B-Valued Free Convolution for Unbounded Operators

TL;DR

This paper extends operator-valued free probability to affiliated unbounded operators, establishing how the Cauchy transform characterizes the algebra and demonstrating the behavior of R-transforms under free addition in finite dimensions.

Contribution

It develops a theory for unbounded affiliated operators in operator-valued free probability, including the behavior of Cauchy and R-transforms, and explores limitations in infinite-dimensional cases.

Findings

Cauchy transform determines the generated algebra.

R-transforms add under free convolution in finite dimensions.

Counterexamples show failure in infinite-dimensional cases.

Abstract

Consider the $\mathcal{B}$-valued probability space $(\mathcal{A}, E, \mathcal{B})$, where $\mathcal{A}$ is a tracial von Neumann algebra. We extend the theory of operator valued free probability to the algebra of affiliated operators $\tilde{\mathcal{A}}$. For a random variable $X \in \tilde{\mathcal{A}}^{sa}$ we study the Cauchy transform $G_{X}$ and show that the operator algebra $(\mathcal{B} \cup \{X\})"$ can be recovered from this function. In the case where $\mathcal{B}$ is finite dimensional, we show that, when $X, Y \in \tilde{\mathcal{A}}^{sa}$ are assumed to be $\mathcal{B}$-free, the $\mathcal{R}$-transforms are defined on universal subsets of the resolvent and satisfy $$ \mathcal{R}_{X} + \mathcal{R}_{Y} = \mathcal{R}_{X + Y}. $$ Examples indicating a failure of the theory for infinite dimensional $\mathcal{B}$ are provided. Lastly, we show that the class of functions that arise as the Cauchy transform of affiliated operators is, in a natural way, the closure of the set of Cauchy transforms of bounded operators.