**Books**

- [1] (With N. P. Brown)
**C*-algebras and finite-dimensional approximations**.

Graduate Studies in Mathematics, 88. American Mathematical Society, 2008, 509 pp.

**Papers with short abstracts and errata**

- [47]
**A remark on fullness of some group measure space von Neumann algebras**.

Preprint. arXiv:1602.02654We give examples of non-singular group actions whose group measure space von Neumann algebras are full factors.

- [46]
**A functional analysis proof of Gromov's polynomial growth theorem**.

Preprint. arXiv:1510.04223We give a functional analysis proof of Gromov's polynomial growth theorem, along the lines of Shalom and Chifan--Sinclair.

- [45] (With R. Okayasu and R. Tomatsu)
**Haagerup approximation property via bimodules**.

Math. Scand., to appear. arXiv:1501.06293We prove that the von Neumann algebra of a locally compact quantum group with the Haagerup property has the Haagerup approximation property. The converse also holds if the locally compact quantum group is strongly inner amenable.

- [44] (With E. Breuillard, M. Kalantar, and M. Kennedy)
**C*-simplicity and the unique trace property for discrete groups**.

Preprint. arXiv:1410.2518We prove the simplicity of the reduced C*-algebra of a large class of discrete groups using a new characterization of this property in terms of boundary actions. We also prove that any tracial state on any reduced group C*-algebra is concentrated on the amenable radical.

- [43] (With D. Blecher)
**Real positivity and approximate identities in Banach algebras**.

Pacific J. Math., 277 (2015), 1--59. doi:10.2140/pjm.2015.277.1 arXiv:1405.5551We study a notion of positivity in approximately unital Banach algebras.

- [42] (With G. Pisier)
**A continuum of C*-norms on B(H) \otimes B(H) and related tensor products**.

Glasg. Math. J., 58 (2016), 433--443. doi:10.1017/S0017089515000257 arXiv:1404.7088We show that there are at least continuum many C*-norms on B(H) \otimes B(H). They give rise to as many injective tensor product functors for C*-algebras in Kirchberg's sense.

- [41] (With M. Rørdam and Y. Sato)
**Elementary amenable groups are quasidiagonal**.

Geom. Funct. Anal., 25 (2015), 307--316. doi:10.1007/s00039-015-0315-x arXiv:1404.3462We show that the group C*-algebra of any elementary amenable group is quasidiagonal. This is an offspring of recent progress in the classification theory of nuclear C*-algebras.

- [40] (With M. Mimura, H. Sako, and Y. Suzuki)
**Group approximation in Cayley topology and coarse geometry, Part III: Geometric property (T)**.

Algebr. Geom. Topol., 15 (2015) 1067--1091. doi:10.2140/agt.2015.15.1067 arxiv:1402.5105We study when the disjoint union of finite Cayley graphs of bounded degree has geometric property (T) and relate it to the behavior of Kazhdan's property (T) in the space of marked groups. We also explore cohomology theories for modules over coarse spaces.

- [39]
**Noncommutative real algebraic geometry of Kazhdan's property (T)**.

J. Inst. Math. Jussieu, 15 (2016), 85--90. doi:10.1017/S1474748014000309 arxiv:1312.5431A characterization of Kazhdan's property (T) is given in terms of noncommutative real algebraic geometry. This implies that property (T) is provable (semidecidable) and suggests the possibility of proving property (T) for a given group by computers.

- [38] (With Y. Choi and I. Farah)
**A nonseparable amenable operator algebra which is not isomorphic to a C*-algebra**.

Forum Math. Sigma, 2 (2014), e2 (12 pages). doi:10.1017/fms.2013.6 arxiv:1309.2145It is proved that there is an amenable operator algebra which is not isomorphic to a C*-algebra. Our example is nonseparable and the existence of a separable example remains an open problem.

- [37]
**Dixmier approximation and symmetric amenability for C*-algebras**.

J. Math. Sci. Univ. Tokyo, 20 (2013), 349--374. arxiv:1304.3523We study some general properties of tracial C*-algebras. In the first part, we use Dixmier type approximation theorem to characterize symmetric amenability for C*-algebras. In the second part, we consider continuous bundles of tracial von Neumann algebras and classify some of them.

- [36]
**About the Connes Embedding Conjecture---algebraic approaches---**.

Jpn. J. Math., 8 (2013), 147--183. doi:10.1007/s11537-013-1280-5 arXiv:1212.1700This is an expanded lecture note of the author's lectures for "Masterclass on sofic groups and applications to operator algebras" (University of Copenhagen, 5--9 November 2012). It is about algebraic aspects of the Connes Embedding Conjecture.

