@Mellon; @MM; @AbateRaissy] (the list is not exhaustive) - are known. This note gives a version of this theorem for noncommutative self-maps of the noncommutative upper half-plane of a von Neumann algebra $\mathcal A$. The result builds on the recent literature in the field - see [@AMcY; @ATY2; @PTD], and falls under the programme aiming to find the noncommutative versions of classical complex analysis results - see for example [@AKV0; @AKV; @AMc1; @AMc3; @AMc2].

In the second section we state our main result, and provide the required background. The third section is dedicated to proving a Schwarz lemma-type result for noncommutative functions. In this same section we give a simple (not necessarily original, though) proof of the classical Julia-Wolff-Carathéodory Theorem in order to make this article self-contained, and some lemmas that make use of it. Finally, in the last section we prove our main result.

Noncommutative functions and the Julia-Carathéodory Theorem
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Noncommutative functions
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Noncommutative sets and functions originate in [@taylor0; @taylor]. We largely follow [@ncfound] in our presentation below. We refer to this excellent monograph for details on, and proofs of, the statements that follow.

First a notation: if $S$ is a nonempty set, we denote by $M_{m\times 