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The Generalized Weyl-von Neumann Theorem and C*-algebra Extensions

In: Algebraic Methods in Operator Theory

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  • Huaxin Lin

    (The State University of New York at Buffalo, Department of Mathematics)

Abstract

An important problem in the C*-algebra theory is to classify the extensions of $$0 \to A \to E \to B \to 0$$ of B by A, where A, B, E are C*-algebras. We will assume that both E and B are unital and the surjective map from E to B is also unital. Furthermore, we assume that extensions are essential, i.e. A may be viewed as an essential ideal of E. The BDF-theory classifies those extensions when A = K and B = C(X), where K is the C*-algebra of compact operators on an infinite dimensional and separable Hilbert space and X is a compact metric space ([BDF1] and [BDF2]). Since early 1970’s, the C*-algebra extension theory and the KK-theory have been developed rapidly (we are not attempting to give a complete list of references but refer the reader to [Bl] for references). With the Universal Coefficient Theorem (see [RS,1.17]), for example, one can compute Ext(A, B) in many cases. However, unlike the original BDF-theory, in general, Ext (A, B) does not provide enough information for classifying these extensions.

Suggested Citation

  • Huaxin Lin, 1994. "The Generalized Weyl-von Neumann Theorem and C*-algebra Extensions," Springer Books, in: Raúl E. Curto & Palle E. T. Jørgensen (ed.), Algebraic Methods in Operator Theory, pages 134-143, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-0255-4_15
    DOI: 10.1007/978-1-4612-0255-4_15
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