Stable isomorphism of hereditary subalgebras ofC∗-algebras

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®3Tis isomorphic to5®^ where .-SΓis the C*-algebra of compact operators on a separable infinite-dimensional Hubert space.In the special case where A = CCg)^~and B = pAp for some projection p in the double centralizer algebra, M(A), of A such that p commutes with C®lc 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 C of C to the strong grassmanian.Thus p defines a continuous field of Hubert spaces on C 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-Hof mann theorem which is needed to justify some of our remarks.