10.30495/maca.2022.1946498.1040

Automatic continuity of almost Jordan derivations on special Jordan Banach algebras

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  • Ganesa Moorthy Chinnadurai1,
  • Siva Gurusamy1,
  1. Department of Mathematics, Alagappa University, Karaikudi-630 003, India

Published in Issue 2022-03-21

How to Cite

Automatic continuity of almost Jordan derivations on special Jordan Banach algebras. (2022). Mathematical Analysis and Its Contemporary Applications, 4(2). https://doi.org/10.30495/maca.2022.1946498.1040

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Abstract

The following is the question form of Kaplansky conjecture of 1958. Is every derivation on semisimple Banach algebra continuous? Kaplansky conjecture was proved by Johnson and Sinclair in 1968. The concept of almost Jordan derivations on Jordan Banach algebras is introduced in this article. Also, Kaplansky conjecture is extended to Jordan Banach algebras as an open question: Is every almost Jordan derivations on semisimple Jordan Banach algebras continuous? Moreover, a partial answer to this open question is derived in the sense that every almost Jordan derivation T on semisimple special Jordan Banach algebras Ω+, with an additional condition on Ω+, is continuous.

Keywords

  • Automatic continuity,
  • Almost derivation,
  • Banach algebras
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References

  1. B. Aupetit, Recent trends in the field of Jordan Banach algebras, Funct. Anal. Oper. Theory, 30(1994), 9-19.
  2. H. G. Dales, Banach Algebras and Automatic Continuity, London Mathematical Society Monographs 24, Clarendon Press, Oxford, 2000.
  3. B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math., 90 (1968), 1067-1073.
  4. P. Jordan, J. V. Neumann and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math., 35(1934), 29-64.
  5. I. Kaplansky, Derivations, Banach algebras seminar on analytic functions, Vol. II, Institute for Advanced Study, Princeton, 1958.
  6. K. McCrimmon and R. D. Schafer, On a class of noncommutative Jordan algebras, Proc. Nat. Acad. Sci. USA, 56(1966), 1-4.
  7. K. McCrimmon, The radical of a Jordan algebra, Proc. Nat. Acad. Sci. USA, 59(1969), 671678.
  8. K. McCrimmon, On Herstein’s theorems relating Jordan and associative algebras, J. Algebra, 13(1969), 382-392.
  9. A. A. Mohammed, Y. Abdulljabar and N. A. Abdulraziq, Jewell Sinclair theorem and automatic continuity of a derivation, J. Edu. Sci., 23(2010), 63-69.
  10. A. A. Mohammed and S. M. Ali, On Villena’s theorem of automatic continuity of essentially defined derivations on semisimple Banach algebras, Int. J. Math. Anal., 7(2013), 2931-2939.
  11. A. A. Mohammed and S. M. Ali, On the nonassociative Jewell Sinclair theorem, J. Pure Appl. Math. Adv. Appl., 11(2014), 137-146.
  12. A. A. Mohammed, On automatic continuity of closable derivations, Int. J. Math. Anal., 8(2014), 1161-1164.
  13. A. R. Villena, Derivations on Jordan Banach algebras, Studia Math., 118(1996), 205-229.