# Range inclusion results for derivations on noncommutative Banach algebras

Studia Mathematica (1993)

- Volume: 105, Issue: 2, page 159-172
- ISSN: 0039-3223

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topRunde, Volker. "Range inclusion results for derivations on noncommutative Banach algebras." Studia Mathematica 105.2 (1993): 159-172. <http://eudml.org/doc/215992>.

@article{Runde1993,

abstract = {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.},

author = {Runde, Volker},

journal = {Studia Mathematica},

keywords = {range inclusion; noncommutative Banach algebras; derivation; radical; quasinilpotent},

language = {eng},

number = {2},

pages = {159-172},

title = {Range inclusion results for derivations on noncommutative Banach algebras},

url = {http://eudml.org/doc/215992},

volume = {105},

year = {1993},

}

TY - JOUR

AU - Runde, Volker

TI - Range inclusion results for derivations on noncommutative Banach algebras

JO - Studia Mathematica

PY - 1993

VL - 105

IS - 2

SP - 159

EP - 172

AB - 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.

LA - eng

KW - range inclusion; noncommutative Banach algebras; derivation; radical; quasinilpotent

UR - http://eudml.org/doc/215992

ER -

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