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Fractal entropies and dimensions for microstate spaces, II

Research paper by Kenley Jung

Indexed on: 10 Dec '03Published on: 10 Dec '03Published in: Mathematics - Operator Algebras



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.