Abstract
A large class of Banach algebras is identified allowing for non-trivial zero sums of idempotents, hence failing to be spectrally regular. Belonging to it are the C*-algebras known under the name Cuntz algebras. Other Banach algebras lying in the class are those of the form L(X) with X a (non-trivial) Banach space isomorphic to a (finite) direct sum of at least two copies of X. There do exist (somewhat exotic) Banach spaces for which L(X) is spectrally regular.
Mathematics Subject Classification (2010). Primary 46H99, 47C15; Secondary 30G30, 46E15.
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Dedicated to Leonia Lerer, in celebration of his seventieth birthday
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Bart, H., Ehrhardt, T., Silbermann, B. (2013). Zero Sums of Idempotents and Banach Algebras Failing to be Spectrally Regular. In: Kaashoek, M., Rodman, L., Woerdeman, H. (eds) Advances in Structured Operator Theory and Related Areas. Operator Theory: Advances and Applications, vol 237. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0639-8_7
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