Invariant projections for operators that are free over the diagonal

TL;DR

This paper investigates the regularity and invariant projections of sums of free, self-adjoint noncommutative random variables over a commutative algebra, using subordination functions to characterize these projections.

Contribution

It introduces a new characterization of invariant projections for sums of free variables over a diagonal, extending understanding of their distributional regularity.

Findings

Characterization of invariant projections via subordination functions

Insights into the distributional regularity of free sums

Extension of previous work on free probability over diagonal

Abstract

Motivated by recent work of Au, C{\'e}bron, Dahlqvist, Gabriel, and Male, we study regularity properties of the distribution of a sum of two selfad-joint random variables in a tracial noncommutative probability space which are free over a commutative algebra. We give a characterization of the invariant projections of such a sum in terms of the associated subordination functions.