Fractal entropies and dimensions for microstate spaces, II

Details Fractal entropies and dimensions for microstate spaces, II Fractal entropies and dimensions for microstate spaces, II

2003-12-11

arxiv.org/abs/math/0312223v1

Mathematics

Operator Algebras

9 pages

Scientific paper

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.

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