A noncommutative version of the Julia-Wolff-Caratheodory Theorem

TL;DR

This paper extends the classical Julia-Wolff-Carathéodory Theorem to noncommutative settings, specifically for self-maps of noncommutative half-planes within von Neumann algebras, using geometric properties similar to classical hyperbolic spaces.

Contribution

It introduces a noncommutative version of the theorem that applies to von Neumann algebra settings, relying on geometric properties of noncommutative half-planes.

Findings

Established a noncommutative Julia-Wolff-Carathéodory Theorem

Demonstrated geometric properties of noncommutative half-planes

Connected classical and noncommutative hyperbolic geometries

Abstract

The classical Julia-Wolff-Carath{\'e}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.