Operator-Valued Chordal Loewner Chains and Non-Commutative Probability

Abstract

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 ("Löwner's equation from a non-commutative probability perspective", J. Theoretical Prob., 2004) and Scheißinger ("The Chordal Loewner Equation and Monotone Probability Theory", Inf. Dim. Anal., Quantum Probability, and Related Topics, 2017). We show that the Loewner equation ${\partial }_t{F}_t(z)=D{F}_t(z)[V_t(z)]$, 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 $t\to +\infty$ when ${\nu }_t$ has uniformly bounded support.

arXiv commentRevised, 92 pages (different format than v1)