Games on AF-Algebras

We analyze Cs-algebras, particularly AF-algebras, and their K0-groups in the context of the infinitary logic Lw1w. Given two separable unital AF-algebras A and B, and considering their K0-groups as ordered unital groups, we prove that K0(A) =w-α K0(B) implies A =α B, where M =β N means that M and N agree on all sentences of quantifier rank at most β. This implication is proved using techniques from Elliott s classification of separable AF-algebras, together with an adaptation of the Ehrenfeucht-Fraïssé game to the metric setting. We use moreover this result to build a family Aα=α1 of pairwise non-isomorphic separable simple unital AF-algebras which satisfy Aα =α Aβ for every a < β. In particular, we obtain a set of separable simple unital AF-algebras of arbitrarily high Scott rank. Next, we give a partial converse to the aforementioned implication, showing that A ⊗ K =w+2•α+2 B ' K implies K0(A) =a K0(B), for every unital C∗-algebras A and B.