A Microstates Approach to Relative Free Entropy

We define and study a relative free entropy quantity, analogous in its properties to Voiculescu's relative free entropy Chi^*(...:B). Our definition uses matricial microstates, unlike his definition, which involves non-commutative Hilbert transform. We prove a change of variable formula and certain maximization results for our quantity. We also exhibit a connection between the free entropy of a matrix with operator entries, relative to the algebra of scalar matrices, with the free entropy of the entries of the matrix.