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1993 | 105 | 2 | 159-172
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Range inclusion results for derivations on noncommutative Banach algebras

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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.
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Bibliografia
  • [B-D] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Ergeb. Math. Grenzgeb. 80, Springer, 1973.
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  • [Sin1] A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1967), 166-170.
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  • [Tho 1] M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128 (1988), 435-460.
  • [Tho 2] M. P. Thomas, Primitive ideals and derivations on noncommutative Banach algebras, preprint, 1991.
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  • [Vuk 2] J. Vukman, A result concerning derivations on Banach algebras, ibid., 971-976.
  • [Yoo] B. Yood, Continuous homomorphisms and derivations on Banach algebras, in: F. Greenleaf and D. Gulick (eds.), Banach Algebras and Several Complex Variables, Contemp. Math. 32, Amer. Math. Soc., 1984, 279-284.
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