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⁎-algebra A. We define an A-valued chordal Loewner chain as a subordination chain of analytic self-maps of the A-valued upper half-plane, such that each Ft is the reciprocal Cauchy transform of an A-valued law μt, such that the mean and variance of μt are continuous functions of t.We relate A-valued Loewner chains to processes with A-valued free or monotone independent increments just as was done in the scalar case by Bauer [8] and Schleißinger [48]. We show that the Loewner equation ∂tFt(z)=DFt(z)[Vt(z)], when interpreted in a certain distributional sense, defines a bijection between Lipschitz mean-zero Loewner chains Ft and vector fields Vt(z) of the form Vt(z)=−Gνt(z) where νt is a generalized A-valued law.Based on the Loewner equation, we derive a combinatorial expression for the moments of μt in terms of νt. We also construct non-commutative random variables on an operator-valued monotone Fock space which realize the laws μt. Finally, we prove a version of the monotone central limit theorem which describes the behavior of Ft as t→+∞ when νt has uniformly bounded support.