Fractal entropies and dimensions for microstates spaces

Using Voiculescu's notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebraic invariant. We compute the free Hausdorff dimension in the cases where the set generates a finite-dimensional algebra or where the set consists of a single selfadjoint. We show that the Hausdorff dimension becomes additive for such sets in the presence of freeness.