Let $X$ be a bounded closed convex subset of a locally convex space $E$. Let $\Gamma(X)$ be the set of continuous convex functionals on $X$ and let $C(X)$ be the set of continuous functionals.
Is $\Gamma(X)-\Gamma(X)$ dense in $C(X)$ ?
While I am interested in this statement in general, I want it for a particular set $X$ that I describe below. Also it would be fine for me to make $\Gamma(X)$ the set of l.s.c. convex functions instead.
Here is a description of my particular $X$. Let $B$ be a complete Boolean algebra with the countable chain property, for any $x\in B$, let $X\triangleq S(B)$ be the set of measure algebra $\mu$ on $B$ such that $\mu(1)=1$. The topology on $S(B)$ we consider is the coarsest such that for any $x\in B$, the mapping $\mu \to \mu(x)$ is continuous.
Any literature about such spaces would be most welcome, I don't have any starting point.
I am interested in this because I want to extend an increasing, (countably) linear functional from $\Gamma$ to $\mathbb R$ to a (countabley) linear functional from $\Gamma-\Gamma$ to $\mathbb R$ and then into a measure. This is related to what I attempt in this thread. See also this post that is related but on more general spaces.