Continuous functions over a pure C*-algebra

Abstract

Let $X$ be a compact metric space, and let $A$ be a pure ${\mathrm{C}}^{\ast }$-algebra. We show that $C(X,A)$ is pure whenever $A$ is simple; or every quotient of $A$ is stably finite (e.g., $A$ has stable rank one). Using permanence properties of pureness, we prove that the tensor product of any such $A$ with any ASH-algebra is pure.

19 pages; added Corollary C on strict comparison for groups of the form GxH with G acylindrically hyperbolic and H virtually abelian