B-Valued Free Convolution for Unbounded Operators

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}$}^{sa}$westudytheCauchytransform$G_{X}$ and show that the operator algebra $(\mathcal{B}\cup \{X\})\mathrm{"}$ 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 $${\mathcal{R}}_X+{\mathcal{R}}_Y={\mathcal{R}}_{X+Y}\mathrm{.}$$ 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.