Computable $K$-theory for $\mathrm{C}^*$-algebras: UHF algebras

Abstract

We initiate the study of the effective content of $K$-theory for $\mathrm{C}^*$-algebras. We prove that there are computable functors which associate, to a computably enumerable presentation of a $\mathrm{C}^*$-algebra $\boldA$, computably enumerable presentations of the abelian groups $K_0(\boldA)$ and $K_1(\boldA)$. When $\boldA$ is stably finite, we show that the positive cone of $K_0(\boldA)$ is computably enumerable. We strengthen the results in the case that $\boldA$ is a UHF algebra by showing that the aforementioned presentation of $K_0(\boldA)$ is actually computable. In the UHF case, we also show that $\boldA$ has a computable presentation precisely when $K_0(\boldA)$ has a computable presentation, which in turn is equivalent to the supernatural number of $\boldA$ being lower semicomputable; we give an example that shows that this latter equivalence cannot be improved to requiring that the supernatural number of $\boldA$ is computable. Finally, we prove that every UHF algebra is computably categorical.