Free analog of pressure and its Legendre transform

The free analog of the pressure is introduced for multivariate noncommutative random variables and its Legendre transform is compared with Voiculescu's microstate free entropy.Comment: Slightly corrected in April 200

Orbital approach to microstate free entropy

Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy $\chi_orb$ for non-commutative self-adjoint random variables (also for "hyperfinite random multi-variables"). Besides its basic properties the relation of $\chi_orb$ with the usual free entropy $\chi$ is shown. Moreover, the dimension counterpart $\delta_{0,orb}$ of $\chi_orb$ is discussed, and we obtain the relation of $\delta_{0,orb}$ with the original free entropy dimension $\delta_0$ with applications to $\delta_0$ itself.Comment: 38 pages; Section 5 was largely improved and Section 6 was adde