Compact operators and the geometric structure of 𝐶*-algebras

Abstract

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