Games on AF-algebras

Abstract

We analyze ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-algebras $A$ and $B$.

29 pages