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

Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras