Let $G$ be a dimension group (i.e. a directed, unperforated abelian group satisfying the Riesz interpolation property) with order unit $u\in G^{+}$. There is a canonical positive group homomorphism $\theta\colon G\to \operatorname{Aff}(S(G,u))$, where $S(G,u)$ is the state space of $G$ and $\operatorname{Aff}(S(G,u))$ are the real-valued affine, continuous functions on $S(G,u)$ defined by $\theta(g)(\sigma) = \sigma(g)$ for all $\sigma\in S(G,u)$.

There is the obvious short exact sequence (of abelian groups): $$ 0 \to \ker\theta \to G\to \theta(G)\to 0. $$ When $G$ is finitely generated, $\theta(G)$ is a finitely-generated, torsion-free abelian group, and thus free. Thus the short exact sequence splits. Also when $G = K_0(C(X))$ for a compact space $X$, we see that $\theta(G) \cong C(X,\mathbb{Z})$, which is also free. And so the short exact sequence splits in this case as well.

My questions are:

- Does this short exact sequence always split? Does it split when $G$ is simple?
- If not, is there a necessary and sufficient condition under which it does split?
- If not, is there a concrete example of a group $G$ where it does not split?

Edit: Corrected (I hope) a mistake noted in the answer.