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Erschienen in:
2015 | OriginalPaper | Buchkapitel
On Three-Space Problems for Certain Classes of \(C^*\)-algebras
verfasst von : A. K. Vijayarajan
Erschienen in: Semigroups, Algebras and Operator Theory
Verlag: Springer India
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Abstract
It is shown that being a GCR algebra is a three-space property for \(C^*\)-algebras using the structure of composition series of ideals present in GCR algebras. A procedure is presented to construct a composition series for a \(C^*\)-algebra from the unique composition series for any GCR ideal and the corresponding GCR quotient being a \(C^*\)-algebra. We deduce as a consequence that, a GCR algebra is a three-space property. While noting that being a CCR algebra is not a three-space property for \(C^*\)-algebras, sufficient additional conditions required on a \(C^*\)-algebra for the CCR property to be a three-space property are also presented. Relevant examples are also presented.