A non-commutative F&M Riesz Theorem

TL;DR

This paper generalizes the classical Riesz theorem to a non-commutative multivariate setting, introducing NC measures and extending Lebesgue decomposition techniques within operator algebra frameworks.

Contribution

It develops a non-commutative version of the Riesz theorem for analytic measures, extending classical measure theory to operator algebras and free probability.

Findings

Established an NC Riesz theorem for analytic measures.

Refined Lebesgue decomposition for positive NC measures.

Developed new order properties of positive NC measures.

Abstract

We extend results on analytic complex measures on the complex unit circle to a non-commutative multivariate setting. Identifying continuous linear functionals on a certain self-adjoint subspace of the Cuntz--Toeplitz $C^*-$algebra, the free disk operator system, with non-commutative (NC) analogues of complex measures, we refine a previously developed Lebesgue decomposition for positive NC measures to establish an NC version of the Frigyes and Marcel Riesz Theorem for `analytic' measures, i.e. complex measures with vanishing positive moments. The proof relies on novel results on the order properties of positive NC measures that we develop and extend from classical measure theory.