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On the Invariance of Primitive Ideals via φ-derivations on Banach Algebras
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  • Journal title : Kyungpook mathematical journal
  • Volume 53, Issue 4,  2013, pp.497-505
  • Publisher : Department of Mathematics, Kyungpook National University
  • DOI : 10.5666/KMJ.2013.53.4.497
 Title & Authors
On the Invariance of Primitive Ideals via φ-derivations on Banach Algebras
Jung, Yong-Soo;
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 Abstract
The noncommutative Singer-Wermer conjecture states that every derivation on a Banach algebra (possibly noncommutative) leaves primitive ideals of the algebra invariant. This conjecture is still an open question for more than thirty years. In this note, we approach this question via some sufficient conditions for the separating ideal of -derivations to be nilpotent. Moreover, we show that the spectral boundedness of -derivations implies that they leave each primitive ideal of Banach algebras invariant.
 Keywords
-derivation;primitive ideal;Jacobson radical;separating space;spectrally bounded;
 Language
English
 Cited by
1.
Some conditions under which Jordan derivations are zero, Journal of Taibah University for Science, 2016  crossref(new windwow)
 References
1.
F. F. Bonsall and J. Duncan, Complete normed algebras, Berlin-Heidelberg-New York, 1973.

2.
M. Bresar and M. Mathieu, Derivations mapping into the radical, III. J. Func. Anal., 133(1995), 21-29. crossref(new window)

3.
M. Bresar and A. R. Villena, The noncommutative Singer-Wermer conjecture and $\phi$-derivations, J. London. Math. Soc., 66(3)(2002), 710-720. crossref(new window)

4.
J. Cusack, Automatic continuity and topologically simple radical Banach algebras, J. London Math. Soc., 16(1977), 493-500.

5.
S. Hejazian and A. R. Janfada, Invariance of primitive ideals by $\phi$-derivations on Banach algebras, Taiwanese. J. Math., 13(4)(2009), 1181-1187.

6.
B. E. Johnson, Continuity of derivations on commutative Banach algebras, Amer. J. Math., 91(1969), 1-10. crossref(new window)

7.
M. Mathieu, Where to find the image of a derivation, Banach Center Publ., 30(1994), 237-249.

8.
C. E. Rickart, General theory of Banach algebra, Van Nostrand, Princeton 1950.

9.
A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc., 20(1969), 166-170.

10.
A. M. Sinclair, Automatic continuity of linear operators, London Math. Soc. Lecture Note Ser. 21(1976).

11.
I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann., 129(1955), 260-264. crossref(new window)

12.
M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math., 128(2)(1988), 435-460. crossref(new window)

13.
M. P. Thomas, Primitive ideals and derivations on non-commutative Banach algebras, Pacific J. Math., 159(1993), 139-152. crossref(new window)