Fractal Entropies and Dimensions for Microstate Spaces, Ii

نویسنده

KENLEY JUNG

چکیده

[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, denoted by H(X), which is related to H(X) in the same way that Hausdorff dimension is related to the critical value of Hausdorff measures. H can be regarded as an interpolated version of Voiculescu’s free entropy χ in the sense that if X consists of n selfadjoints, then H(X) = χ(X) + n 2 log( πe ). This connection seems perfectly natural since χ is defined in terms of Lebesgue measure and Hausdorff n measure is a normalization of n-dimensional Lebesgue measure. In [3] Voiculescu establishes an equation for χ(x) where x is a selfadjoint operator. He shows that if μ is the Borel measure on sp(x) induced by the tracial state, then the free entropy of x is a normalization of the logarithmic energy of μ, i.e.,

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