A noncommutative version of the Julia-Wolff-Carathéodory theorem

The classical Julia–Wolff–Carathéodory theorem characterizes the behaviour of the derivative of an analytic self-map of a unit disk 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, somewhat surprisingly, relies almost entirely on simple geometric properties of noncommutative half-planes, which are quite similar to the geometric properties of classical hyperbolic spaces, but use virtually no elements of analytic function theory in the proofs.