Fractal Entropies and Dimensions for Microstates Spaces
نویسنده
چکیده
Using Voiculescu’s notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebraic invariant. We compute the free Hausdorff dimension in the cases where the set generates a finite dimensional algebra or where the set consists of a single selfadjoint. We show that the Hausdorff dimension becomes additive for such sets in the presence of freeness.
منابع مشابه
Fractal Entropies and Dimensions for Microstate Spaces, Ii
[1] introduced fractal geometric entropies and dimensions for Voiculescu’s microstate spaces ([3], [4]). One can associate to a finite set of selfadjoint elements X in a tracial von Neumann algebra and an α > 0 an extended real number H(X) ∈ [−∞,∞]. H(X) is a kind of asymptotic logarithmic α-Hausdorff measure of the microstate spaces of X. One can also define a free Hausdorff dimension of X, de...
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