  We generalize to the setting of Arveson's maximal subdiagonal subalgebras of
finite von Neumann algebras, the Szeg\"o $L^p$-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.
