Nowhere scattered multiplier algebras

Abstract

We study sufficient conditions under which a nowhere scattered $\mathrm {C}^*$ -algebra $A$ has a nowhere scattered multiplier algebra $\mathcal {M}(A)$ , that is, we study when $\mathcal {M}(A)$ has no nonzero, elementary ideal-quotients. In particular, we prove that a $\sigma$ -unital $\mathrm {C}^*$ -algebra $A$ of (i) finite nuclear dimension, or (ii) real rank zero, or (iii) stable rank one with $k$ -comparison, is nowhere scattered if and only if $\mathcal {M}(A)$ is.