Are there workable conditions that imply that $K_0(A)$ is finitely generated, for a noncommutative unital C* algebra $A$. My actual question is in fact much more specific than this:

Is it possible to give a simpler proof that $K_0$(Cuntz algebra) is finitely generated, without going through Cuntz' proof that $K_0(O_n)$ is cyclic of order $n-1$?