Fractal entropies and dimensions for microstate spaces

TL;DR

This paper introduces fractal dimensions and entropies for microstate spaces of selfadjoint operators in von Neumann algebras, relating them to free entropy and showing their invariance and additivity properties.

Contribution

It defines new fractal dimensions and entropies for microstate spaces, connecting them to free entropy and establishing their algebraic invariance and additive behavior.

Findings

Free Hausdorff dimension is computed for finite-dimensional algebras and single operators.

The new dimensions relate to free entropy and are algebraically invariant.

Free Hausdorff dimension is additive under freeness.

Abstract

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 free Hausdorff dimension becomes additive for such sets in the presence of freeness.