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Title

Range inclusion results for derivations on noncommutative Banach algebras

Authors 1

Affiliations

  1. Department of Mathematics, University of California, Berkeley, California 94720, U.S.A.

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.

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Studia Mathematica
1993/105/2
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Pages:
159-172
Main language of publication
English
Received
1992-08-11
Accepted
1993-01-14
Published
1993
Exact and natural sciences