Compatibility of a noncommutative probability space and a noncommutative probability space with amalgamation

TL;DR

This paper explores the relationship between scalar-valued and operator-valued R-transforms in compatible noncommutative probability spaces, highlighting differences between freeness and amalgamated freeness.

Contribution

It develops R-transform calculus for compatible noncommutative probability spaces and clarifies the distinction between freeness and amalgamated freeness.

Findings

Relation between scalar-valued R-transforms and operator-valued moment series established

Demonstrates a significant gap between freeness and amalgamated freeness

Provides a framework for R-transform calculus in compatible spaces

Abstract

In this paper, we will consider R-transform theory and R-transform calculus for compatible noncommutative probability space and amagamated noncommutative probability space. By doing this, we can realize the relation between scalar-valued R-transforms and operator-valued moment series, under the compatibility. Also, we can see that there is a big gap between freeness and amalgamated freeness.