Fractal entropies and dimensions for microstate spaces, II

Abstract

For a selfadjoint element x in a tracial von Neumann algebra and $\alpha ={\delta }_0(x)$ we compute bounds for ${\mathbb{H}}^{\alpha }(x),$ where ${\mathbb{H}}^{\alpha }(x)$ is the free Hausdorff $\alpha$-entropy of $x.$ The bounds are in terms of $\int {\int }_{{\mathbb{R}}^2-D}\log |y-z|d\mu (y)d\mu (z)$ where $\mu$ is the Borel measure on the spectrum of x induced by the trace and $D\subset {\mathbb{R}}^2$ is the diagonal. We compute similar bounds for the free Hausdorff entropy of a free family of selfadjoints.

9 pages