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

TL;DR

This paper demonstrates that for certain computably presented unital C*-algebras, the associated K-theory groups cannot be computably presented, highlighting fundamental limits in the computability of algebraic invariants.

Contribution

It provides the first example of a computably presented unital C*-algebra with non-computable K-theory groups, revealing intrinsic non-computability in algebraic invariants.

Findings

Existence of a computably presented unital C*-algebra with non-computable K-theory groups

K_0 and K_1 groups can lack computable presentations despite the algebra being computably presented

Highlights fundamental limits of computability in operator algebra invariants

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.