# Existence of a separating affine functional

Let be $S$ a separable(non compact) metric space and $X=C_b(S)$ the set of all bounded continuous functions, then it's topological dual $X^{\star}=rba(S)$ is the set of all regular Borel additive measures endowed with the variation norm. Denote by $\mathscr{P}(S)$ the subset of $rba(S)$ of all additive probability measures, endowed with the weak$^{*}$ topology.

Fix $\mu$ a extremal point of $\mathscr{P}(S)$ and a closed set $\mathcal{F}$ in $\mathscr{P}(S)$ such that the convex hull $co(\mathcal{F})$ do not contains $\mu,$ then it is possible to show that there is an affine functional $\ell_{\mu,\mathcal{F}}:\mathscr{P}(S)\to \mathbb{R}$ such that $\ell_{\mu,\mathcal{F}}(\mu)=0$ and $\ell_{\mu,\mathcal{F}}$ is strictly positive in $\mathcal{F}$

Question 1: Is there some way to use the above stated to get a global result, that is, I want to show that there is a affine functional $\ell_{\mu}:\mathscr{P}(S)\to \mathbb{R}$ such that $\ell_\mu(\mu)=0$ and $\ell_\mu(\nu)>0$ for all $\nu\in \mathscr{P}(S)\setminus \{\mu\}$ ?

Question 2: If not, is there some other approach to show that there is a affine functional $\ell_{\mu}:\mathscr{P}(S)\to \mathbb{R}$ such that $\ell_\mu(\mu)=0$ and $\ell_\mu(\nu)>0$ for all $\nu\in \mathscr{P}(S)\setminus \{\mu\}$ ?

Edit: Following the suggestions of Jochen Glueck I had made some edits in the second paragrapher

• Two remarks: (i) What you claim in the paragraph before Question 1 is not true if $\mu$ is contained in the convex hull of $\mathcal{F}$, so you need an additional assumption such as, e.g., convexity of $\mathcal{F}$, right? – Jochen Glueck Jun 8 '18 at 18:27
• (ii) Concerning your questions: maybe I'm missing something, but assume that $\mu$ is not an extreme point of $\mathscr{P}(S)$. Then $\mu$ can be written as a convex combination of two points in $\mathscr{P}(S) \setminus \{\mu\}$, so if $\ell_\mu(\nu) > 0$ for all $\nu \in \mathscr{P}(S) \setminus \{\mu\}$, then we always have $\ell_\mu(\mu) > 0$. Doesn't this show that a functional with the properties you are looking for does never exist unless $\mu$ is an extreme point of $\mathscr{P}(S)$? – Jochen Glueck Jun 8 '18 at 18:27
• And (iii) let's recall that in general a point with that property is called an exposed point en.wikipedia.org/wiki/Exposed_point – Pietro Majer Jun 8 '18 at 18:34
• As you observe I have to clarify some thinks in the statement, thanks for the warning, I will rewrite the question. – Eduardo Jun 8 '18 at 18:34
• Thanks for point out the mistakes and lack of precision in the statement. – Eduardo Jun 8 '18 at 18:50

Here is a characterization of those $\mu$'s which have the desired property:

Theorem. Let $S$ be an arbitrary topoligcal space, let $C_b(S)$ denote the space of bounded real-valued continuous functios on $S$ (endowed with the supremum norm) and let $\mathscr{P}(S)$ denote the subset of the dual space $C_b(S)'$ consisting of those functionals which have norm one and map the positive cone $C_b(S)_+$ into $[0,\infty)$. For each $\mu \in \mathscr{P}(S)$ the followig assertions are equivalent:

(i) $\mu$ is an extreme point of $\mathscr{P}(S)$, i.e. $\mu$ cannot be written as a convex combination of two functionals in $\mathscr{P}(S) \setminus \{\mu\}$.

(ii) $\mu$ is an exposed point of $\mathscr{P}(S)$, i.e. there exists a functional $g$ in the bi-dual $C_b(S)''$ wich attains its strict miminum on the set $\mathscr{P}(S)$ at the point $\mu$.

(iii) There exists a functional $\tilde g$ in the bi-dual $C_b(S)''$ which is zero at $\mu$ and strictly positive on $\mathscr{P}(S) \setminus \{\mu\}$.

(iv) $\mu$ is a lattice homomorphism, i.e. we have $|\langle \mu, f\rangle| = \langle \mu, |f| \rangle$ for all $f \in C_b(S)$.

(v) $\mu$ is an algebra homomorphism, i.e. we have $\langle \mu, f_1f_2 \rangle = \langle \mu, f_1 \rangle \cdot \langle \mu, f_2 \rangle$ for all $f_1,f_2 \in C_b(S)$.

Proof. "(i) $\Leftrightarrow$ (iv)" This equivalence is a standard argument in the proof of Kakutani's representation theorem for AM-spaces in the theory of Banach lattices; see for instance [H. H. Schaefer: Banach Lattices and Positive Operators (1974), proof of Theorem II.7.4].

