The Bootstrap and von Neumann algebras: The Maximal Intersection Lemma

TL;DR

This paper introduces a generalized entropy concept for operators in von Neumann algebras, inspired by bootstrap methods in statistical inference, establishing the existence of operators with finite entropy under certain conditions.

Contribution

It generalizes microstates free entropy using a new entropy definition and proves the existence of operators with finite entropy in von Neumann algebras.

Findings

Existence of finite entropy operators in von Neumann algebras.

Generalization of free entropy via a new entropy measure.

Application of bootstrap concepts to operator algebras.

Abstract

Given a suitably nested family $Z = \langle Z(m,k,\gamma) \rangle_{m,k \in \mathbb N, \gamma >0}$ of Borel subsets of matrices, and associated Borel measures and rate function, $\mu$, an entropy, $\chi^{\mu}(Z)$, is introduced which generalizes the microstates free entropy in free probability theory. Under weak regularity conditions there exists a finite tuple of operators $X$ in a tracial von Neumann algebra such that \begin{eqnarray*} \chi^{\mu}(X) & \geq & \chi^{\mu}(X \cap Z) & = & \chi^{\mu}(Z)\\ \end{eqnarray*} where $X \cap Z = \langle \Gamma(X;m,k,\gamma) \cap Z(m,k,\gamma) \rangle_{m, k \in \mathbb N, \gamma >0}$. This observation can be used to establish the existence of finite tuples of operators with finite $\chi^{\mu}$-entropy. The intuition and proof come from the bootstrap in statistical inference.