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.
This article is an expanded form of an invited talk delivered on 27 February 2014 during the International Conference on Semigroups, Algebras and Operator Theory (ICSAOT) held during February 26–28, 2014 at the Cochin University of Science and Technology.
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Acknowledgments
This article is based on a joint work with Ms. A.M. Shabna who was CSIR research fellow at the National Institute of Technology Calicut during the work.
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Vijayarajan, A.K. (2015). On Three-Space Problems for Certain Classes of \(C^*\)-algebras. In: Romeo, P., Meakin, J., Rajan, A. (eds) Semigroups, Algebras and Operator Theory. Springer Proceedings in Mathematics & Statistics, vol 142. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2488-4_14
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DOI: https://doi.org/10.1007/978-81-322-2488-4_14
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