Displaying similar documents to “Where to find the image of a derivation”

Aspects of the theory of derivations

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.

Range inclusion results for derivations on noncommutative Banach algebras

Volker Runde (1993)

Studia Mathematica

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Let A be a Banach algebra, and let D : A → A be a (possibly unbounded) derivation. We are interested in two problems concerning the range of D: 1. When does D map into the (Jacobson) radical of A? 2. If [a,Da] = 0 for some a ∈ A, is Da necessarily quasinilpotent? We prove that derivations satisfying certain polynomial identities map into the radical. As an application, we show that if [a,[a,[a,Da]]] lies in the prime radical of A for all a ∈ A, then D maps into the radical. This generalizes...

The canonical test case for the non-commutative Singer-Wermer conjecture

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...

Derivations mapping into the socle, III

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...

Polynomially compact derivations on Banach algebras

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.

Partially defined σ-derivations on semisimple Banach algebras

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...

Weak * -continuous derivations in dual Banach algebras

M. Eshaghi-Gordji, A. Ebadian, F. Habibian, B. Hayati (2012)

Archivum Mathematicum

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Let 𝒜 be a dual Banach algebra. We investigate the first weak * -continuous cohomology group of 𝒜 with coefficients in 𝒜 . Hence, we obtain conditions on 𝒜 for which H w * 1 ( 𝒜 , 𝒜 ) = { 0 } .

Compactness of derivations from commutative Banach algebras

Matthew J. Heath (2010)

Banach Center Publications

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We consider the compactness of derivations from commutative Banach algebras into their dual modules. We show that if there are no compact derivations from a commutative Banach algebra, A, into its dual module, then there are no compact derivations from A into any symmetric A-bimodule; we also prove analogous results for weakly compact derivations and for bounded derivations of finite rank. We then characterise the compact derivations from the convolution algebra ℓ¹(ℤ₊) to its dual. Finally,...