Geodesic
Range inclusion results for derivations on noncommutative Banach algebras
Studia Mathematica, Tome 105 (1993) no. 2, pp. 159-172

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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 a result by M. Mathieu and the author which asserts that every centralizing derivation on a Banach algebra maps into the radical. As far as the second question is concerned, we are unable to settle it, but we obtain a reduction of the problem and can prove the quasinilpotency of Da under commutativity assumptions slightly stronger than [a,Da] = 0.
DOI : 10.4064/sm-105-2-159-172

Volker Runde 1

1 
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Volker Runde. Range inclusion results for derivations on noncommutative Banach algebras. Studia Mathematica, Tome 105 (1993) no. 2, pp. 159-172. doi: 10.4064/sm-105-2-159-172

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