Barnes, B. A.; Katavolos, A. Properties of quasinilpotents in some operator algebras. (English) Zbl 0797.46041 Proc. R. Ir. Acad., Sect. A 93, No. 2, 155-170 (1993). 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 \({\mathcal A}\). Cited in 1 Review MSC: 46L05 General theory of \(C^*\)-algebras Keywords:algebra of all bounded linear operators; finite rank operators; simultaneous triangularisation PDFBibTeX XMLCite \textit{B. A. Barnes} and \textit{A. Katavolos}, Proc. R. Ir. Acad., Sect. A 93, No. 2, 155--170 (1993; Zbl 0797.46041)