K-theory agreement between Roe and quasi-local algebras

Determine whether the Roe algebra and quasi-local algebra of every coarsely embeddable metric space have the same K-theory, in order to clarify whether K-theoretic arguments for ghost projections extend from Roe algebras to quasi-local algebras.

Background

The paper studies ghost projections in quasi-local algebras for sparse metric spaces that coarsely embed into Hilbert space. Existing results establish compactness of ghost projections in the Roe algebra using K-theory, but the authors’ operator-theoretic argument is needed for the quasi-local algebra because the relationship between the relevant K-theories is unresolved. Establishing equality of these K-theories would explain whether the existing Roe-algebra methods can be transferred directly to the quasi-local setting.