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Gerard Murphy (1994)
Banach Center Publications
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We survey some old and new results in the theory of derivations on Banach algebras. Although our overview is broad ranging, our principal interest is in recent results concerning conditions on a derivation implying that its range is contained in the radical of the algebra.
Martin Mathieu (1994)
Banach Center Publications
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With this paper, we intend to provide an overview of some recent work on a problem on unbounded derivations of Banach algebras that still defies solution, the non-commutative Singer-Wermer conjecture. In particular, we discuss several global as well as local properties of derivations entailing quasinilpotency in the image.
Matej Brešar, Yuri V. Turovskii (2009)
Studia Mathematica
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We consider a continuous derivation D on a Banach algebra 𝓐 such that p(D) is a compact operator for some polynomial p. It is shown that either 𝓐 has a nonzero finite-dimensional ideal not contained in the radical rad(𝓐) of 𝓐 or there exists another polynomial p̃ such that p̃(D) maps 𝓐 into rad(𝓐). A special case where Dⁿ is compact is discussed in greater detail.
Tsiu-Kwen Lee, Cheng-Kai Liu (2009)
Studia Mathematica
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Let A be a semisimple Banach algebra with a linear automorphism σ and let δ: I → A be a σ-derivation, where I is an ideal of A. Then Φ(δ)(I ∩ σ(I)) = 0, where Φ(δ) is the separating space of δ. As a consequence, if I is an essential ideal then the σ-derivation δ is closable. In a prime C*-algebra, we show that every σ-derivation defined on a nonzero ideal is continuous. Finally, any linear map on a prime semisimple Banach algebra with nontrivial idempotents is continuous if it satisfies...
Tsiu-Kwen Lee (2005)
Studia Mathematica
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We discuss range inclusion results for derivations on noncommutative Banach algebras from the point of view of ring theory.
Marc P. Thomas (2009)
Studia Mathematica
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It is a famous conjecture that every derivation on each Banach algebra leaves every primitive ideal of the algebra invariant. This conjecture is known to be true if, in addition, the derivation is assumed to be continuous. It is also known to be true if the algebra is commutative, in which case the derivation necessarily maps into the (Jacobson) radical. Because I. M. Singer and J. Wermer originally raised the question in 1955 for the case of commutative Banach algebras, the conjecture...
Nadia Boudi, Peter Šemrl (2010)
Studia Mathematica
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Let A be a Banach algebra, and let d: A → A be a continuous derivation such that each element in the range of d has a finite spectrum. In a series of papers it has been proved that such a derivation is an inner derivation implemented by an element from the socle modulo the radical of A (a precise formulation of this statement can be found in the Introduction). The aim of this paper is twofold: we extend this result to the case where d is not necessarily continuous, and we give a complete...
Dumitru D. Draghia (1995)
Extracta Mathematicae
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Niels Groenbaek (1989)
Studia Mathematica
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Nadia Boudi, Said Ouchrif (2009)
Studia Mathematica
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We study generalized derivations G defined on a complex Banach algebra A such that the spectrum σ(Gx) is finite for all x ∈ A. In particular, we show that if A is unital and semisimple, then G is inner and implemented by elements of the socle of A.
El Harti, R. (2004)
International Journal of Mathematics and Mathematical Sciences
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K. Beidar, Matej Brešar (2000)
Studia Mathematica
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The problem of when derivations (and their powers) have the range in the Jacobson radical is considered. The proofs are based on the density theorem for derivations.
Park, Kyoo-Hong, Jung, Yong-Soo, Bae, Jae-Hyeong (2002)
International Journal of Mathematics and Mathematical Sciences
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Naoual Kouki, Mohamed Ali Toumi (2015)
Topological Algebra and its Applications
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In this paper we prove that the image of a nth order derivation on real commutative Banach ℓ-algebras with positive squares is contained in the set of nilpotent elements.
Graham Allan, Allan Sinclair (1976)
Studia Mathematica
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Bernard Aupetit, Martin Mathieu (2000)
Studia Mathematica
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We prove that the separating space of a Lie homomorphism from a Banach algebra onto a Banach algebra is contained in the centre modulo the radical.
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...