamated free products over Cartan subalgebras
  Trans. Amer. Math. Soc., Vol.355, No. 1 (2003), 349-371.
  abstract
  
  Abstract: Ultra-product algebras associated with amalgamated free products over Cartan subalgebras are investigated. As applications, their Connes' $\chi$-groups are computed in terms of ergodic theory, and also we clarify what condition makes them full factors (i.e., their inner automorphism groups become closed).
  Comment: I studied Connes's automorphism anaysis and Popa's maximal amenability paper, probably during my first stay in Berkeley (fall '98-- spring '99). Then I tried to apply those techniques to general amalgamated free products. This experience was conductive to [22,23] almost a decade from then. This paper was written at MSRI in 2000.


• [6] Amalgamated free product over Cartan subalgebra, II. Supplementary Results & Examples
  Advanced Studies in Pure Mathematics, Vol. 38 (2004), 239-265.
  [math.OA/0211164] abstract
  
  Abstract: Supplementary results obtained after the completion of our previous paper are given together with discussing some examples. A quick review of the previous paper is also included.
  Comment: This paper was written at Hiroshima in Summer 2000.


• [5] On the fixed-point algebra under a minimal free product-type action of the quantum group $\mathrm{SU}_q(n)$
  International Mathematics Research Notices, Vol. 2000, No.1 (2000), 35-56.
  abstract
  
  Abstract: In this note, a minimal action of $\mathrm{SU}_q(n)$ is investigated as a natural continuation of our previous work. The type (in the sense of Murray-von Neumann and Connes) of the fixed-point algebra is determined. As a simple application, an example of a pair of a type II$_1$ factor and its irreducible subfactor with an arbitrary index greater than four is constructed.
  Comment: Following Izumi's comment to my talk about [1] in Jan. 1997, I studied subfactors a little bit, and wrote this paper. I finished writing this paper at UCB in Jan. 1999. [1]--[5] were done during my graduate study Apr. 1996 - Mar. 1999.


• [4] A relation between certain interpolated Cuntz algebras and interpolated free group factors , (with Yasuo Watatani)
  Proc. Amer. Math. Soc., Vol. 128, No.5 (2000), 1397-1404.
  abstract
  
  Abstract: We investigate von Neumann algebras generated by the real parts of generators of Toeplitz extensions of interpolated Cuntz algebras on sub-Fock spaces. We show that some of them are isomorphic to interpolated free group factors. For example, in case of the golden number the corresponding free group factor rank is 3/2.
  Comment: I got an observation when I attended in Watatani's seminar talk about his joint work with Katayama and Matsumoto on $C^*$-algebras associated with $\beta$-shifts. The observation was subsequently developed by him to this short note. This paper was written in 1997.


• [3] Remarks on free products with respect to non-tracial states
  Math. Scand., Vol. 88, No.1 (2001), 111-125.
  abstract
  
  Abstract: We give some results on the questions of factoriality and type classification for free product von Neumann algebras. We also give a result on (normal) conditional expectations from free product von Neumann algebras onto thier subalgebras.
  Comment: I investigated the so-called ILP paper in 1996. A part of this work is its output. This work was probably done in 1997.


• [2] Amalgamated free product over Cartan subalgebra
  Pacific J. Math., Vol.191, No.2 (1999), 359-392.
  abstract
  
  Abstract: We study amalgamated free products of factors over their common Cartan subalgebras. We will show that the resulting amalgamated free product is a factor as long as given factors are non-type I and furthermore its (smooth) flow of weights is determined.
  Comment: This is my Ph.D thesis. I spent almost two years (the mid '95-- the mid '97) on this work.


• [1] A minimal action of the compact quantum group $\mathrm{SU}_q(n)$ on a full factor
  J. Math. Soc. Japan Vol.51, No. 2 (1999), 449-461.
  abstract
  
  Abstract: Based on the free product construction we show that a certain full factor of type III$_{q^2}$ admits a minimal coaction of the compact quantum group $\mathrm{SU}_q(n)$. Minimal coactions of compact Kac algebras are also investigated by the same technique.
  Comment: I already got an essential observation in 1995. However, I hadn't realized that it is worth until the mid 1996. This paper enabled me to continue with math.


• Co-authors (16):
  Yoshiki Aibara (1), Kei Hasegawa (1), Kanae Hatano (1), Fumio Hiai (9), Cyril Houdayer (2), Masaki Izumi (1), Shinya Kato (1), Roberto Longo (1), Miguel Martin (1), Takuho Miyamoto (1), Denis Petz (2), Dimitri Shlyakhtenko (1), Yoh Tanimoto (1), Reiji Tomatsu (1), Shuhei Wada (1), Yasuo Watatani (1)
• Journals (46):
  Soc: Scand, PAMS., ASPM, TAMS, CMB, Abel, BLMS, JLMS, MPCPS, Compos, Irish-Acad, CJM, Acad.-Canada, Studia, JAustMS
  Univ.: Pacific, Illinois, Johns Hopkins(AJM), Indiana, Fourier, Szeged, Muenster
  Publishers: IMRN, Crelle, IJM, Math. Ann., Adv. Math., MRL, Arch. Math.
  Domestic: JMSJ, RIMS, Nagoya, PJA, Kyoto
  FA: JOT, JFA, Positivity, JNCG, IEOT
  PR: QP-WN, PTRF, IDAQP, AIHP-PR, JOTP
  MP: CMP, SIGMA
  
• Topics:
  von Neumann algebra: [1-10,12,15,17,22-26,30-34,37]
  Free probability: [11,13,14,16,18,19,27-29,35,36,39]
  Random matrix: [11,13,16,36,39]
  Quantum group: [1,5,8,40,43,44]
  Non-selfadjoint operator algebra: [20,21,38]
  Subfactor: [5,8]
  Non-commutative $L^p$-space: [46]
  Operator theory: [41,42,45]
  $\infty$-dim representation theory: [40,43,44]
  Orbit equivalence: [15]
  AQFT: [37]
  QIT: [46]

Informal Publications and Reports (Unpublished versions & Notes)

• Inequalities related to free entropy derived from random matrix approximation, (with F. Hiai and D. Petz)
  Unpublished preprint. (2003). This is divided into [11,13] in the above list. [math.OA/0310453]
  
  
• Free Talagrand Inequality, pdf.
  Oberwolfach Reports No.15/2005, "Free Probability Theory", 857--861.
  Summary mainly on the joint work with Fumio Hiai [14].
  
  
• On orbital free entropy dimension, pdf.
  This is an outtake of one of my private notes for the proceedings of the RIMS conference, Sep. 07.
  We give a direct proof to the general upper bound of orbital free entropy dimension.
  
  
• Quick review on property (X), pdf.
  This is an outtake of my private notes for the proceedings of the RIMS conference, Jun. 08.
  We give a short survey on some techniques that are useful for proving the uniqueness of preduals.
  


• Free product factor and bicentralizer problem, pdf.
  This is originally a preprint circulated to restricted persons.
  We confirmed that any type III$_1$ free product factor has the trivial asymptotic bicentralizer.
  Only the Connes--Stormer transitivity and two results in [22] are used there. After that, Cyril Houdayer found that his technology gives a much simpler proof, which is included in [32].