The Bootstrap and von Neumann algebras: The Maximal Intersection Lemma

Abstract

Given a suitably nested family $Z=⟨Z(m,k,\gamma ){⟩}_{m,k\in \mathbb{N},\gamma >0}$ 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*} χ^μ(X) & \geq & χ^μ(X \cap Z) & = & χ^μ(Z)\\ \end{eqnarray*} where $X\cap Z=⟨\Gamma (X;m,k,\gamma )\cap Z(m,k,\gamma ){⟩}_{m,k\in \mathbb{N},\gamma >0}$. 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.

10 pages

References in corpus (1)

• Concavification of free entropy