A Game-Theoretic Unital Classification Theorem for $C^*$-Algebras

TL;DR

The paper analyzes the descriptive-set-theoretic complexity of KK-equivalence for unital separable C*-algebras and proves that KK-equivalence is analytic, yielding the analyticity of the UCT class. It develops a comprehensive framework using metric infinitary logic, back-and-forth DST games, and Borel categories to relate KK-theory to first-order-like invariants, notably the KT_u invariant, and to enable a game-theoretic refinement of unital classification. A key achievement is the DST transfer that translates equivalence at the level of invariants into equivalence of algebras under refined quantifier ranks, and vice versa, via universal functors from Borel categories. The results illuminate the descriptive-set-theoretic structure of the Elliott classification landscape, showing that classification phenomena can be captured by analytic relations and Borel mechanisms, while also highlighting open questions about the limits of analyticity for UCT and KK-equivalence. Overall, the work provides a rigorous bridge between model-theoretic methods and KK-theory, with potential implications for understanding the boundaries of classifiability in operator algebras.

Abstract

We study the complexity of the $KK$-equivalence relation on unital $C^*$-algebras, in the sense of descriptive set theory. We prove that $KK$-equivalence is analytic, which in turn shows that the set of separable $C^*$-algebras satisfying the UCT is analytic. This allows us to prove a game-theoretic refinement of the unital classification theorem: there is a transfer of strategies between Ehrenfeucht-Fraïssé games (of various lengths) on classifiable $C^*$-algebras and their invariants.

Theorems & Definitions (95)

• Theorem 1.1
• Theorem 1.2: games_on_AF_algebras
• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 4
• Theorem 5
• Definition 2.2
• Definition 2.3
• Example 2.4