The Bootstrap and von Neumann algebras: The Maximal Intersection Lemma

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.