Noncommutative function theory and unique extensions

David P. Blecher¹ ; Louis E. Labuschagne²

¹ Department of Mathematics University of Houston Houston, TX 77204-3008, U.S.A.
² Department of Mathematical Sciences P.O. Box 392 0003 Unisa, South Africa

Studia Mathematica, Tome 178 (2007) no. 2, pp. 177-195

Résumé

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.

Détail

DOI : 10.4064/sm178-2-4

Keywords: generalize setting arvesons maximal subdiagonal subalgebras finite von neumann algebras szeg p distance estimate classical theorems riesz gleason whitney kolmogorov byproduct completes noncommutative analog famous cycle theorems characterizing function algebraic generalizations infty sample other results prove kaplansky density result large class these algebras necessary condition every completely contractive homomorphism unital subalgebra * algebra have unique completely positive extension