On Vector-Valued Characters for Noncommutative Function Algebras

Abstract

Let A be a closed subalgebra of a $$C^*$$-algebra, that is a norm-closed algebra of Hilbert space operators. We generalize to such operator algebras several key theorems and concepts from the theory of classical function algebras. In particular we consider several problems that arise when generalizing classical function algebra results involving characters (nontrivial homomorphisms from the algebra into the scalars). For example, the Jensen inequality, the related Bishop–Ito–Schreiber theorem, and the theory of Gleason parts. Inspired by Arveson’s work on noncommutative Hardy spaces, we replace characters (classical function algebra case) by D-characters; certain completely contractive homomorphisms $$\Phi : A \rightarrow D$$, where D is a $$C^*$$-subalgebra of A. Using Brown’s measure and a potential theoretic balayage argument we prove a partial noncommutative Jensen inequality appropriate for $$C^*$$-algebras with a tracial state. We also show that this Jensen inequality characterizes D-characters among the module maps. Other advances include a theory of noncommutative Gleason parts appropriate for D-characters, which uses Harris’ noncommutative hyperbolic metric and Schwarz–Pick inequality, and other ingredients. As an application of Gleason parts we show that in the antisymmetric case, one is guaranteed the existence of a ‘quantum’ Wermer embedding function, and also of non-trivial compact Hankel operators, whenever the Gleason part of the canonical trace is rich in tracial states.