Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On the Invariance of Primitive Ideals via φ-derivations on Banach Algebras

  • Received : 2012.07.03
  • Accepted : 2013.09.17
  • Published : 2013.12.23
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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 ${\phi}$-derivations to be nilpotent. Moreover, we show that the spectral boundedness of ${\phi}$-derivations implies that they leave each primitive ideal of Banach algebras invariant.

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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. https://doi.org/10.1006/jfan.1995.1116
  3. M. Bresar and A. R. Villena, The noncommutative Singer-Wermer conjecture and $\phi$-derivations, J. London. Math. Soc., 66(3)(2002), 710-720. https://doi.org/10.1112/S0024610702003721
  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. https://doi.org/10.11650/twjm/1500405500
  6. B. E. Johnson, Continuity of derivations on commutative Banach algebras, Amer. J. Math., 91(1969), 1-10. https://doi.org/10.2307/2373262
  7. M. Mathieu, Where to find the image of a derivation, Banach Center Publ., 30(1994), 237-249. https://doi.org/10.4064/-30-1-237-249
  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. https://doi.org/10.1007/BF01362370
  12. M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math., 128(2)(1988), 435-460. https://doi.org/10.2307/1971432
  13. M. P. Thomas, Primitive ideals and derivations on non-commutative Banach algebras, Pacific J. Math., 159(1993), 139-152. https://doi.org/10.2140/pjm.1993.159.139

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