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.