Erratum: The remark after Theorem 7 is not accurate, because it is not so "routine" to prove (1) <=> (2) in the real case. See doi:10.1016/j.aim.2013.10.020. - [35]
**Tsirelson's problem and asymptotically commuting unitary matrices**.

J. Math. Phys., 54 (2013), 032202 (8 pages). doi:10.1063/1.4795391 arXiv:1211.2712We consider quantum correlations of bipartite systems having a slight interaction, and reinterpret Tsirelson's problem (and hence Kirchberg's and Connes's conjectures) in terms of finite-dimensional asymptotically commuting positive operator valued measures.

- [34] (With W. B. Johnson
and G. Schechtman)
**A quantitative version of the commutator theorem for zero trace matrices**.

Proc. Nat. Acad. Sci., 110 (2013), 19251--19255. doi:10.1073/pnas.1202411109 arXiv:1202.0986It is well-known that every trace zero matrix

*A*is a commutator: there are*B*and*C*such that*A=BC-CB*. In this paper, we address the problem of bounding the norms of*B*and*C*in terms of the norm and the size of*A*. - [33] (With G. Godefroy)
**Free Banach spaces and the approximation properties**.

Proc. Amer. Math. Soc., 142 (2014), 1681--1687. doi:10.1090/S0002-9939-2014-11933-2 arXiv:1201.0847We characterize the metric spaces whose free space has the bounded approximation property through a Lipschitz analogue of the local reflexivity principle. We show that there exist compact metric spaces whose free spaces fail the approximation property.

- [32] (With N. Monod and
A. Thom)
**Is an irng singly generated as an ideal?**

Internat. J. Algebra Comput., 22 (2012), 1250036 (6 pages). doi:10.1142/S0218196712500361 arXiv:1112.1802In this note, we address the problem whether every finitely generated idempotent ideal of a ring is singly generated as an ideal.

- [31]
**A remark on contractible Banach algebras**.

Kyushu J. Math., 67 (2013), 51--53. doi:10.2206/kyushujm.67.51 arXiv:1110.6216It is proved that a contractible Banach algebra acting on a Banach space with the uniform approximation property is finite-dimensional.

- [30]
**Metric spaces with subexponential asymptotic dimension growth**.

Internat. J. Algebra Comput., 22 (2012), 1250011 (3 pages). doi:10.1142/S0218196711006777 arxiv:1108.3165We prove that a metric space with subexponential asymptotic dimension growth has Yu's property A.

- [29]
**Examples of groups which are not weakly amenable**.

Kyoto J. Math., 52 (2012), 333--344. doi:10.1215/21562261-1550985 arXiv:1012.0613We prove that weak amenability of a locally compact group imposes a strong condition on its amenable closed normal subgroups. This extends non weak amenability results of Haagerup (1988) and Ozawa--Popa (2010). A von Neumann algebra analogue is also obtained.

Erratum: In Proof of Theorem A, before the displayed inequality, Ad_g should mean the standard unitary operator u_g on L^2(N) that implements the automorphism Ad_g of the group von Neumann algebra LN. Concretely, (u_g \zeta)(x) = d_g^{1/2}\zeta(g^{-1}xg), where d_g is the positive real number such that Ad_g m = d_g m for the Haar measure m on N. - [28] (With M. Burger and
A. Thom)
**On Ulam stability**.

Israel J. Math., 193 (2013), 109--129. doi:10.1007/s11856-012-0050-z arxiv:1010.0565An ε-represenation of a (discrete) group is a map into the unitary group on a Hilbert space which is multiplicative up to an error ε. We study when an ε-represenation is a perturbation of an exact representation and, if it is the case, whether such an exact representation is unique.

- [27]
**Quasi-homomorphism rigidity with noncommutative targets**.

J. Reine Angew. Math., 655 (2011), 89--104. doi:10.1515/crelle.2011.034 arXiv:0911.3975As a strengthening of Kazhdan's property (T), property (TT) was introduced by Burger and Monod. In this paper, we add more rigidity and introduce property (TTT). Partially upgrading a result of Burger and Monod, we will prove that SL(

*n*,**R**) with*n*at least 3 and their lattices have property (TTT). - [26] (With N. Monod)
**The Dixmier problem, lamplighters and Burnside groups**.

