Range inclusion results for derivations on noncommutative Banach algebras

Volker Runde

Studia Mathematica (1993)

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

Abstract

top
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

top

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

top
  1. [B-D] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Ergeb. Math. Grenzgeb. 80, Springer, 1973. 
  2. [B-V] M. Brešar and J. Vukman, Derivations on noncommutative Banach algebras, Arch. Math. (Basel) 59 (1992), 363-370. Zbl0807.46049
  3. [Cus] J. Cusack, Automatic continuity and topologically simple radical Banach algebras, London Math. Soc. 16 (1977), 493-500. Zbl0398.46042
  4. [Gar] R. V. Garimella, Continuity of derivations on some semiprime Banach algebras, Proc. Amer. Math. Soc. 99 (1987), 289-292. Zbl0617.46056
  5. [G-W] K. R. Goodearl and R. B. Warfield Jr., Primitivity in differential operator rings, Math. Z. 180 (1982), 503-524. Zbl0495.16002
  6. [Hel] A. Ya. Helemskii, The Homology of Banach and Topological Algebras, Math. Appl. (Soviet Ser.) 41, Kluwer Acad. Publ., 1989 (translated from the Russian). 
  7. [Jia] X. Jiang, Remarks on automatic continuity of derivations and module derivations, Acta Math. Sinica (N.S.) 4 (1988), 227-233. Zbl0673.46026
  8. [Joh] B. E. Johnson, Continuity of derivations on commutative Banach algebras, Amer. J. Math. 91 (1969), 1-10. Zbl0181.41103
  9. [J-S] B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, ibid. 90 (1968), 1067-1073. Zbl0179.18103
  10. [Klei] D. C. Kleinecke, On operator commutators, Proc. Amer. Math. Soc. 8 (1957), 535-536. Zbl0079.12904
  11. [M-M] M. Mathieu and G. J. Murphy, Derivations mapping into the radical, Arch. Math. (Basel) 57 (1991), 469-474. Zbl0714.46038
  12. [M-R] M. Mathieu and V. Runde, Derivations mapping into the radical, II, Bull. London Math. Soc. 24 (1992), 485-487. Zbl0733.46023
  13. [Pos] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100. Zbl0082.03003
  14. [Rick] C. E. Rickart, General Theory of Banach Algebras, The University Series in Higher Mathematics, D. van Nostrand, 1960. 
  15. [Run] V. Runde, Automatic continuity of derivations and epimorphisms, Pacific J. Math. 147 (1991), 365-374. Zbl0666.46052
  16. [Shi] F. V. Shirokov, Proof of a conjecture of Kaplansky, Uspekhi Mat. Nauk 11 (1956), 167-168 (in Russian). 
  17. [Sin1] A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1967), 166-170. 
  18. [Sin2] A. M. Sinclair, Automatic Continuity of Linear Operators, London Math. Soc. Lecture Note Ser. 21, Cambridge University Press, 1976. 
  19. [S-W] I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260-264. Zbl0067.35101
  20. [Tho 1] M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128 (1988), 435-460. Zbl0681.47016
  21. [Tho 2] M. P. Thomas, Primitive ideals and derivations on noncommutative Banach algebras, preprint, 1991. 
  22. [Vuk 1] J. Vukman, On derivations in prime rings, Proc. Amer. Math. Soc. 116 (1992), 877-884. Zbl0792.16034
  23. [Vuk 2] J. Vukman, A result concerning derivations on Banach algebras, ibid., 971-976. 
  24. [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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.