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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.

A noncommutative version of the Julia–Wolff–Carathéodory theorem