# Journal of Operator Theory

Volume 90, Issue 2, Autumn 2023 pp. 569-604.

A non-commutative F. and M. Riesz Theorem**Authors**: Michael T. Jury (1), Robert T.W. Martin (2), Edward J. Timko (3)

**Author institution:**(1) Department of Mathematics, University of Florida, U.S.A.

(2) Department of Mathematics, University of Manitoba, Canada

(3) Department of Mathematics, Georgia Institute of Technology, U.S.A.

**Summary:**We extend results on complex analytic 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.

**DOI:**http://dx.doi.org/10.7900/jot.2022jan20.2397

**Keywords:**non-commutative disc algebra, F. and M. Riesz Theorem, operator systems, Cuntz and Cuntz--Toeplitz algebras, non-commutative measure theory

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