Computable $K$-theory for $\mathrm{C}^*$-algebras: UHF algebras
Christopher Eagle, Isaac Goldbring, Timothy McNicholl, Russell Miller
TL;DR
The paper develops an effective framework for K-theory of separable C*-algebras by introducing computable and c.e. presentations and constructing computable functors $K_0$ and $K_1$ that output computably enumerable presentations of $K_0(oldsymbol{A})$ and $K_1(oldsymbol{A})$, respectively. For UHF algebras, it strengthens these results: $K_0(oldsymbol{A}^ullet)$ is computable, the positive cone is computable, and computability of $oldsymbol{A}$ is equivalent to the computability of $K_0(oldsymbol{A})$ and to the supernatural number $oldsymbol{ extepsilon}_{oldsymbol{A}}$ being lower semicomputable (with a sharp example showing this need not imply full computability). It also shows that every UHF algebra is computably categorical, meaning any two computable presentations are computably isomorphic. The constructions rely on effective Murray–von Neumann theory, the Grothendieck construction, and the suspension, and lay groundwork for an effective theory of AF algebras in follow-up work.
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.