Extreme points of certain banach spaces related to conditional expectations

Let (X, F, mu) be a complete probability space and let B be a sub-sigma-algebra of F. We consider the extreme points of the closed unit ball B(A) of the normed space A whose points are the elements of L-infinity(X, F, mu) with the norm parallel tofparallel to = parallel toPhi(\f\)parallel to(infinity), where Phi is the probabilistic conditional expectation operator determined by B. No B-measurable function is an extreme point of the closed unit ball of A, and in certain cases there are no extreme points of B(A). For an interesting family of examples, depending on a parameter n, we characterize the extreme points of the unit ball and show that every element of the open unit ball is a convex combination of extreme points. Although in these examples every element of the open ball of radius 1/n can be shown to be a convex combination of at most 2n extreme points by elementary arguments, our proof for the open unit ball requires use of the, lambda-function of Aron and Lohman. In the case of the open unit ball, we only obtain estimates for the number of extreme points required in very special cases, e.g. the B-measurable functions, where 2n extreme points suffice.