---
abstract: 'The classical Julia-Wolff-Carathéodory 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 the geometric properties of classical hyperbolic spaces.'
address: |
    CNRS, Institut de Mathématiques de Toulouse\
    118 Route de Narbonne\
    F-31062 Toulouse Cedex 09, France.
author:
- Serban Teodor Belinschi
title: 'A noncommutative version of the Julia-Wolff-Carathéodory Theorem'
---

Introduction
============

The classical Julia-Wolff-Carathéodory Theorem describes the behaviour of the derivative of an analytic self-map of the unit disc $\mathbb D$ or of the upper half-plane $\mathbb C^+$ of the complex plane $\mathbb 
C$ at certain boundary points. Numerous generalizations, to self-maps of balls or polydisks in $\mathbb C^n$, analytic functions with values in spaces of linear operators, analytic self-maps on domains in spaces of operators or in more general Banach spaces etc. - see for example [@Rudin; @Fan; @Jafari; @Wlo; @Abate; 