Conditional Free Entropy Dimension

Let M be a finite yon Neumann algebra that can be embedded into the ultrapower of the hyperfinite II1factor and X,Y be two finite sets of normal operators in M. Using Voiculescu’s notion of matricial microstates we introduce a quantityδ0(X|Y),which measures the freeness of X given Y.We establish some basic properties of δ0(X|Y).We prove that if the von Neumann algebra generated by Y is hyperfinite,then δ0(X,Y)=δ0(Y)+δ0(X|Y).(where Y is the(fraCtal)free entropy dimension of X in the sense of [1],which is Voiculsecu’s modified free entropy dimension[19]when X is a finite set of self-adjoint operators).As an application,we prove that if X1,…,Xn and Y1,…,Yn are finite sets of normal operators in M such that{Xi,Xi*}"={Yi,Yi*)"is hyperfinite for1≤i≤n,thenδ0(X1,…,Xn)=δ0(Y1,…,Yn).As another application,we obtain Jung’s hyperfinite inequality[9]for free entroyp dimension:If.X,Y,Z are finite sets of normal operators in M and X generates a hyperfinite von Neumann algebra,then δ0(X,Y,Z)≤δ0(X,Y)+δ0(X,Z)-δ0(X).For two Haar unitary operators U,W in M,we introduce a quantity δ’M(u|W),which also measures the freeness of U given W. This class of finite von Neumann algebras include many well-known examples.