Operator-Valued Chordal Loewner Chains and Non-Commutative Probability

We adapt the theory of chordal Loewner chains to the operator-valued matricial upper-half plane over a $C^{\ast }$-algebra $\mathcal{A}$. We define an $\mathcal{A}$-valued chordal Loewner chain as a subordination chain of analytic self-maps of the $\mathcal{A}$-valued upper half-plane, such that each $F_t$ is the reciprocal Cauchy transform of an $\mathcal{A}$-valued law ${\mu }_t$, such that the mean and variance of ${\mu }_t$ are continuous functions of $t$. We relate $\mathcal{A}$-valued Loewner chains to processes with $\mathcal{A}$-valued free or monotone independent independent increments just as was done in the scalar case by Bauer (“L"owner’s equation from a non-commutative probability perspective”, J. Theoretical Prob., 2004) and Schei{\ss}inger (“The Chordal Loewner Equation and Monotone Probability Theory”, Inf. Dim. Anal., Quantum Probability, and Related Topics, 2017). We show that the Loewner equation ${\mathrm{\partial }}_t{F}_t(z)=D{F}_t(z)[V_t(z)]$, when interpreted in a certain distributional sense, defines a bijection between Lipschitz mean-zero Loewner chains $F_t$ and vector fields $V_t(z)$ of the form $V_t(z)=-G_{{\nu }_t}(z)$ where ${\nu }_t$ is a generalized $\mathcal{A}$-valued law. Based on the Loewner equation, we derive a combinatorial expression for the moments of ${\mu }_t$ in terms of ${\nu }_t$. We also construct non-commutative random variables on an operator-valued monotone Fock space which realize the laws ${\mu }_t$. Finally, we prove a version of the monotone central limit theorem which describes the behavior of $F_t$ as $t\to +\mathrm{\infty }$ when ${\nu }_t$ has uniformly bounded support.