  We establish several deep existence criteria for conditional expectations on
von Neumann algebras, and then apply this theory to develop a noncommutative
theory of representing measures of characters of a function algebra. Our main
cycle of results describes what may be understood as a `noncommutative
Hoffman-Rossi theorem' giving the existence of weak* continuous `noncommutative
representing measures' for so-called D-characters. These results may be viewed
as `module' Hahn-Banach extension theorems for weak* continuous
`characters'into possibly noninjective von Neumann algebras. In closing we
introduce the notion of `noncommutative Jensen measures' and show that as in
the classical case, representing measures of logmodular algebras are Jensen
measures. The proofs of the two main cycles of results rely on the delicate
interplay of Tomita-Takesaki theory, noncommutative Radon-Nikodym derivatives,
Connes cocycles, Haagerup noncommutative Lp-spaces, Haagerup's reduction
theorem, etc.
