We begin by reviewing the theory of normal conditional expectations on von Neumann algebras. Second, we describe several new contributions to this theory, such as establishing a correspondence between normal conditional expectations onto a subalgebra, and `noncommutative weight' functions affiliated with a centralizer in the algebra. Third, inspired by Arveson’s noncommutative Hardy space theory, we introduce a new kind of noncommutative representing measures of ‘non-commutative characters’ on noncommutative function algebras. Finally, we consider and give successively more general solutions to the problem of the ‘existence of representing measures’ in the weak* case. That is, when do there exist weak* continuous positive extensions of non-commutative characters on such noncommutative function algebras?

(Joint work with L. E. Labuschagne.)