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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 $n-1$?

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  • $\begingroup$ It's easy to see that class of the identity has order $n-1$. I'd be curious to see if there is some easy proof that it is finitely generated that doesn't go through showing the class of the identity generates the group. $\endgroup$
    – Benjamin Steinberg
    Commented Apr 27, 2018 at 15:22
  • $\begingroup$ Regarding the C*-algebra of an ample groupoid having an infinite unit space, such as ${\cal O}_n$, there are zillions of projections taking the form of the characteristic function of a compact open set. It is therefore highly unlikely that these algebras ever turn out to have a finitely generated $K_0$ group. When they do, it is partly due to the hard work of the clopen bissections identifying projections in this big set. $\endgroup$
    – Ruy
    Commented Apr 28, 2018 at 22:33

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