Free Diffusions and Property Ao

for a suitably locally convex multivariable *-polynomial V . Indeed, they were able to establish that such an SDE has a unique stationary distribution μV satisfying the Schwinger-Dyson equation μV ⊗ μV (∂iP ) = μV (DiV P ) where ∂i is the non-commutative partial difference quotient and Di is the cyclic partial derivative. By using the fact that they also had convergence in norm to this distribution from all initial data, they were able to show that the von Neumann algebra MV generated by operators with joint law μV is a factor with the Haagerup property. They also proved that MV has finite free entropy dimension and hence is prime and has no Cartan subalgebras. All of this provided evidence for the conjecture of Voiculescu that MV is isomorphic to a free group factor. Recall that a von Neumann algebra M ⊆ B(H) is said to have property AO if there are ultraweakly dense C subalgebras A ⊆ M and B ⊆ M ′ with A locally reflexive and such that the *-homomorphism Φ : A⊗B → B(H)/K given by