The main theorem is that if
B is a hereditary C∗-subalgebra of A which is not contained in any proper closed
two-sided ideal, then under a suitable separability hypothesis A ⊗𝒦 is isomorphic
to B ⊗𝒦, where 𝒦 is the C∗-algebra of compact operators on a separable
infinite-dimensional Hilbert space. In the special case where A = C ⊗𝒦 and B = pAp
for some projection p in the double centralizer algebra, M(A), of A such that p
commutes with C ⊗ 1 ⊂ M(A), the theorem follows from a result of Dixmier and
Douady [12]. In fact p must be defined by a continuous mapping from the spectrum
Ĉ of C to the strong grassmanian. Thus p defines a continuous field of Hilbert spaces
on Ĉ and [12] implies that the countable direct sum of this field with itself is trivial.
Our proof amounts to an abstraction of [12]. The theorem also leads to an
abstraction and generalization of some results of Douglas, Fillmore, and us on
extensions of C∗-algebras ([6, §3]). The final section of the paper contains a
generalization of the Dauns-Hofmann theorem which is needed to justify some of our
remarks.
