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.
@article{bwmeta1.element.bwnjournal-article-smv105i2p159bwm,
author = {Volker Runde},
title = {Range inclusion results for derivations on noncommutative Banach algebras},
journal = {Studia Mathematica},
volume = {104},
year = {1993},
pages = {159-172},
zbl = {0810.46044},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv105i2p159bwm}
}
Runde, Volker. Range inclusion results for derivations on noncommutative Banach algebras. Studia Mathematica, Tome 104 (1993) pp. 159-172. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv105i2p159bwm/
[00000] [B-D] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Ergeb. Math. Grenzgeb. 80, Springer, 1973.
[00001] [B-V] M. Brešar and J. Vukman, Derivations on noncommutative Banach algebras, Arch. Math. (Basel) 59 (1992), 363-370. | Zbl 0807.46049
[00002] [Cus] J. Cusack, Automatic continuity and topologically simple radical Banach algebras, London Math. Soc. 16 (1977), 493-500. | Zbl 0398.46042
[00003] [Gar] R. V. Garimella, Continuity of derivations on some semiprime Banach algebras, Proc. Amer. Math. Soc. 99 (1987), 289-292. | Zbl 0617.46056
[00004] [G-W] K. R. Goodearl and R. B. Warfield Jr., Primitivity in differential operator rings, Math. Z. 180 (1982), 503-524. | Zbl 0495.16002
[00005] [Hel] A. Ya. Helemskii, The Homology of Banach and Topological Algebras, Math. Appl. (Soviet Ser.) 41, Kluwer Acad. Publ., 1989 (translated from the Russian).
[00006] [Jia] X. Jiang, Remarks on automatic continuity of derivations and module derivations, Acta Math. Sinica (N.S.) 4 (1988), 227-233. | Zbl 0673.46026
[00007] [Joh] B. E. Johnson, Continuity of derivations on commutative Banach algebras, Amer. J. Math. 91 (1969), 1-10. | Zbl 0181.41103
[00008] [J-S] B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, ibid. 90 (1968), 1067-1073. | Zbl 0179.18103
[00009] [Klei] D. C. Kleinecke, On operator commutators, Proc. Amer. Math. Soc. 8 (1957), 535-536. | Zbl 0079.12904
[00010] [M-M] M. Mathieu and G. J. Murphy, Derivations mapping into the radical, Arch. Math. (Basel) 57 (1991), 469-474. | Zbl 0714.46038
[00011] [M-R] M. Mathieu and V. Runde, Derivations mapping into the radical, II, Bull. London Math. Soc. 24 (1992), 485-487. | Zbl 0733.46023
[00012] [Pos] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100. | Zbl 0082.03003
[00013] [Rick] C. E. Rickart, General Theory of Banach Algebras, The University Series in Higher Mathematics, D. van Nostrand, 1960.
[00014] [Run] V. Runde, Automatic continuity of derivations and epimorphisms, Pacific J. Math. 147 (1991), 365-374. | Zbl 0666.46052
[00015] [Shi] F. V. Shirokov, Proof of a conjecture of Kaplansky, Uspekhi Mat. Nauk 11 (1956), 167-168 (in Russian).
[00016] [Sin1] A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1967), 166-170.
[00017] [Sin2] A. M. Sinclair, Automatic Continuity of Linear Operators, London Math. Soc. Lecture Note Ser. 21, Cambridge University Press, 1976.
[00018] [S-W] I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260-264. | Zbl 0067.35101
[00019] [Tho 1] M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128 (1988), 435-460. | Zbl 0681.47016
[00020] [Tho 2] M. P. Thomas, Primitive ideals and derivations on noncommutative Banach algebras, preprint, 1991.
[00021] [Vuk 1] J. Vukman, On derivations in prime rings, Proc. Amer. Math. Soc. 116 (1992), 877-884. | Zbl 0792.16034
[00022] [Vuk 2] J. Vukman, A result concerning derivations on Banach algebras, ibid., 971-976.
[00023] [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.