Noncommutative function theory and unique extensions
Tom 178 / 2007
Studia Mathematica 178 (2007), 177-195 MSC: Primary 46L51, 46L52, 47L75; Secondary 46J15, 46K50, 47L45. DOI: 10.4064/sm178-2-4
We generalize, to the setting of Arveson's maximal subdiagonal subalgebras of finite von Neumann algebras, the Szegő $L^p$-distance estimate and classical theorems of F. and M. Riesz, Gleason and Whitney, and Kolmogorov. As a byproduct, this completes the noncommutative analog of the famous cycle of theorems characterizing the function algebraic generalizations of $H^\infty $ from the 1960's. 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 every completely contractive homomorphism on a unital subalgebra of a $C^*$-algebra to have a unique completely positive extension.