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arXiv · 0707.1355

Graphs of functions and vanishing free entropy

Kenley Jung
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Abstract

Suppose X is an n-tuple of selfadjoint elements in a tracial von Neumann algebra M. If z is a selfadjoint element in M and for some selfadjoint element y in the von Neumann algebra generated by X $δ_0(y, z) < δ_0(y) + δ_0(z)$, then $χ(X \cup \{z\}) = -\infty$ (here $χ$ and $δ_0$ denote the microstates free entropy and free entropy dimension, respectively). In particular, if z lies in the von Neumann algebra generated by X, then $χ(X \cup \{z\}) = -\infty$. The statement and its proof are motivated by geometric-measure-theoretic results on graphs of functions. A similar statement for the nonmicrostates free entropy is obtained under the much stronger hypothesis that z lies in the algebra generated by X.

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Kenley Jung. 2007-07-10. Graphs of functions and vanishing free entropy. https://arxiv.org/abs/0707.1355

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