Let $X$ be a topological space or a scheme, and let $K^0(X)$ be $K$-group of vector bundles of $X$. One may ask when an element $x$ of $K^0(X)$ is represented by an actual vector bundle, and not just a formal sum. The set of such classes form a cone in $K^0(X)$. Are there any known statements about the structure of this cone?

For example, consider the case $X = \mathbb{P^1}$. Since any vector bundle (at least when we are over a field) is isomorphic to a direct sum of bundles of the form $\mathcal O(k)$, and Euler exact sequence gives us the relation $[\mathcal O(-2)]=2[\mathcal O(-1)]-[\mathcal O]$, we have that $K^0(\mathbb P^1)=\mathbb Z[t]/t^2$, where $t$ is the class of $[\mathcal O] - [\mathcal O(-1)]$. So $[\mathcal O(k)]=1+kt$ in this presentation, and we see that the positive cone is the cone of elements such that free term is bigger than zero. In particular, it is not finitely generated (as a semigroup).

Do we have a description of a positive cone in other nontrivial ($K^0(X) \ne \mathbb Z$) cases? Can we say that sometimes it is finitely generated? Can we at least describe this cone for $K^0(\mathbb P^1) = \mathbb Z[t]/t^{n+1}$?

We can obtain one necessary condition for a class $x$ to lie in this cone by employing lambda operations on $K^0$: If $x$ is positive, then $\lambda^n(x) = 0$ for $n$ big enough. However, I doubt that this is a sufficient condition --- is there an example of not a positive class satisfying this?

Of course, this question makes sense for $K^0$ of any exact category, so it would be interesting to know whether something can be said in these another contexts. For example, if we are considering $K^0$ of $C^*$-algebras, then a huge number of explicit examples arises for approximately finite algebras (which are direct limits of direct sums of matrix algebras).

Thank you!