Fractal entropies and dimensions for microstate spaces, II

Abstract

For a selfadjoint element x in a tracial von Neumann algebra and $α= δ_0(x)$ we compute bounds for $\mathbb H^α(x),$ where $\mathbb H^α(x)$ is the free Hausdorff $α$-entropy of $x.$ The bounds are in terms of $\int \int_{\mathbb R^2 -D} \log |y-z| dμ(y) dμ(z)$ where $μ$ 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.