EXTREME POINTS OF CERTAIN BANACH SPACES RELATED TO CONDITIONAL EXPECTATIONS

Abstract

Let $(X,\mathcal{F},\mu)$ be a complete probability space and let $\mathcal{B}$ be a sub- $\sigma$ -algebra of $\mathcal{F}$ . We consider the extreme points of the closed unit ball $\mathbb{B}(\mathcal{A})$ of the normed space $\mathcal{A}$ whose points are the elements of $L^\infty(X,\mathcal{F},\mu)$ with the norm $\Vert\, f \Vert = \Vert\Phi(\vert\, f \vert)\Vert_\infty$ , where $\Phi$ is the probabilistic conditional expectation operator determined by $\mathcal{B}$ . No $\mathcal{B}$ - measurable function is an extreme point of the closed unit ball of $\mathcal{A}$ , and in certain cases there are no extreme points of $\mathbb{B}(\mathcal{A})$ . 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 $\frac{1}{n}$ 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 $\mathcal{B}$ -measurable functions, where $2n$ extreme points suffice.