Vol. 71, No. 2, 1977

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Stable isomorphism of hereditary subalgebras of Cāˆ—-algebras

Lawrence Gerald Brown

Vol. 71 (1977), No. 2, 335ā€“348

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.

Mathematical Subject Classification 2000
Primary: 46L05
Received: 22 November 1976
Published: 1 August 1977
Lawrence Gerald Brown