Free entropies, Free Fisher information, Free Stochastic differential equations, with applications to von Neumann algebras

This work extends our knowledge of free entropies, free Fisher information and free stochastic differential equations in three directions. First, we prove that a W*-probability space generated by more than 2 self-adjoints with finite non-microstate tree Fisher information doesn't have property Gamma of Murray and von Neumann (especially is not amenable). This is an analog of a well-known result of Voiculescu for microstate free entropy. We also prove factoriality in case of finite non-microstate entropy. Second, we study a general free stochastic differential equation with unbounded coefficients ("stochastic PDE"), and prove stationarity of solutions in well-chosen cases. This leads to a computation of microstates free entropy dimension in case of Lipschitz conjugate variable. Finally, we introduce a non-commutative path space approach to solve general stationary free Stochastic differential equations. By defining tracial states on a non-commutative analog 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. Among applications, we prove a non-commutative Talagrand inequality for non-microstate free entropy (relative to a subalgebra B and a symmetric completely positive map eta : B → B). We also use those new deformations in conjunction with Popa's deformation/rigidity theory. 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 L2 Betti number has no Cartan subalgebra.