**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**
(Reprints available upon request.)

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

Preprint. 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.

- [28] (With M. Burger and
A. Thom)
**On Ulam stability**.

Israel J. Math., accepted. 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., to appear. 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. 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. 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.

- [24]
**An example of a solid von Neumann algebra**.

Hokkaido Math. J., 38 (2009), 557--561. 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. 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. 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. 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. 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. 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. 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]
**About the QWEP conjecture**.

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

- [16]
**A Kurosh type theorem for type II_1 factors**.

Int. Math. Res. Not., Volume 2006, Article ID97560. 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. - [15] (With
S. Popa)
**Some prime factorization results for type II_1 factors**.

Invent. Math., 156 (2004), 223--234. 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*. - [14]
**Solid von Neumann algebras**.

Acta Math., 192 (2004), 111--117. 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. - [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. 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. 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. 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. 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. 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*$_\infty$ structure of nuclear*C**-algebras

Math. Ann., 325 (2003), 449--483. 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. 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. 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. 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.We 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.We 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. 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. 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**

- [Notes4]
**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.

- [Notes3]
**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. - [Notes2]
**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.

- [Notes1]
**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 at TAMU [1-3], at Institut Henri Poincaré [4-6 and 8 in part], at Université Paris 6 [7], at MSRI [none], at UC Berkeley [12-13 and 16-17], at UCLA [14-15,22-23,25 in part], at HIM [27-28], and at University of Tokyo [9-11,18-21,24-26,29].

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