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We adapt the theory of chordal Loewner chains to the operator-valuedmatricial upper-half plane over a $C^*$-algebra $\mathcal{A}$. We define an$\mathcal{A}$-valued chordal Loewner chain as a subordination chain of analyticself-maps of the $\mathcal{A}$-valued upper half-plane, such that each $F_t$ isthe reciprocal Cauchy transform of an $\mathcal{A}$-valued law $\mu_t$, suchthat 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 justas was done in the scalar case by Bauer ("L\"owner's equation from anon-commutative probability perspective", J. Theoretical Prob., 2004) andSchei{\ss}inger ("The Chordal Loewner Equation and Monotone ProbabilityTheory", Inf. Dim. Anal., Quantum Probability, and Related Topics, 2017).  We show that the Loewner equation $\partial_t F_t(z) = DF_t(z)[V_t(z)]$, wheninterpreted in a certain distributional sense, defines a bijection betweenLipschitz 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 $\mathcal{A}$-valuedlaw.  Based on the Loewner equation, we derive a combinatorial expression for themoments of $\mu_t$ in terms of $\nu_t$. We also construct non-commutativerandom variables on an operator-valued monotone Fock space which realize thelaws $\mu_t$. Finally, we prove a version of the monotone central limit theoremwhich describes the behavior of $F_t$ as $t \to +\infty$ when $\nu_t$ hasuniformly bounded support.

Operator-Valued Chordal Loewner Chains and Non-Commutative Probability