J. Funct. Anal., 258 (2010), 255--259. doi:j.jfa.2009.06.029 arXiv:0902.4585J. Dixmier asked in 1950 whether every non-amenable group admits uniformly bounded representations that cannot be unitarised. We provide such representations upon passing to extensions by abelian groups. As a corollary, we deduce that certain Burnside groups are non-unitarisable, answering a question of G. Pisier.

- [25] (With S. Popa)
**On a class of II_1 factors with at most one Cartan subalgebra II**.

Amer. J. Math., 132 (2010), 841--866. doi:10.1353/ajm.0.0121 arXiv:0807.4270This is a continuation of our previous paper studying the structure of Cartan subalgebras of von Neumann factors of type II_1. We provide more examples of II_1 factors having either zero, one or several Cartan subalgebras. We also prove a rigidity result for some group measure space II_1 factors.

Erratum: Remark 5.5 is not justified, but the result is recovered by Sinclair (doi:10.1016/j.jfa.2010.12.017). - [24]
**An example of a solid von Neumann algebra**.

Hokkaido Math. J., 38 (2009), 557--561. doi:10.14492/hokmj/1258553976 arXiv:0804.0288We prove that the group-measure-space von Neumann algebra

*L*^{∞}(*T*^{2}) \rtimes SL(2,*Z*) is solid. The proof uses topological amenability of the action of SL(2,*Z*) on the Higson corona of*Z*^{2}. - [23] (With
S. Popa)
**On a class of II_1 factors with at most one Cartan subalgebra**.

Ann. of Math. (2), 172 (2010), 713--749. doi:10.4007/annals.2010.172.713 arXiv:0706.3623We prove that the von Neumann subalgebra generated by the normalizer of any diffuse amenable subalgebra of a free group factor is again amenable. We also prove that certain group measure space factor has a unique Cartan subalgebra, up to unitary conjugacy.

- [22]
**Weak amenability of hyperbolic groups**.

Groups Geom. Dyn., 2 (2008), 271--280. doi:10.4171/GGD/40 arXiv:0704.1635We prove that hyperbolic groups are weakly amenable. This partially extends the result of Cowling and Haagerup showing that lattices in simple Lie groups of real rank one are weakly amenable.

- [21]
**Boundary amenability of relatively hyperbolic groups**.

Topology Appl., 153 (2006), 2624--2630. doi:10.1016/j.topol.2005.11.001 math.GR/0501555Let

*K*be a fine hyperbolic graph and*G*be a group acting on*K*with finite quotient. We prove that*G*is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups are exact. - [20]
**Boundaries of reduced free group C*-algebras**.

Bull. London Math. Soc., 39 (2007), 35--38. doi:10.1112/blms/bdl003 math.OA/0411474We prove that the crossed product C*-algebra of a free group with its boundary naturally sits between the reduced group C*-algebra and its injective envelope.

- [19]
**Weakly exact von Neumann algebras**.

J. Math. Soc. Japan, 59 (2007), 985--991. doi:10.1215/21562261-1550985 math.OA/0411473We study the structure of weakly exact von Neumann algebras and give a local characterization of weak exactness. As a corollary, we prove that a discrete group is exact if and only if its group von Neumann algebra is weakly exact.

- [18]
**A note on non-amenability of**.*B(\ell_p)*for*p=1,2*

Internat. J. Math., 15 (2004), 557--565. doi:10.1142/S0129167X04002430 math.FA/0401122This is an expository note on non-amenabilty of the Banach algebra

*B(\ell_p)*for*p=1,2*. These were proved respectively by Connes (*p=2*) and Read (*p=1*) via very different methods. We give a single proof which reproves both. - [17]
**A Kurosh type theorem for type II_1 factors**.

Int. Math. Res. Not., Volume 2006, Article ID97560 (21 pages). doi:10.1155/IMRN/2006/97560 math.OA/0401121We prove a Kurosh type theorem for free-product type II_1 factors. In particular, if

*M = LF_2 \otimes R*, then the free-product type II_1 factors*M*...*M*are all prime and pairwise non-isomorphic.

Erratum: A part of Remark 4.8 is not justified, but the result is recovered by Isono (doi:10.1016/j.jfa.2012.10.021). - [16] (With
S. Popa)
**Some prime factorization results for type II_1 factors**.

Invent. Math., 156 (2004), 223--234. doi:10.1007/s00222-003-0338-z math.OA/0302240We prove several unique prime factorization results for tensor products of type II_1 factors comming from (subgroups of) hyperbolic groups. In particular, we prove that the tensor product type II_1 factor of

*n*free groups cannot be embedded in that of*m*free groups with*m < n*. - [15]
**Solid von Neumann algebras**.

