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 $B$.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2022
- DOI:
- 10.48550/arXiv.2204.04087
- arXiv:
- arXiv:2204.04087
- Bibcode:
- 2022arXiv220404087D
- Keywords:
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- Mathematics - Logic;
- Mathematics - Operator Algebras;
- 03C98;
- 46L05;
- 03C75
- E-Print:
- 29 pages