  Let A be a closed subalgebra of a C*-algebra, that is a closed algebra of
Hilbert space operators. We generalize to such operator algebras $A$ 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 ((contractive) homomorphisms into
the scalars) on the algebra. For example, the Jensen inequality, the related
Bishop-Ito-Schreiber theorem, and the theory of Gleason parts. We will usually
replace characters (classical function algebra case) by D-characters, certain
completely contractive homomorphisms $\Phi : A \to D$, where D is a
C*-subalgebra of A. We also consider some D-valued variants of the classical
Gleason-Whitney theorem.
