What does countable weakly unperforated $\textbf{simple}$ ordered group mean here?

The second part of the main theorem states that, given any countable weakly unperforated simple ordered group $G_0$ with order unit $e$, any countable abelian group $G_1$, and any metrizable Choquet simple $T$ and any surjective continuous affine map $s : T \to S_e(G_0)$ (the state space of $G_0$), there exists a unital $C^∗$-algebra $A ∈ > \mathcal N_1$ such that $$ \text{Ell}(A) = (K_0(A), K_0(A)_+, [1_A], > K_1(A), T(A), r_A) = (G_0,(G_0)_+, e, G_1, T, s)$$

From A classification of finite simple amenable $\mathcal Z$-stable $C^*$-algebras

The usuall definition of a simple group is a group that has no non-trivial normal subgroups. However, the $K_0$ group of a $C^*$-algebra is necessarily abelian, and a simple abelian group is necessarily finite. So what simplicity could it refer to here?

Solution 1:

The following is from Blackadar's "$K$-Theory for Operator Algebras", 2nd edition, definition 6.2.1:

Fix and ordered group $(G,G_+)$. A subgroup $I\subset G$ is called an order ideal if whenever $x\in G$, $y\in I$, $x\leq y$ implies $x\in I$. We then say that $(G,G_+)$ is simple if its only order ideals are $0$ and $G$.

This is equivalent to another common definition, namely that every $u\in G_+$ is an order unit, meaning that for every $x\in G$ then there is some $n\in\mathbb Z_{>0}$ such that $x\leq nu$.