Definable $\mathrm{K}$-homology of separable C*-algebras

Abstract

In this paper we show that the K-homology groups of a separable C*-algebra can be enriched with additional descriptive set-theoretic information, and regarded as definable groups. Using a definable version of the Universal Coefficient Theorem, we prove that the corresponding definable K-homology is a finer invariant than the purely algebraic one, even when restricted to the class of UHF C*-algebras, or to the class of unital commutative C*-algebras whose spectrum is a 1-dimensional connected subspace of R 3 .