2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/110243We generalize to the setting of Arveson's maximal subdiagonal subalgebras of finite von Neumann algebras, the Szegö $L^p$-distance estimate, and classical theorems of F. and M. Riesz, Gleason and Whitney, and Kolmogorov. In so doing, we are finally able to provide a complete noncommutative analog of the famous cycle of theorems characterizing the function theoretic generalizations of $H^\infty$. A sample of our other results: we prove a Kaplansky density result for a large class of these algebras, and give a necessary condition for when every completely contractive homomorphism on a unital subalgebra of a C*-algebra possesses a unique completely positive extension.15 pagesOperator AlgebrasFunctional AnalysisPrimary 46L51, 46L52, 47A15, Secondary 46J15, 46K50, 47L45text