Multilinear function series and transforms in free probability theory
Operator Algebras
2007-05-23 v2 Combinatorics
Rings and Algebras
Abstract
The algebra Mul[[B]] of formal multilinear function series over an algebra B and its quotient SymMul[[B]] are introduced, as well as corresponding operations of formal composition. In the setting of Mul[[B]], the unsymmetrized R- and T-transforms of random variables in B-valued noncommutative probability spaces are introduced. These satisfy properties analogous to the usual R- and T-transforms, (the latter being just the reciprocal of the S-transform), but describe all moments of a random variable, not only the symmetric moments. The partially ordered set of noncrossing linked partitions is introduced and is used to prove properties of the unsymmetrized T-transform.
Cite
@article{arxiv.math/0504361,
title = {Multilinear function series and transforms in free probability theory},
author = {Ken Dykema},
journal= {arXiv preprint arXiv:math/0504361},
year = {2007}
}
Comments
54 pages. (In the revised version, only minor changes were made.)