Compact operators and the geometric structure of $C^\ast$-algebras

Given a $C^\ast$-algebra $\mathcal {A}$ and an element $A\in \mathcal{A}$, we give necessary and sufficient geometric conditions equivalent to the existence of a representation $(\phi ,\mathcal {H})$ of $\mathcal {A}$ so that $\phi (A)$ is a compact or a finite-rank operator. The implications of these criteria on the geometric structure of $C^\ast$-algebras are also discussed.