# Conditionally Free Reduced Products of Hilbert Spaces

## TL;DR

This paper introduces a new product of Hilbert spaces that generalizes the reduced free product within conditionally free probability, unifying various notions of independence and providing key applications and properties.

## Contribution

It constructs a unified product of Hilbert spaces for conditionally free probability, enabling new insights and results in the theory of non-commutative independence.

## Key findings

- Existence of conditionally free and free copies of algebras
- Simplified proof of the linearization property of the cR-transform
- Application of the construction to prove properties of conditionally free algebras

## Abstract

We present a product of pairs of pointed Hilbert spaces that, in the context of Boz\.ejko, Leinert and Speicher's theory of conditionally free probability, plays the role of the reduced free product of pointed Hilbert spaces, and thus gives a unified construction for the natural notions of independence defined by Muraki. We additionally provide important applications of this construction. We prove that, assuming minor restrictions, for any pair of conditionally free algebras there are copies of them that are conditionally free and also free, a property that is frequently assumed (as hypothesis) to prove several results in the literature. Finally, we give a short proof of the linearization property of the $^cR$-transform (the analog of Voiculescu's $R$-transform in the context of conditionally free probability).