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 $C^*$-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 $\partial_t F_t(z) = DF_t(z)[V_t(z)]$, 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 $t \to +\infty$ when $\nu_t$ has uniformly bounded support.