2022/01/19 by Jury, Michael T., Martin, Robert T. W., Timko, Edward J.
#46L52 (Secondary) #47A13 (Primary) #47L25 #47L55 #47L75 #47L80 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2201.07393
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.