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

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.KeywordsSeparable Hilbert SpaceApproximate IdentityTrivial ExtensionAlgebra ExtensionRiesz Decomposition PropertyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.