We initiate the study of the effective content of K K -theory for C ∗ C^* -algebras. We prove that there are computable functors which associate, to a computably enumerable presentation of a C ∗ C^* -algebra A A, computably enumerable presentations of the abelian groups K 0 (A) K₀ (A) and K 1 (A) K₁ (A). When A A is stably finite, we show that the positive cone of K 0 (A) K₀ (A) is computably enumerable. We strengthen the results in the case that A A is a Uniformly Hyperfinite (UHF) algebra by showing that the aforementioned presentation of K 0 (A) K₀ (A) is actually computable. In the UHF case, we also show that A A has a computable presentation precisely when K 0 (<mml: mi mathvariant="
Eagle et al. (Mon,) studied this question.