Proposition 14

The set of extreme points of Pos$_{\leqslant 1}(G)$ is equal to the union of the set of extreme points of Pos$_1(G)$ and of the zero function. Moreover, the closed convex hull of the set of extreme points of Pos$_1(G)$ contains Pos$_1(G)$.

By cor. of prop. 13, the set Pos$_{\leqslant 1}(G)$ is a cap of the pointed convex cone Pos(G), which is defined by the gauge $\varphi \mapsto \varphi (e)$ (EVT, II, p. 61, def. 3 and prop. 4). Its extreme points are therefore the zero function and the elements $\varphi$ of Pos$_1(G)$ belonging to the extreme generators of Pos(G) (EVT, II, p. 62, cor. 1). These are the extreme points of Pos$_1(G)$ (EVT, II, p. 61, prop. 3). The first assertion follows.

Let us prove the second assertion. Let $\varphi \in$ Pos$_1(G)$. Since the set Pos$_{\leqslant 1}(G)$ is compact for the weak topology (cor. of prop. 13), there exists a filter $\mathfrak{F}$ on the convex hull of the extreme points of Pos$_{\leqslant 1}(G)$ which converges to $\varphi$ (EVT, II, p. 59, th. 1). Since the closed balls of the Banach space $L^{\infty}(G)$ are closed for the weak topology and since $\|\varphi \|_{\infty}= 1$, we have lim$_{\psi ,\mathfrak{F}}\|\psi \|_{\infty}= 1$ (indeed, otherwise there would exist a real number $c <1$ and a filter $\mathfrak{G}$ on the closed ball of radius $c$ in $L^{\infty}(G)$ which would be finer than $\mathfrak{F}$, which would imply that $\varphi$ belongs to this closed ball).

From the description of the extreme points of Pos$_{\leqslant 1}$(G), every element $\psi$ of the convex hull of the extreme points of Pos$_{\leqslant 1}(G)$ is a function of positive type on G of the form

where I is a finite set, $\psi_i$ is an extreme point of Pos$_1(G)$ for every $i\in I$, and $t_i\in [0,1]$. We have $\sum_it_i=\psi (e) =\|\psi \|_{\infty}\leqslant 1$ (lemma 4 of V, p. 446). If $\psi \not = 0$, the function

therefore belongs to the convex hull of the extreme points of Pos$_1(G)$. Since lim$_{\psi ,\mathfrak{F}}\|\psi \|^{-1}_{\infty}\psi =\varphi$, we conclude that $\varphi$ belongs to the closed convex hull of the extreme points of Pos$_1(G)$. The assertion follows.