[Paper Review] Fractal entropies and dimensions for microstate spaces

Kenley Jung|ArXiv.org|Dec 1, 2002

Random Matrices and Applications4 references3 citations

TL;DR

This paper introduces fractal geometric invariants—free Hausdorff dimension and entropy—for microstate spaces of selfadjoint operators in tracial von Neumann algebras, using Voiculescu's matricial microstate framework. It establishes that a modified free Hausdorff dimension is an algebraic invariant, computes it for finite-dimensional algebras and single selfadjoints, and proves additivity under freeness, linking microstate geometry to von Neumann algebra structure.

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.

Motivation & Objective

• To develop fractal geometric invariants—specifically free Hausdorff dimension and entropy—for microstate spaces of selfadjoint operators in tracial von Neumann algebras.
• To address the open problem of whether free entropy dimension is an invariant of the generated von Neumann algebra by analyzing the asymptotic geometry of microstate spaces.
• To establish that a modified version of free Hausdorff dimension is an algebraic invariant, independent of the choice of generators.
• To compute free Hausdorff dimension explicitly in cases where the generators form a finite-dimensional algebra or consist of a single selfadjoint operator.
• To explore the relationship between microstate geometry and structural properties of the generated von Neumann algebra, such as the existence of minimal projections.

Proposed method

• Uses Voiculescu’s matricial microstate framework to define free Hausdorff dimension and entropy via asymptotic packing and covering of microstate sets.
• Applies classical Hausdorff measure theory to microstate spaces in matrix algebras, adapting the notion of dimension and entropy to noncommutative settings.
• Introduces a modified free Hausdorff dimension, denoted $\overline{\mathbb{H}}$, which is proven to be invariant under change of generators.
• Employs Radon probability measures on microstate spaces and applies results from random matrix theory and free probability to control measure concentration.
• Uses local isometric smooth manifolds within microstate sets to bound packing numbers and establish lower bounds on Hausdorff measure.
• Applies results from [11] on concentration of measure and free independence to construct microstate sets with controlled dimension and entropy.

Experimental results

Research questions

• RQ1 Is the free Hausdorff dimension of a set of selfadjoint operators invariant under change of generators for the von Neumann algebra they generate?
• RQ2 Can the free Hausdorff dimension be explicitly computed when the operators generate a finite-dimensional algebra?
• RQ3 Does the free Hausdorff dimension become additive when the operators are freely independent, particularly in the finite-dimensional and single-operator cases?
• RQ4 How does the free Hausdorff dimension relate to the free entropy dimension $\delta_0$?
• RQ5 What structural properties of the generated von Neumann algebra can be inferred from the value of the free Hausdorff dimension, such as the existence of minimal projections?

Key findings

• The modified free Hausdorff dimension $\overline{\mathbb{H}}$ is an algebraic invariant, meaning it depends only on the von Neumann algebra generated and not on the choice of generators.
• For a single selfadjoint operator, the free Hausdorff dimension equals the free entropy dimension, and $\mathbb{H}(z_1) < 1$ implies the generated von Neumann algebra has a minimal projection.
• When the $n$-tuple generates a finite-dimensional unital $C^*$-algebra, the free Hausdorff dimension equals the free entropy dimension and is additive under direct sum decomposition.
• In the presence of freeness, the free Hausdorff dimension becomes additive for $n$-tuples that generate finite-dimensional algebras or are single selfadjoints.
• The $n$-dimensional free Hausdorff measure satisfies $\mathbb{H}^{n}(z_1,\ldots,z_n) = \chi(z_1,\ldots,z_n) + \frac{n}{2}\log(\frac{2n}{\pi e})$, establishing a non-trivial link to free entropy.
• For any $\alpha \in \mathbb{R}_+$, the $\alpha$-free Hausdorff entropy $\mathbb{H}^\alpha$ is finite if and only if $\alpha$ is at least the free Hausdorff dimension, and $\mathbb{H}^\alpha(z_1,\ldots,z_n) > -\infty$ when $\alpha = \mathbb{H}(z_1,\ldots,z_n)$.