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Inventiones Mathematicae | 2002

Grothendieck’s theorem for operator spaces

Gilles Pisier; Dimitri Shlyakhtenko

Abstract.We prove several versions of Grothendieck’s Theorem for completely bounded linear maps T:E→F*, when E and F are operator spaces. We prove that if E, F are C*-algebras, of which at least one is exact, then every completely bounded T:E→F* can be factorized through the direct sum of the row and column Hilbert operator spaces. Equivalently T can be decomposed as T=Tr+Tc where Tr (resp. Tc) factors completely boundedly through a row (resp. column) Hilbert operator space. This settles positively (at least partially) some earlier conjectures of Effros-Ruan and Blecher on the factorization of completely bounded bilinear forms on C*-algebras. Moreover, our result holds more generally for any pair E, F of “exact” operator spaces. This yields a characterization of the completely bounded maps from a C*-algebra (or from an exact operator space) to the operator Hilbert space OH. As a corollary we prove that, up to a complete isomorphism, the row and column Hilbert operator spaces and their direct sums are the only operator spaces E such that both E and its dual E* are exact. We also characterize the Schur multipliers which are completely bounded from the space of compact operators to the trace class.


Crelle's Journal | 2005

L 2-homology for von Neumann algebras

Alain Connes; Dimitri Shlyakhtenko

The aim of this paper is to introduce a notion of L-homology in the context of von Neumann algebras. Finding a suitable (co)homology theory for von Neumann algebras has been a dream for several generations (see [KR71a, KR71b, JKR72, SS95] and references therein). One’s hope is to have a powerful invariant to distinguish von Neumann algebras. Unfortunately, little positive is known about the Kadison-Ringrose cohomology H∗ b (M,M), except that it vanishes in many cases. Furthermore, there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von Neumann algebra. Our interest in developing an L-cohomology theory was revived by the introduction of Lcohomology invariants in the field of ergodic equivalence relations in the paper of Gaboriau [Gab02]. His results in particular imply that L-Betti numbers β (2) i (Γ) of a discrete group are the same for measure-equivalent groups (i.e., for groups that can generate isomorphic ergodic measure-preserving equivalence relations). Parallels between the “worlds” of von Neumann algebras and measurable equivalence relations have been noted for a long time (starting with the parallel between the work of Murray and von Neumann [MvN] and that of H. Dye [Dy]). Thus there is hope that an invariant of a group that “survives” measure equivalence will survive also “von Neumann algebra equivalence”, i.e., will be an invariant of the von Neumann algebra of the group. The original motivation for our construction comes from the well understood analogy between the theory of II1-factors and that of discrete groups, based on the theory of correspondences [Con, Con94]. This analogy has been remarkably efficient to transpose analytic properties such as “amenability” or “property T” from the group context to the factor context [Con80] [CJ] and more recently in the breakthrough work of Popa [Popa] [Con03]. We use the theory of correspondences together with the algebraic description of L-Betti numbers given by Luck. His definition involves the computation of the algebraic group homology with coefficients in the group von Neumann algebra. Following the guiding idea that the category of bimodules over a von Neumann algebra is the analogue of the category of modules over a group, we are led to the following algebraic definition of L-homology of a von Neumann algebra M :


arXiv: Operator Algebras | 2001

EXACTNESS OF CUNTZ–PIMSNER C*-ALGEBRAS

Ken Dykema; Dimitri Shlyakhtenko

Let H be a full Hilbert bimodule over a C*-algebra A. We show that the Cuntz-Pimsner C*-algebra associated to H is exact if and only if A is exact. Using this result, we give alternative proofs for exactness of reduced amalgamated free products of exact C*-algebras. In the case that A is a finite dimensional C*-algebra, we also show that the Brown-Voiculescu topological entropy of Bogljubov automorphisms of the Cuntz-Pimsner algebra associated to an A,A Hilbert bimodule is zero.


International Mathematics Research Notices | 2002

Operator-valued distributions. I. Characterizations of freeness

Alexandru Nica; Dimitri Shlyakhtenko; Roland Speicher

Let


International Mathematics Research Notices | 2004

Some estimates for non-microstates free entropy dimension with applications to q-semicircular families

Dimitri Shlyakhtenko

M


Canadian Journal of Mathematics | 2001

R-Diagonal Elements and Freeness With Amalgamation

Alexandru Nica; Dimitri Shlyakhtenko; Roland Speicher

be a


Communications in Mathematical Physics | 2012

Loop models, random matrices and planar algebras

Alice Guionnet; Vaughan F. R. Jones; Dimitri Shlyakhtenko; Paul Zinn-Justin

B


Acta Mathematica | 2002

Topological entropy of free product automorphisms

Nathanial P. Brown; Ken Dykema; Dimitri Shlyakhtenko

-probability space. Assume that


Proceedings of the National Academy of Sciences of the United States of America | 2000

Prime type III factors

Dimitri Shlyakhtenko

B


Transactions of the American Mathematical Society | 2015

Freely independent random variables with non-atomic distributions

Dimitri Shlyakhtenko; Paul Skoufranis

itself is a

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Alice Guionnet

Massachusetts Institute of Technology

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Sorin Popa

University of California

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Stephen Curran

University of California

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Cyril Houdayer

École normale supérieure de Lyon

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