Acta Math., 192 (2004), 111--117. doi:10.1007/BF02441087 math.OA/0302082We prove that the relative commutant of a diffuse von Neumann subalgebra in a hyperbolic group von Neumann algebra is always injective. It follows that any von Neumann subalgebra in a hyperbolic group von Neumann algebra is either injective or non-Gamma. The proof is based on C*-algebra theory.

Erratum: The proof of Proposition 7 requires N_0=M. While this is enough for the application, I do not know if Proposition 7 is valid as it is stated. - [14]
**About the QWEP conjecture**.

Internat. J. Math., 15 (2004), 501--530. doi:10.1142/S0129167X04002417 math.OA/0306067A survey on the QWEP conjecture and Connes' Approximate Embedding Problem, mainly focusing on Kirchberg's work.

- [13] (With M. A. Rieffel)
**Hyperbolic group C*-algebras and free-product C*-algebras as compact quantum metric spaces**.

Canad. J. Math., 57 (2005), 1056--1079. doi:10.4153/CJM-2005-040-0 math.OA/0302310Any length function on a group

*G*defines a metric on the state space of the reduced C*-algbera of*G*. We show that if*G*is a hyperbolic group with any word length, then the topology from this metric coincides with the weak*-topology. Thus, the state space equipps a structure of a "compact quantum metric space". - [12]
**There is no separable universal II_1-factor**.

Proc. Amer. Math. Soc., 132 (2004), 487--490. doi:10.1090/S0002-9939-03-07127-2 math.OA/0210411Gromov constructed uncountably many discrete groups with Kazhdan's property (T). We show that no separable II_1-factor can contain all these groups in its unitary group. It follows there is no separable universal II_1-factor. We also show that the full group C*-algebras of some of these groups fail the lifting property.

- [11]
**Homotopy invariance of AF-embeddability**.

Geom. Funct. Anal., 13 (2003), 216--222. doi:10.1007/s000390300005 math.OA/0201191Homotopy invariance of AF-embeddability in the class of separable exact C*-algebras is proved.

- [10]
**An application of expanders to**.*B(\ell_2)\otimes B(\ell_2)*

J. Funct. Anal., 198 (2003), 499--510. doi:10.1016/S0022-1236(02)00107-6 math.OA/0110151We prove that the minimal tensor product of

*B(\ell_2)*with itself does not have the WEP (weak expectation property). We also give an example of an inclusion of comapct metrizable spaces with compatible SL(3,Z)-actions, for which the six-term exact sequence in equivariant KK-theory fails. - [9] (With A. Kishimoto and S. Sakai)
**Homogeneity of the pure state space of a separable C*-algebra**.

Canad. Math. Bull., 46 (2003), 365--372. doi:10.4153/CMB-2003-038-3 math.OA/0110152We prove that two pure states on a separable C*-algebra are translated each other by an asymptotic inner automorphism iff their GNS representations have the same kernel.

- [8] (With
M. Junge and
Z.-J. Ruan)
**On**.*OL*$_\infty$ structures of nuclear C*-algebras

Math. Ann., 325 (2003), 449--483. doi:10.1007/s00208-002-0384-7 math.OA/0206061We study the underlying local operator space stuctures of nuclear C*-algebras. In particular, it is shown that a C*-algebra is nuclear iff it is an

*OL*$_\infty$ space. - [7]
**Amenable actions and exactness for discrete groups**.

C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 691--695. doi:10.1016/S0764-4442(00)00248-2 math.OA/0002185We show that a discrete group is exact iff its left translation action on the Stone-Čech compactification is amenable. Combined with a result of Gromov, this proves the existence of non exact discrete groups.

- [6]
**On the set of finite-dimensional subspaces of preduals of von Neumann algebras**.

C. R. Acad. Sci. Paris Ser. I Math., 331 (2000), 309--312. doi:10.1016/S0764-4442(00)01646-3 dviWe show that for each

*d*, the metric space of all*d*-dimensional subspaces of non-commutative*L*_{1}-spaces is compact (in its natural topology induced from the cb-distance). - [5]
**Almost completely isometric embeddings between preduals of von Neumann algebras**.

