Extensions of C*-Algebras by a Small Ideal

Abstract

Abstract We classify all essential extensions of the form $$ \begin{align*} &0 \rightarrow {\mathcal{W}} \rightarrow {D} \rightarrow A \rightarrow 0,\end{align*}$$where ${\mathcal {W}}$ is the unique separable simple C*-algebra with a unique tracial state, which is $KK$-contractible and has finite nuclear dimension, and $A$ is a separable amenable ${\mathcal {W}}$-embeddable C*-algebra, which satisfies the Universal Coefficient Theorem (UCT). We actually prove more general results. We also classify a class of amenable $C^*$-algebras, which have only one proper closed ideal ${\mathcal {W}}.$