Orbital approach to microstate free entropy

Abstract

Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy $χ_orb$ for non-commutative self-adjoint random variables (also for "hyperfinite random multi-variables"). Besides its basic properties the relation of $χ_orb$ with the usual free entropy $χ$ is shown. Moreover, the dimension counterpart $δ_{0,orb}$ of $χ_orb$ is discussed, and we obtain the relation of $δ_{0,orb}$ with the original free entropy dimension $δ_0$ with applications to $δ_0$ itself.