Graphs of functions and vanishing free entropy

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 $\delta_0(y, z) < \delta_0(y) + \delta_0(z)$, then $\chi(X \cup \{z\}) = -\infty$ (here $\chi$ and $\delta_0$ denote the microstates free entropy and free entropy dimension, respectively). In particular, if z lies in the von Neumann algebra generated by X, then $\chi(X \cup \{z\}) = -\infty$. 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.