R-Transforms of Free Joint Distributions and Non-crossing Partitions

✍ Scribed by Alexandru Nica

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 909 KB
• Volume: 135
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.0011

✦ Synopsis

We introduce an extension of the notion of R-transform, defined by D. Voiculescu, to ( free) joint distributions, i.e., normalized linear functionals on algebras of noncommutative polynomials in several indeterminates. We point out that the R-transform has good behavior with respect to the operations of free product and free convolution of joint distributions. We prove that the explicit computation of the inverse-R-transform of a joint distribution is done via a formula of summation over the lattice of non-crossing partitions, which shows that the R-transform is the operator-theoretic counterpart of the ``free cumulants'' considered in the combinatorial approach of R. Speicher. Moreover, in connection to the equivariance of the multidimensional R-transform under rotations, we show that a natural free analogue of a classical result about rotations of independent random variables is holding.

1996 Academic Press, Inc. assume that they are normalized by + 1 (1)=+ 2 (1)=1. The usual formula defining their covolution gets the expression

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