Games on AF-algebras

TL;DR

This paper explores the logical equivalences between AF-algebras and their K_0-groups, establishing a connection between algebraic invariants and model-theoretic properties, and constructing a hierarchy of non-isomorphic AF-algebras with high Scott rank.

Contribution

It proves a new logical equivalence relation between AF-algebras and their K_0-groups, and constructs a family of AF-algebras with arbitrarily high Scott rank.

Findings

Logical equivalence of K_0-groups implies algebraic equivalence of AF-algebras.

Constructed a hierarchy of non-isomorphic AF-algebras with high Scott rank.

Established a partial converse relating tensor products and K_0-group equivalences.

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-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 $B$.