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 (Lowners 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 $V_t(z) = -G_{ u_t}(z)$ where $ u_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 $ u_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 $ u_t$ has uniformly bounded support.