Abstract
Abstract
Let A be a separable (not necessarily unital) simple
$C^*$
-algebra with strict comparison. We show that if A has tracial approximate oscillation zero, then A has stable rank one and the canonical map
$\Gamma $
from the Cuntz semigroup of A to the corresponding lower-semicontinuous affine function space is surjective. The converse also holds. As a by-product, we find that a separable simple
$C^*$
-algebra which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map
$\Gamma $
is surjective.
Publisher
Canadian Mathematical Society