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Automatic Continuity of Almost Derivations on Frechet $Q$-Algebras

Year 2022, Volume: 5 Issue: 2, 88 - 91, 30.06.2022
https://doi.org/10.33434/cams.1032315

Abstract

In 1971 R. L. Carpenter proved that every derivation on a semisimple commutative Frechet algebra with identity is continuous. The concept of almost derivations on Frechet algebras is introduced in this article. Also, R. L. Carpenter result motivates us to ask an open question: Is every almost derivation on semisimple commutative Frechet algebras continuous?. Moreover, a partial answer to this open question is derived in the sense that every almost derivation T on semisimple commutative Frechet Q-algebras A, with an additional condition on A, is continuous. Furthermore, an example is provided to illustrate our main result.

References

  • [1] R. L. Carpenter, Continuity of derivations in F-algebras, Amer. J. Math., 93 (1971), 500-502.
  • [2] H. G. Dales, Banach Algebras and Automatic Continuity, London Mathematical Society Monographs 24, Clarendon Press, Oxford, 2000.
  • [3] M. Fragoulopoulou, Topological Algebras with Involution, Elsevier Science, Amsterdam, 2005.
  • [4] T. G. Honary, M. Omidi and A. H. Sanatpour, Automatic continuity of almost multiplicative maps between Fr´echet algebras, Bull. Korean Math. Soc., 41 (2015), 1497-1509.
  • [5] B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math., 90 (1968), 1067-1073.
  • [6] I. Kaplansky, Derivations, Banach algebras seminar on analytic functions, Vol. II, Institute for Advanced Study, Princeton, 1958.
  • [7] A. Mallios, Topological Algebras, North-Holland Publishing Co., Amsterdam, 1986.
  • [8] A. A. Mohammed, Y. Abdulljabar and N. A. Abdulraziq, Jewell Sinclair theorem and automatic continuity of a derivation, J. Edu. and Sci., 23 (2010), 63-69.
  • [9] A. A. Mohammed and S. M. Ali, On Villena’s theorem of automatic continuity of essentially defined derivations on semisimple Banach algebras, Int. Journal of Math. Analysis., 7 (2013), 2931-2939.
  • [10] A. A. Mohammed and S. M. Ali, On the nonassociative Jewell Sinclair theorem, J. Pure and Applied Mathematics: Advances and Applications, 11 (2014), 137-146.
  • [11] A. A. Mohammed, On automatic continuity of closable derivations, Int. Journal of Math. Analysis., 8 (2014), 1161-1164.
  • [12] C. G. Moorthy and G. Siva, Automatic continuity of almost Jordan derivations on special Jordan Banach algebras, Mathematical Analysis and its Contemporary Applications., 4 (2022), 11-16.
Year 2022, Volume: 5 Issue: 2, 88 - 91, 30.06.2022
https://doi.org/10.33434/cams.1032315

Abstract

References

  • [1] R. L. Carpenter, Continuity of derivations in F-algebras, Amer. J. Math., 93 (1971), 500-502.
  • [2] H. G. Dales, Banach Algebras and Automatic Continuity, London Mathematical Society Monographs 24, Clarendon Press, Oxford, 2000.
  • [3] M. Fragoulopoulou, Topological Algebras with Involution, Elsevier Science, Amsterdam, 2005.
  • [4] T. G. Honary, M. Omidi and A. H. Sanatpour, Automatic continuity of almost multiplicative maps between Fr´echet algebras, Bull. Korean Math. Soc., 41 (2015), 1497-1509.
  • [5] B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math., 90 (1968), 1067-1073.
  • [6] I. Kaplansky, Derivations, Banach algebras seminar on analytic functions, Vol. II, Institute for Advanced Study, Princeton, 1958.
  • [7] A. Mallios, Topological Algebras, North-Holland Publishing Co., Amsterdam, 1986.
  • [8] A. A. Mohammed, Y. Abdulljabar and N. A. Abdulraziq, Jewell Sinclair theorem and automatic continuity of a derivation, J. Edu. and Sci., 23 (2010), 63-69.
  • [9] A. A. Mohammed and S. M. Ali, On Villena’s theorem of automatic continuity of essentially defined derivations on semisimple Banach algebras, Int. Journal of Math. Analysis., 7 (2013), 2931-2939.
  • [10] A. A. Mohammed and S. M. Ali, On the nonassociative Jewell Sinclair theorem, J. Pure and Applied Mathematics: Advances and Applications, 11 (2014), 137-146.
  • [11] A. A. Mohammed, On automatic continuity of closable derivations, Int. Journal of Math. Analysis., 8 (2014), 1161-1164.
  • [12] C. G. Moorthy and G. Siva, Automatic continuity of almost Jordan derivations on special Jordan Banach algebras, Mathematical Analysis and its Contemporary Applications., 4 (2022), 11-16.
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ganesa Moorthy C

Sıva G

Publication Date June 30, 2022
Submission Date December 4, 2021
Acceptance Date June 23, 2022
Published in Issue Year 2022 Volume: 5 Issue: 2

Cite

APA C, G. M., & G, S. (2022). Automatic Continuity of Almost Derivations on Frechet $Q$-Algebras. Communications in Advanced Mathematical Sciences, 5(2), 88-91. https://doi.org/10.33434/cams.1032315
AMA C GM, G S. Automatic Continuity of Almost Derivations on Frechet $Q$-Algebras. Communications in Advanced Mathematical Sciences. June 2022;5(2):88-91. doi:10.33434/cams.1032315
Chicago C, Ganesa Moorthy, and Sıva G. “Automatic Continuity of Almost Derivations on Frechet $Q$-Algebras”. Communications in Advanced Mathematical Sciences 5, no. 2 (June 2022): 88-91. https://doi.org/10.33434/cams.1032315.
EndNote C GM, G S (June 1, 2022) Automatic Continuity of Almost Derivations on Frechet $Q$-Algebras. Communications in Advanced Mathematical Sciences 5 2 88–91.
IEEE G. M. C and S. G, “Automatic Continuity of Almost Derivations on Frechet $Q$-Algebras”, Communications in Advanced Mathematical Sciences, vol. 5, no. 2, pp. 88–91, 2022, doi: 10.33434/cams.1032315.
ISNAD C, Ganesa Moorthy - G, Sıva. “Automatic Continuity of Almost Derivations on Frechet $Q$-Algebras”. Communications in Advanced Mathematical Sciences 5/2 (June 2022), 88-91. https://doi.org/10.33434/cams.1032315.
JAMA C GM, G S. Automatic Continuity of Almost Derivations on Frechet $Q$-Algebras. Communications in Advanced Mathematical Sciences. 2022;5:88–91.
MLA C, Ganesa Moorthy and Sıva G. “Automatic Continuity of Almost Derivations on Frechet $Q$-Algebras”. Communications in Advanced Mathematical Sciences, vol. 5, no. 2, 2022, pp. 88-91, doi:10.33434/cams.1032315.
Vancouver C GM, G S. Automatic Continuity of Almost Derivations on Frechet $Q$-Algebras. Communications in Advanced Mathematical Sciences. 2022;5(2):88-91.

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