Noncommutative function theory and unique extensions

2006/03/17 by David P. Blecher, Blecher, David P., Louis E. Labuschagne +1
Mathematics · #46K50 #46L52 #47A15 #47L45 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary 46L51 #Secondary 46J15 #math.FA #math.OA #msc:46J15 #msc:46K50 #msc:46L51 #msc:46L52 #msc:47A15 #msc:47L45

paper · pdf · doi:10.48550/arxiv.math/0603437

15 pages

arxiv created 2006/03/17 · arxiv updated 2009/12/01

Abstract

We 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^∞. 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.

Related

• Characterizations of noncommutative H^∞
• Von Neumann algebraic H^p theory
• Boundaries, Bundles and Trace Algebras
• Null projections and noncommutative function theory in operator algebras
• The noncommutative weak Extension Principle