Non-computability of $K$-theory for computably presented C*-algebras

TL;DR

The paper establishes the existence of a computably presentable unital C*-algebra whose K-theory groups $K_0$ and $K_1$ do not admit computable presentations. It achieves this by encoding a fixed noncomputable c.e. set $R$ into a direct-sum construction of C*-algebras, and then using suspension and unitization to transfer noncomputability into $K$-theory. Starting from a noncomputable group $G$ with $G \cong K_0(\mathbf{A})$ for some computably presentable $\mathbf{A}$, the authors build a chain of algebras $\mathbf{B}$, $\mathbf{C} = S\mathbf{B}$, and $\mathbf{A} = \mathbf{B} \oplus \mathbf{C}$, followed by its unitization $\mathbf{D}$, to obtain a unital algebra with noncomputable $K$-theory. This work clarifies the limits of effective K-theory in the computable-presentations setting and raises questions about finite algebras and uniform computability of prescribed $K$-groups.

Abstract

We give an example of a unital C*-algebra $\mathbf{A}$ with a computable presentation and for which neither $K_0(\mathbf{A})$ nor $K_1(\mathbf{A})$ has a computable presentation.

Theorems & Definitions (19)

• Proposition 1.1
• proof
• Remark 1.2
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Corollary 2.4