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

TL;DR

This paper explores the complexity of classifying unital $C^*$-algebras using game theory, showing that $KK$-equivalence is an analytic relation and establishing a strategy transfer in Ehrenfeucht-Fra"issé games for classifiable algebras.

Contribution

It introduces a game-theoretic refinement of the unital classification theorem, linking strategies in Ehrenfeucht-Fra"issé games to algebraic invariants.

Findings

$KK$-equivalence is shown to be an analytic relation.

The set of separable $C^*$-algebras satisfying the UCT is analytic.

Strategies in Ehrenfeucht-Fra"issé games can be transferred between algebras and their invariants.

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-Fra\"iss\'e games (of various lengths) on classifiable $C^*$-algebras and their invariants.