Games on AF-Algebras

Abstract

Abstract We analyze $\textrm {C}^{\ast }$-algebras, particularly AF-algebras, and their $K_{0}$-groups in the context of the infinitary logic $\mathcal {L}_{\omega _{1} \omega }$. Given two separable unital AF-algebras $A$ and $B$, and considering their $K_{0}$-groups as ordered unital groups, we prove that $K_{0}(A) \equiv _{\omega \cdot \alpha } K_{0}(B)$ implies $A \equiv _{\alpha } B$, where $M \equiv _{\beta } N$ means that $M$ and $N$ agree on all sentences of quantifier rank at most $\beta $. 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 $\{ A_{\alpha } \}_{\alpha < \omega _{1}}$ of pairwise non-isomorphic separable simple unital AF-algebras which satisfy $A_{\alpha } \equiv _{\alpha } A_{\beta }$ for every $\alpha < \beta $. 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 $A \otimes \mathcal {K} \equiv _{\omega + 2 \cdot \alpha +2} B \otimes \mathcal {K}$ implies $K_{0}(A) \equiv _{\alpha } K_{0}(B)$, for every unital $\textrm {C}^{\ast }$-algebras $A$ and $B$.