J. Funct. Anal., 186 (2001), 329--341. doi:10.1006/jfan.2001.3796 dviWe show an operator analogue of Dor's theorem on embedding of

*L*_{1}-spaces. Namely, if a non-commutative*L*_{1}-space is embedded in another non-commutative*L*_{1}-space almost completely isometrically, then it is complemented almost completely contractively. - [4] (with P.-W. Ng)
**A characterization of completely 1-complemented subspaces of non-commutative**.*L*^{1}-spaces

Pacific J. Math., 205 (2002), 171--195. doi:10.2140/pjm.2002.205.171We show that a subspace of a non-commutative

*L*_{1}-space is completely isometrically complemented iff it is completely isometric to a corner of a non-commutative*L*_{1}-space. We also show that separable abstract non-commutative*L*_{1}-spaces (called*OL*_{1+}-spaces) are indeed non-commutative*L*_{1}-spaces. - [3] (With
E. G. Effros
and Z.-J. Ruan)
**On injectivity and nuclearity for operator spaces**.

Duke Math. J., 110 (2001) 489--521. doi:10.1215/S0012-7094-01-11032-6We show that a dual injective operator space is a corner of an injective von Neumann algebra. In particular, an operator space is nuclear iff it is locally reflexive and its second dual is injective. We also show that (isometrically) exact operator spaces are always locally reflexive.

- [2]
**A non-extendable bounded linear map between C*-algebras**.

Proc. Edinburgh Math. Soc., 44 (2001), 241--248. doi:10.1017/S0013091599000978 dviWe present an example of a bounded linear map from a C*-algebra into

*B(H)*which is non-extendable. As an application, we find Q-spaces which are not exact. - [1]
**On the lifting property for universal C*-algebras of operator spaces**.

J. Op. Theory, 46 (2001), 579--591. Published version dviWe define and investigate OLLP (Operator Local Lifting Property) for operator spaces, which is analogous to the LLP (Local Lifting Property) for C*-algebras.

**Notes & Conference Proceedings**

- [7]
**A remark on amenable von Neumann subalgebras in a tracial free product**.

Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), 104. doi:10.3792/pjaa.91.104 arXiv:1501.06373We present a short proof of a result of C. Houdayer. Namely, for any nontrivial tracial free product M = M_1*M_2 of finite von Neumann algebras M_i and any amenable subalgebra A of M such that A \cap M_1 is diffuse, we prove that A is in fact contained in M_1.

- [6]
**Lecture on the Furstenberg boundary and C*-simplicity**.

pdfThis is a handout for the lecture at the domestic "Annual Meeting of Operator Theory and Operator Algebras" at Toyo University in December 2014.

- [5]
**Hyperlinearity, sofic groups and applications to group theory**.

pdfThis is a handwritten note prepared for the lectures with the same title at ``Approximation Properties of Discrete Groups and Operator Spaces'' at TAMU in August 2009.

- [4]
**A comment on free group factors**.

Noncommutative harmonic analysis with applications to probability II. Banach Center Publ., 89 (2010), 241--245. pdfLet

*M*be a finite von Neumann algebra acting on the standard Hilbert space*L*^{2}(*M*). We look at the space of those bounded operators on*L*^{2}(*M*) that are compact as operators from*M*into*L*^{2}(*M*). The case where*M*is the free group factor is particularly interesting. - [3]
**Amenable Actions And Applications**.

International Congress of Mathematicians. Vol. II, 1563--1580, Eur. Math. Soc., Zürich, 2006. pdfThis is a survey on amenable actions and exact groups for the ICM 2006.

- [2]
**An Invitation to the Similarity Problems (after Pisier)**.

Surikaisekikenkyusho Kokyuroku, 1486 (2006), 27--40. pdfThis is a handout for the minicourse given in RIMS workshop ``Operator Space Theory and its Applications'' in January 2006, where I surveyed some aspects of Similarity Problems for representations of operator algebras and discrete groups, highlighting Pisier's theorem that Strong Similarity Property is equivalent to amenability.

- [1]
**Nuclearity of reduced amalgamated free product C*-algebras**.

Surikaisekikenkyusho Kokyuroku, 1250 (2002), 49--55. dviWe prove that the reduced amalgamated free product of two nuclear C*-algebras is again nuclear provided that either (i) one of states is pure (with trivial amalgamation), or (ii) one of GNS-representations contains the `compact operators'.

These researches were carried out while I was staying at Texas A&M University [1-3; 30, 33-34]; Institut Henri Poincaré [4-6, 8; 31-32; 42]; Université Paris 6 [7]; UC Berkeley [12-13; 16-17]; UCLA [14-15; 22-23, 25]; Hausdorff Institute [27-28]; University of Tokyo [9-11, 18-21, 24-26, 29]; University of Copenhagen [35-36]; RIMS [37-41,44-45]; Fields Institute [43].

Back to the main Index.