A Free Entropy Dimension Lemma

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…

关于子代数的条件作用——冯·诺依曼代数中的熵对偶性与广义量子斯坦引理

Conditioning on Subalgebras - Entropy Duality and Generalized Quantum Stein's Lemma in von Neumann Algebras

该论文将条件熵对偶性推广至任意冯·诺依曼代数子代数，并建立无限维量子斯坦引理，为子代数相对熵提供复合假设检验中的操作解释。

Minimal faces and Schur's Lemma for embeddings into R^U

In the context of N. Brown's Hom(N,R^U), we establish that given π: N \rightarrow R^U, the dimension of the minimal face containing [π] is one less than the dimension of the center of the relative commutant of π. We also show the "convex independence" of extreme points in the sense that the convex hull of n extreme points is an n-vertex simplex. Along the way, we establish a version of Schur's Lemma for embeddings of II$_1$-factors.