# 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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## Abstract

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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 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.

## How to cite

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Runde, 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 -

## References

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