Graphs of Functions and Vanishing Free Entropy
نویسنده
چکیده
Suppose X = {x1, . . . , xn} is an n-tuple of selfadjoint elements in a tracial von Neumann algebra M . If z = z∗ ∈ M and for some y = y∗ in the von Neumann algebra generated by X δ0(y, z) < δ0(y) + δ0(z), then χ(X ∪ {z}) = −∞ (here χ and δ0 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 ∪ {z}) = −∞. The statement and its proof are motivated by geometric-measuretheoretic 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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