This is a question that is a bit outside my usual mathematical comfort zone, but I feel like an expert might know the answer.

Recall that a dimension group is an ordered abelian group $G$ with positive cone $G^+$ that arises as a limit of ordered groups of the form $\mathbb Z^d$ with the componentwise order. It is called simple if every element in $G^+\setminus\{0\}$ is an order unit.

If one considers a distinguished order unit $u\in G^+$, then one defines the state space $\mathcal{S}(G,G^+,u)$ as the set of all states (duh), meaning all order-preserving homomorphisms $s: G\to\mathbb R$ with $s(u)=1$. If $G$ is countable, then the topology of pointwise convergence will give the state space the structure of a metrizable Choquet simplex.

It is a well-known fact that every metrizable Choquet simplex $T$ can be realized as the state space of a simple dimension group $(G,G^+,u)$. However, it is also well-known that this is highly non-unique, and in fact there are many possible choices one can make in general. I am interested to know whether one can ask the following in addition:

Let $T$ be a metrizable Choquet simplex and let $f: T\to (0,1)$ be an affine continuous function. Does there exist a simple dimension group $(G,G^+,u)$ with distinguished order unit together with another element $a\in G^+$ such that $T$ is affinely homeomorphic to $\mathcal{S}(G,G^+,u)$ in such a way that $f$ corresponds to the assignment $[s\mapsto s(a)]$?

If this happens to be false in general, are there any natural sufficient conditions on the function $f$?

A note for C*-algebraists: What this question really asks is whether there exists a unital simple AF algebra $A$ having $T$ as its tracial state space and a projection $p\in A$ such that the function $f$ corresponds to the point evaluation $[\tau\mapsto\tau(p)]$.