To a real Hilbert space and a one-parameter group of orthogonal transformations we associate a C * -algebra which admits a free quasi-free state. This construction is a freeprobability analog of the construction of quasi-free states on the CAR and CCR algebras. We show that under certain conditions, our C * -algebras are simple, and the free quasi-free states are unique.The corresponding von Neumann algebras obtained via the GNS construction are free analogs of the Araki-Woods factors. Such von Neumann algebras can be decomposed into free products of other von Neumann algebras. For non-trivial one-parameter groups, these von Neumann algebras are type III factors. In the case the one-parameter group is nontrivial and almost-periodic, we show that Connes' Sd invariant completely classifies these algebras.

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 = T r + T c where T r (resp. T c ) 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.

No abstract

By changing to an orthogonal basis, we give a short proof that the subfactor of the graded algebra of a planar algebra reproduces the planar algebra.

Abstract. By solving a free analog of the Monge-Ampère equation, we prove a noncommutative analog of Brenier's monotone transport theorem: if an n-tuple of self-adjoint non-commutative random variables Z 1 , . . . , Z n satisfies a regularity condition (its conjugate variables ξ 1 , . . . , ξ n should be analytic in Z 1 , . . . , Z n and ξ j should be close to Z j in a certain analytic norm), then there exist invertible non-commutative functions F j of an n-tuple of semicircular variables S 1 , . . . , S n , so that Z j = F j (S 1 , . . . , S n ). Moreover, F j can be chosen to be monotone, in the sense that F j = D j g and g is a non-commutative function with a positive definite Hessian. In particular, we can deduce that. Thus our condition is a useful way to recognize when an n-tuple of operators generate a free group factor. We obtain as a consequence that the q-deformed free group factors Γ q (R n ) are isomorphic (for sufficiently small q, with bound depending on n) to free group factors. We also partially prove a conjecture of Voiculescu by showing that free Gibbs states which are small perturbations of a semicircle law generate free group factors. Lastly, we show that entrywise monotone transport maps for certain Gibbs measure on matrices are well-approximated by the matricial transport maps given by free monotone transport.