Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9495484 | Journal of Functional Analysis | 2005 | 35 Pages |
Abstract
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.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Kenley Jung,