"(iii) $\Rightarrow$ (ii)" Obvious.

"(ii) $\Rightarrow$ (iii)" Let $g$ be as in (ii), let $1_S \in C_b(S) \subseteq C_b(S)''$ denote the constant function with value $1$ and set $\tilde g = g - \langle g,\mu\rangle 1_S$. Then $\tilde g$ fulfils the properties claimed in (iii) (to see this, note that $\langle 1_S, \nu \rangle = \|\nu\| = 1$ for all $\nu \in \mathscr{P}(S)$).

"(iii) $\Rightarrow$ (i)" Obvious.

"(i) $\Rightarrow$ (iii)" Assume that $\mu$ is an extreme point of $\mathscr{P}(S)$. It follows from Kakutani's representation theorem for AM-spaces [op. cit.] that there exists a Banach lattice isomorphism $\Psi$ between the space $C(K)$ of all real-valued continuous functions on some compact Hausdorff space $K$ and the space $C_b(S)$ and that this isomorphism can be chosen to fulfil $\Psi(1_K) = 1_S$. Now the extreme point property of $\mu$ implies that $\Psi'(\mu) \in C(K)'$ is a delta functional, i.e. there exists a point $\omega \in K$ such that $\Psi'(\mu) = \delta_\omega$, where $\langle \delta_\omega,f\rangle = f(\omega)$ for all $f \in C(K)$; this is again a standard fact in Banach lattice theory (see e.g. [P. Meyer-Nieberg: Banach Lattices (1991), Proposition 2.1.2(i)]). The space of all bounded Borel measurable functions on $K$ is contained in the bi-dual of $C(K)$, so the indicator function $1_{K \setminus \{\omega\}}$ is an element of the bi-dual $C(K)''$ which does not vanish on any positive normalized functional on $C(K)$ except for $\delta_\omega = \Psi'(\mu)$. Consequently, $\tilde g := \Psi'' 1_{K \setminus \{\mu\}} \in C_b(S)''$ does the job in (iii).

"(v) $\Rightarrow$ (iv)" For each $f \in C_b(S)$ we have \begin{align*} |\langle \mu, f\rangle|^2 = \langle \mu,f\rangle^2 = \langle \mu, f^2\rangle = \langle \mu, |f|^2 \rangle = \langle \mu,|f|\rangle^2, \end{align*} which implies (iv).

"(iv) $\Rightarrow$ (v)" This can be shown by an approximation procedure which is, for instance, explained in [T. Eisner, B. Farkas, M. Haase, R. Nagel: Operator Theoretic Aspects of Ergodic Theory (2015), supplement to Chapter~7] (though in a slightly different context).

The proof of the theorem is complete.

Important remark. The proof of the implication "(i) $\Rightarrow$ (iii)" above should not trick us into believing that every extreme point $\mu$ of $\mathscr{P}(S)$ is of the form $\mu = \delta_\omega$ for some $\omega \in S$. In fact, the compact space $K$ is usually much larger than $S$ (if, for instance, $S$ is a locally compact space, then the space $K$ is the Stone–Čech compactification $\beta S$ of $S$).

In fact, extremal points of $\mathcal{P}(S)$ are exactly the deltas (if $\operatorname{supp}(\mu)$ is not a single point $p$, it can easily be written as a mean of two probability measures). In this case, any nonnegative $f\in C_b(S)$ vanishing exactly at $p$ easily does the job, because then $\int_S fd\nu>0$ for all $\nu\in \mathcal{P}(S)\setminus\{\mu\}.$

• I tend to believe that your right, but I can't find an argument right now that shows that there are no further extremal points. If, for instance, $S = \mathbb{N}$ (i.e. $C_b(S) = \ell^\infty$), then every ultra filter limit defines an extreme point of $\mathcal{P}(S)$ (and in fact these are the only extreme points). Your answer says that such an ultra filter is an extremal point if and only if the ultra filter is not free. That "feels" right, but I think a rigorous argument (or a reference) would be nice ;-). – Jochen Glueck Jun 8 '18 at 21:12
• If the deltas are the only extreme points, then we have no genuinely additive (not $\sigma$-additive) probability measure as an extreme point? – Eduardo Jun 8 '18 at 21:26
• Sorry, I see now I misread the definition of $\mathcal{P}(S)$ (as "$\sigma$-additive") – Pietro Majer Jun 8 '18 at 21:29
• I thought about it once again, and it seems that my above comment is not correct. Since @Eduardo allows $\ell_\mu$ to be an element of the bi-dual of $C_b(S)$ rather than only an element of $C_b(S)$, it follows that all extreme points of $\mathscr{P}(S)$ have the desired property. Moreover, the extreme points of $\mathscr{P}(S)$ are exactly the characters on the $C^*$-algebra $C_b(S)$. – Jochen Glueck Jun 8 '18 at 22:39
• @JochenGlueck Could you elaborate your comment or provide me a reference? Thanks in advance. – Eduardo Jun 9 '18 at 16:06