# Fatou's Theorem for Non-commutative Measures

## TL;DR

This paper extends Fatou's Theorem from classical complex analysis to the setting of non-commutative measures and harmonic functions in several matrix variables, using operator algebra techniques.

## Contribution

It introduces a non-commutative analogue of Fatou's Theorem for positive harmonic functions and measures in several non-commuting variables, expanding classical results to operator algebra context.

## Key findings

- Established non-commutative Fatou's Theorem for matrix-variable harmonic functions
- Defined positive non-commutative measures as linear functionals on operator algebras
- Extended classical boundary limit results to non-commutative setting

## Abstract

A classical theorem of Fatou asserts that the Radon-Nikodym derivative of any finite positive Borel measure, $\mu$, with respect to Lebesgue measure on the complex unit circle, is recovered as the non-tangential limits of its Poisson transform in the complex unit disk. This positive harmonic Poisson transform is the real part of an analytic function whose Taylor coefficients are in fixed proportion to the conjugate moments of $\mu$. Replacing Taylor series in one variable by power series in several non-commuting variables, we show that Fatou's Theorem and related results have natural extensions to the setting of positive harmonic functions in an open unit ball of several non-commuting matrix-variables, and a corresponding class of positive \emph{non-commutative (NC) measures}. Here, an NC measure is any positive linear functional on a certain self-adjoint unital subspace of the Cuntz-Toeplitz algebra, the $C^*-$algebra generated by the left creation operators on the full Fock space.