A noncommutative version of the Julia-Wolff-Caratheodory Theorem

Abstract

The classical Julia-Wolff-Carath{é}odory Theorem characterizes the behaviour of the derivative of an analytic self-map of a unit disc or of a half-plane of the complex plane at certain boundary points. We prove a version of this result that applies to noncommutative self-maps of noncommutative half-planes in von Neumann algebras at points of the distinguished boundary of the domain. Our result, somehow surprisingly, relies almost entirely on simple geometric properties of noncommutative half-planes, which are quite similar to certain geometric properties of classical hyperbolic spaces.

Several corrections, additions and modifications. Lemma 4.2 corrected and several new computations added. Closer to final form, but still preliminary version: comments/corrections are welcome