# Multilinear function series in conditionally free probability with amalgamation

## TL;DR

This paper extends the concept of conditional freeness in non-commutative probability by developing positivity results, a central limit theorem, and an analogue of the R-transform using multilinear function series.

## Contribution

It introduces a multilinear function series approach to conditionally free probability with amalgamation, providing new tools and results in the field.

## Key findings

- Positivity results for conditional freeness
- A version of the central limit theorem
- An analogue of the conditionally free R-transform

## Abstract

As in the cases of freeness and monotonic independence, the notion of conditional freeness is meaningful when complex-valued states are replaced by positive conditional expectations. In this framework, the paper presents several positivity results, a version of the central limit theorem and an analogue of the conditionally free R-transform constructed by means of multilinear function series.