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Properties of quasinilpotents in some operator algebras. (English) Zbl 0797.46041

Summary: Let \({\mathcal A}\) be a subalgebra of the algebra of all bounded linear operators on a Banach space, and let \({\mathcal F}\) be the finite rank operators in \({\mathcal A}\). It is shown that provided \({\mathcal F}\) is sufficiently large in \({\mathcal A}\), then \({\mathcal A}/\text{rad}({\mathcal A})\) is abelian if and only if \({\mathcal F}/\text{rad}({\mathcal F})\) is abelian. A number of other conditions are given on \({\mathcal A}\) or \({\mathcal F}\) which are equivalent to the commutativity of \({\mathcal A}/\text{rad}({\mathcal A})\). In addition, these properties are related to the simultaneous triangularisation of the algebra \({\mathcal A}\).

MSC:

46L05 General theory of \(C^*\)-algebras
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