Books
Papers with short abstracts (Reprints available upon request.)
We 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.
This 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: A remark after Theorem 7 is not accurate, because it is not so "routine" to prove (1) <=> (2) in the real case. See arXiv:1303.3711.
We 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.
It 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.
We 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.
In this note, we address the problem whether every finitely generated idempotent ideal of a ring is singly generated as an ideal.
It is proved that a contractible Banach algebra acting on a Banach space with the uniform approximation property is finite-dimensional.
We prove that a metric space with subexponential asymptotic dimension growth has Yu's property A.
We 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.
An 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.
As 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).
J. 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.
This 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 was recovered by Sinclair (arXiv:1009.2247).
We prove that the group-measure-space von Neumann algebra L∞(T2) \rtimes SL(2,Z) is solid. The proof uses topological amenability of the action of SL(2,Z) on the Higson corona of Z2.
We 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.
We 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.
Let 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.
We 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.
We 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.
This 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.
We 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 was recovered by Isono (arXiv:1206.0388).
We 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.
We 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 in its generality.
A survey on the QWEP conjecture and Connes' Approximate Embedding Problem, mainly focusing on Kirchberg's work.
Any 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".
Gromov 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.
Homotopy invariance of AF-embeddability in the class of separable exact C*-algebras is proved.
We 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.
We 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.
We 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.
We 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.
We show that for each d, the metric space of all d-dimensional subspaces of non-commutative L1-spaces is compact (in its natural topology induced from the cb-distance).
We show an operator analogue of Dor's theorem on embedding of L1-spaces. Namely, if a non-commutative L1-space is embedded in another non-commutative L1-space almost completely isometrically, then it is complemented almost completely contractively.
We show that a subspace of a non-commutative L1-space is completely isometrically complemented iff it is completely isometric to a corner of a non-commutative L1-space. We also show that separable abstract non-commutative L1-spaces (called OL1+-spaces) are indeed non-commutative L1-spaces.
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.
We 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.
We define and investigate OLLP (Operator Local Lifting Property) for operator spaces, which is analogous to the LLP (Local Lifting Property) for C*-algebras.
Notes
This 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.
Let M be a finite von Neumann algebra acting on the standard Hilbert space L2(M). We look at the space of those bounded operators on L2(M) that are compact as operators from M into L2(M). The case where M is the free group factor is particularly interesting.
This 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.
We 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; 30, 33], at Institut Henri Poincaré [4-6, 8; 31-32], at Université Paris 6 [7], at UC Berkeley [12-13; 16-17], at UCLA [14-15; 22-23, 25], at HIM [27-28], and at University of Tokyo [9-11, 18-21, 24-26, 29].
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