$K$-theory of cluster $C^*$-algebras

It is proved that the $K_0$-group of a cluster $C^*$-algebra is isomorphic to the corresponding cluster algebra. As a corollary, one gets a shorter proof of the positivity conjecture for cluster algebras. As an example, we consider a cluster $C^*$-algebra $\mathbb A(1,1)$ coming from triangulation of an annulus with one marked point on each boundary component.