Noncommutative function theory and unique extensions

Original · EN

We generalize to the setting of Arveson's maximal subdiagonal subalgebras of finite von Neumann algebras, the Szegö Lᵖ-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∞. 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.