Univalent Subordination Chains in Reflexive Complex Banach Spaces

Graham, Ian; Hamada, Hidetaka; Kohr, Gabriela; Kohr, Mirela

doi:10.1090/conm/591/11829

Cited by 10 publications

(1 citation statement)

References 31 publications

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“…The theory of Loewner chains has been extended to domains in C n and even reflexive Banach spaces [7], motivated by similar complex-analytic applications. Our adaptation of chordal Loewner chains to the operator-valued upper halfspace is motivated instead by its applications to non-commutative probability.…”

Section: Chordal Loewner Chainsmentioning

confidence: 99%

Operator-valued chordal Loewner chains and non-commutative probability

Jekel

2020

Journal of Functional Analysis

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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 halfplane, such that each F t is the reciprocal Cauchy transform of an A-valued law µ t , such that the mean and variance of µ t are continuous functions of t.We relate A-valued Loewner chains to processes with A-valued free or monotone independent independent increments just as was done in the scalar case by Bauer [1] and Scheißinger [2].We show that the Loewner equation, when interpreted in a certain distributional sense, defines a bijection between Lipschitz mean-zero Loewner chains F t and vector fieldsBased on the Loewner equation, we derive a combinatorial expression for the moments of µ t in terms of ν t . We also construct non-commutative random variables on an operator-valued monotone Fock space which realize the laws µ t . Finally, we prove a version of the monotone central limit theorem which describes the behavior of F t as t → +∞ when ν t has uniformly bounded support.Remark 2.11. Note that by the previous lemma and some basic results on L ∞ Boch , there is an isometric inclusion ι : L ∞ Boch ([0, T ], X ) → L(L 1 [0, T ], X ) given by, so in the sequel we will regard L ∞ Boch ([0, T ], X ) as a subspace of L(L 1 [0, T ], X ).If we had a bounded function R : [0, T ] × [0, T ] → X denoted R(s, t), then could define the diagonal restriction R(t, t). We claim that under appropriate hypotheses, this operation still makes sense when R(s, ·) is an element of L(L 1 [0, T ], X ) rather than a bounded function [0, T ] → X . For this to be rigorous, we must view R as a map [0, T ] → L(L 1 [0, T ], X ). Lemma 2.12 (Diagonal restriction). There exists a unique linear map diag : L ∞ Boch ([0, T ], L(L 1 [0, T ], X )) → L(L 1 [0, T ], X ) such that

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Section: Chordal Loewner Chainsmentioning

confidence: 99%