𝐶*-algebras with weak* angelic dual balls

Abstract

Some characterizations of hereditary C ∗ C^* -subalgebras of weakly Rickart C ∗ C^* -algebras with weak ∗ ^* angelic dual balls are given. In particular, it turns out that weak ∗ ^* angelicity, weak ∗ ^* sequentiality and weak ∗ ^* sequential compactness of dual balls are the same on this class, and that they are characterized by using ideals of compact elements in those algebras. As an application, it is shown that a weakly Rickart C ∗ C^* -algebra has weak ∗ ^* angelic dual ball if and only if it coincides with the ideal of compact elements, which happens if and only if the algebra is finite dimensional.