Fractal entropies and dimensions for microstate spaces
摘要
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 free Hausdorff dimension becomes additive for such sets in the presence of freeness.
引用
@article{arxiv.math/0212013,
title = {Fractal entropies and dimensions for microstate spaces},
author = {Kenley Jung},
journal= {arXiv preprint arXiv:math/0212013},
year = {2007}
}
备注
25 pages, minor corrections, lifting of restrictive conditions for the computation of dimension of a single selfadjoint, additional lemma in section 6