Concavification of free entropy

We introduce a modification of Voiculescu's free entropy which coincides with the liminf variant of Voiculescu's free entropy on extremal states, but is a concave upper semi-continuous function on the trace state space. We also extend the orbital free entropy of Hiai, Miyamoto and Ueda to non-hyperfinite multivariables and prove freeness in case of additivity of Voiculescu's entropy (or vanishing of our extended orbital entropy).Comment: 28 pages, final version : details added in several proofs and relations to other variants of free entropy explaine

A Free Stochastic Partial Differential Equation

We get stationary solutions of a free stochastic partial differential equation. As an application, we prove equality of non-microstate and microstate free entropy dimensions under a Lipschitz like condition on conjugate variables, assuming also R^\omega\ embeddability. This includes an N-tuple of q-Gaussian random variables e.g. for |q|N\leq 0.13.Comment: 57 pages, final published version, slightly weaker results with improved expositio

A non-commutative Path Space approach to stationary free Stochastic Differential Equations

By defining tracial states on a non-commutative analogue of a path space, we construct Markov dilations for a class of conservative completely Markov semigroups on finite von Neumann algebras. This class includes all symmetric semigroups. For well chosen semigroups (for instance with generator any divergence form operator associated to a derivation valued in the coarse correspondence) those dilations give rise to stationary solutions of certain free SDEs previously considered by D. Shlyakhtenko. Among applications, we prove a non-commutative Talagrand inequality for non-microstates free entropy (relative to a subalgebra B and a completely positive map \eta:B\to B). We also use those new deformations in conjunction with Popa's deformation/rigidity techniques. For instance, combining our results with techniques of Popa-Ozawa and Peterson, we prove that the von Neumann algebra of a countable discrete group with CMAP and positive first L^2 Betti number has no Cartan subalgebras.Comment: 75 pages; new results : more resolutions of SDEs from our dilations, free Talagrand inequality generalized to relative case; (slightly) improved exposition in section 2, typos correcte

The simplex of tracial quantum symmetric states

We show that the space of tracial quantum symmetric states of an arbitrary unital C*-algebra is a Choquet simplex and is a face of the tracial state space of the universal unital C*-algebra free product of A with itself infinitely many times. We also show that the extreme points of this simplex are dense, making it the Poulsen simplex when A is separable and nontrivial. In the course of the proof we characterize the centers of certain tracial amalgamated free product C*-algebras.Comment: 14 pages; version 2 has an improved proof of one result and updated citations to the revised version of [4