Graphs of functions and vanishing free entropy

TL;DR

This paper explores conditions under which the free entropy of a set of operators becomes negative infinity, linking geometric measure theory with free probability and entropy concepts in von Neumann algebras.

Contribution

It establishes new results connecting free entropy and free entropy dimension with geometric properties of graphs of functions within von Neumann algebras.

Findings

If δ₀(y, z) < δ₀(y) + δ₀(z), then χ(X ∪ {z}) = -∞

When z is in the algebra generated by X, the free entropy χ(X ∪ {z}) is -∞

Results are motivated by geometric-measure-theoretic properties of graphs of functions

Abstract

Suppose X is an n-tuple of selfadjoint elements in a tracial von Neumann algebra M. If z is a selfadjoint element in M and for some selfadjoint element y in the von Neumann algebra generated by X $\delta_0(y, z) < \delta_0(y) + \delta_0(z)$, then $\chi(X \cup \{z\}) = -\infty$ (here $\chi$ and $\delta_0$ denote the microstates free entropy and free entropy dimension, respectively). In particular, if z lies in the von Neumann algebra generated by X, then $\chi(X \cup \{z\}) = -\infty$. The statement and its proof are motivated by geometric-measure-theoretic results on graphs of functions. A similar statement for the nonmicrostates free entropy is obtained under the much stronger hypothesis that z lies in the algebra generated by X.