Graham R. Allan (2008)
Studia Mathematica
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We give a simple complex-variable proof of an old result of Zemánek and Le Page on the radical of a Banach algebra. Incidentally, the argument also proves a recent result of Harris and Kadison.
Zemánek, Jaroslav
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W. Bade, P. Curtis, A. Sinclair (2000)
Studia Mathematica
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Let A be a commutative unital Banach algebra and let A/ℛ be the quotient algebra of A modulo its radical ℛ. This paper is concerned with raising bounded groups in A/ℛ to bounded groups in the algebra A. The results will be applied to the problem of splitting radical extensions of certain Banach algebras.
Jaroslav Zemánek (1982)
Banach Center Publications
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Bernard Aupetit (1982)
Banach Center Publications
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A. Katavolos, C. Stamatopoulos (1998)
Studia Mathematica
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It is proved that the set Q of quasinilpotent elements in a Banach algebra is an ideal, i.e. equal to the Jacobson radical, if (and only if) the condition [Q,Q] ⊆ Q (or a similar condition concerning anticommutators) holds. In fact, if the inner derivation defined by a quasinilpotent element p maps Q into itself then p ∈ Rad A. Higher commutator conditions of quasinilpotents are also studied. It is shown that if a Banach algebra satisfies such a condition, then every quasinilpotent element...
Matej Brešar, Peter Šemrl (1996)
Studia Mathematica
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Let A be a complex unital Banach algebra. We characterize elements belonging to Γ(A), the set of elements central modulo the radical. Our result extends and unifies several known characterizations of elements in Γ(A).