Games on AF-algebras
Ben De Bondt, Andrea Vaccaro, Boban Velickovic, Alessandro Vignati

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 -algebras, particularly AF-algebras, and their -groups in the context of the infinitary logic . Given two separable unital AF-algebras and , and considering their -groups as ordered unital groups, we prove that implies , where means that and agree on all sentences of quantifier rank at most . 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 of pairwise non-isomorphic separable simple unital AF-algebras which satisfy for every . In particular, we…
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