# Conditional expectations through Boolean cumulants and subordination -- towards a better understanding of the Lukacs property in free probability

## TL;DR

This paper introduces a Boolean cumulants approach to compute conditional expectations in free probability, simplifying proofs and providing explicit calculations related to the Lukacs property.

## Contribution

It develops a new method using Boolean cumulants and subordination to explicitly calculate conditional expectations in free probability, enhancing understanding of the Lukacs property.

## Key findings

- Boolean cumulants enable explicit conditional expectation calculations
- Simplified proofs of known results using Boolean cumulants and subordination
- Enhanced understanding of the Lukacs property in free probability

## Abstract

We use here a recent idea of studying functions of free random variables using Boolean cumulants. We develop idea of explicit calculations of conditional expectation using Boolean cumulants. We demonstrate Boolean cumulants approach allows to calculate explicitly some conditional expectations of functions in free random variables. We present how Boolean cumulants together with subordination simplify proofs of some results which are known in research literature.