Let {Aα: α ∈ A} be a family of
C∗-algebras (resp., W∗-algebras). For α0∈ A, we let Pα0:⊕αAα→ Aα0 denote the
canonical coordinate projection of ⊕αAα onto Aα0. If B is a C∗-(resp., W∗-)
subalgebra of ⊕αAα, we say that B splits if B =⊕αPα(B). In this note, we give
conditions both necessary and sufficient for B to split. In the C∗-category, these
conditions are given in terms of separation properties of the spectrum and primitive
ideal space of B, and in the W∗-category, the conditions are expressed in terms of
disjointness of certain subsets of the center of B. We also give examples to show that
these conditions cannot be weakened, and are hence the best possible of their
kind.
