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

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.