The Bootstrap and von Neumann algebras: The Maximal Intersection Lemma

Given a suitably nested family $Z = \langle Z(m,k,γ) \rangle_{m,k \in \mathbb N, γ>0}$ of Borel subsets of matrices, and associated Borel measures and rate function, $μ$, an entropy, $χ^μ(Z)$, 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 $X \cap Z = \langle Γ(X;m,k,γ) \cap Z(m,k,γ) \rangle_{m, k \in \mathbb N, γ>0}$. This observation can be used to establish the existence of finite tuples of operators with finite $χ^μ$-entropy. The intuition and proof come from the bootstrap in statistical inference.

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