Authentic Publications (Published Papers & Preprints)

• [24] Some analysis on amalgamated free products of von Neumann algebras in non-tracial setup
  Preprint. [arXiv:1203.1806]
  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.


• [23] On type III$_1$ factors arising as free products
  Math. Res. Lett., Vol.18, No.5 (2011), 909--920. [arXiv:1101.4991]
  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.


• [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: We give a complete answer to the questions of factoriality, type classification and fullness for arbitrary free product von Neumann algebras.


• [21] On the predual of non-commutative $H^\infty$
  Bull. London Math. Soc., Vol.43, No.5 (2011), 886--896. [arXiv:1002.3672]
  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 $M_\star$.


• [20] On peak phenomena for non-commutative $H^\infty$
  Math. Ann., Vol.343, No.2 (2009), 421--429. [arXiv:0802.3449]
  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.


• [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: 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.


• [18] Notes on microstate free entropy of projections, (with Fumio Hiai)
  Publ. RIMS, Vol.44, No.1 (2008), 49--89. [math.OA/0605633]
  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: 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.


• [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: 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: 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.


• [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: 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.


• [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: 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.


• [12] HNN extensions of von Neumann algebras
  J. Funct. Anal., Vol. 225, No. 2 (2005), 383--426. [math.OA/0312439]
  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.


• [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: 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.


• [10] Free product actions and their applications
  Quantum Probability and White Noise Analysis, Vol. 16 (2003), 388-411. [math.OA/0211168]
  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: 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.


• [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: 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.


• [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: 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).


• [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: 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.


• [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: 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.


• [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: 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 .


• [3] Remarks on free products with respect to non-tracial states
  Math. Scand., Vol. 88, No.1 (2001), 111--125.
  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.


• [2] Amalgamated free product over Cartan subalgebra
  Pacific J. Math., Vol.191, No.2 (1999), 359--392.
  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.


• [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: 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.

Research Topics (roughly classified):

• Free products and related constructions in operator algebras: [2],[3],[6],[7],[12],[17],[22],[23],[24]
• (Quantum) group actions on von Neumann algebras and subfactors: [1],[5],[8],[9],[10]
• Free probability; especially free entropy: [11],[13],[14],[16],[18],[19]
• Banach space study on non-commutative function spaces: [20],[21]
• Miscellaneous: [4],[15]

Informal Publications and Reports (Unpublised versions & Notes)

• Inequalities related to free entropy derived from random matrix approximation , (with F. Hiai and D. Petz)
  Unpublished preprint. (2003). Parts of this are 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.