Games on AF-algebras - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2022

Games on AF-algebras

Résumé

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-Fra\"iss\'e 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 $B$.

Dates et versions

hal-03796268 , version 1 (04-10-2022)

Identifiants

Citer

Ben de Bondt, Andrea Vaccaro, Boban Veličković, Alessandro Vignati. Games on AF-algebras. 2022. ⟨hal-03796268⟩
14 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More