$R$-diagonal pairs - a common approach to Haar unitaries and circular elements

Details $R$-diagonal pairs - a common approach to Haar unitaries and circular elements $R$-diagonal pairs - a common approach to Haar unitaries and circular elements

1996-04-30

arxiv.org/abs/funct-an/9604012v1

Mathematics

Operator Algebras

LaTeX2e, to appear in The Fields Institute Communications

Scientific paper

In the free probability theory of Voiculescu two of the most frequently used *-distributions are those of a Haar unitary and of a circular element. We define an $R$-diagonal pair as a generalization of these distributions by the requirement that their two-dimensional $R$-transform (or free cumulants) have a special diagonal form. We show that the class of such $R$-diagonal pairs has an absorption property under nested multiplication of free pairs. This implies that in the polar decomposition of such an element the polar part and the absolute value are free. Our calculations are based on combinatorial statements about non-crossing partitions, in particular on a canonical bijection between the set of intervals of NC(n) and the set of 2-divisible partitions in NC(2n). In a forthcoming paper the theory of $R$-diagonal pairs will be used to solve the problem of the free commutator.

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