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

Abstract: 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* continuousnoncommutative representing measures' for so-called D-characters. These results may also be viewed as module' Hahn-Banach extension theorems for weak* continuouscharacters' 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.