Let $A$ be a unital $C^*$-algebra. In what follows, I use the standard notation of $K$-theory for $C^*$-algebras. That is, $P_\infty(A):=\bigcup_{n\in\mathbb{N}} P_n(A)$, where $P_n(A)$ is the set of projections in $\mathbb{M}_n(A)$, $D(A)=P_\infty(A)/\sim_0$, and $D(A)$ is a semigroup with $[p]+[q]:=[p\oplus q]$, where $p\oplus q=\begin{pmatrix} p & 0\\ 0 & q\end{pmatrix}.$
Let $S\subseteq D(A)$ be the subsemigroup generated by projections in $A$, i.e, by $P_1(A)$. Do we have some abstract description of $S$? Can we describe $C^*$-algebras $A$ for which actually $S=D(A)$? On the other hand, what is some simple example of $p\in D(A)\setminus S$, for some $A$.
Or analogoulsy, let $G\leq K_0(A)$ be the subgroup generated by such projections, i.e. subgroup of $K_0(A)$ generated by the set $\{[p]-[q]\colon p,q\in P_1(A)\}$. When $G=K_0(A)$ and when is it different?