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

TL;DR

This paper introduces $R$-diagonal pairs as a unifying framework for Haar unitaries and circular elements in free probability, revealing their structural properties and implications for free independence.

Contribution

It defines $R$-diagonal pairs with a special diagonal form of their free cumulants and proves their stability under nested multiplication, linking them to free polar decompositions.

Findings

$R$-diagonal pairs generalize Haar unitaries and circular elements.

They exhibit an absorption property under free multiplication.

The theory is based on combinatorial analysis of non-crossing partitions.

Abstract

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.