Some additive mappings on Banach \({\ast}\)-algebras with derivations

Authors

Jae-Hyeong Bae - Humanitas College, Kyung Hee University, Yongin 17104, Republic of Korea
Ick-Soon Chang - Department of Mathematics, Chungnam National University, 99 Daehangno, Yuseong-gu, Daejeon 34134, Republic of Korea

Abstract

We take into account some additive mappings in Banach \(\ast\)-algebras with derivations. We will first study the conditions for additive mappings with derivations on Banach \(\ast\)-algebras. Then we prove some theorems involving linear mappings on Banach $\ast$-algebras with derivations. So derivations on \(C^{\ast}\)-algebra are characterized.

Keywords

Banach \(\ast\)-algebra, \(C^{\ast}\)-algebra, additive mapping with involution, derivation

References

[1] R. P. Agarwal, R. Saadati, A. Salamati, Approximation of the multiplicatives on random multi-normed space, J. Inequal. Appl., 2017 (2017), 10 pages.
[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64–66.
[3] Z. Baderi, R. Saadati, Generalized stability of Euler-Lagrange quadratic functional equation in random normed spaces under arbitrary t-norms, Thai J. Math., 14 (2016), 585–590.
[4] R. Badora, On approximate ring homomorphisms, J. Math. Anal. Appl., 276 (2002), 589–597.
[5] R. Badora, On approximate derivations, Math. Inequal. Appl., 9 (2006), 167–173.
[6] F. F. Bonsall, J. Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer- Verlag, New York-Heidelberg, (1973).
[7] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16 (1949), 385–397.
[8] M. Brešar, J. Vukman, On some additive mappings in rings with involution, Aequationes Math., 38 (1989), 178–185.
[9] M. Brešar, J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc., 110 (1990), 7–16.
[10] Y. J. Cho, C.-K. Park, T. M. Rassias, R. Saadati, Stability of functional equations in Banach algebras, Springer, Cham, (2015).
[11] M. N. Daif, M. S. Tammam El-Sayiad, On generalized derivations of semiprime rings with involution, Int. J. Algebra, 1 (2007), 551–555.
[12] P. Găvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431–436.
[13] M. E. Gordji, Nearly involutions on Banach algebras, A fixed point approach, Fixed Point Theory, 14 (2013), 117–123.
[14] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), 222–224.
[15] B. E. Johnson, A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math., 90 (1968), 1067– 1073.
[16] P. Kannappan, Functional equations and inequalities with applications, Springer Monographs in Mathematics, Springer, New York, (2009).
[17] G. V. Milovanović(ed.), T. M. Rassias (ed.), Analytic number theory, approximation theory, and special functions, In honor of Hari M. Srivastava, Springer, New York, (2014).
[18] C.-K. Park, G. A. Anastassiou, R. Saadati, S.-S. Yun, Functional inequalities in fuzzy normed spaces, J. Comput. Anal. Appl., 22 (2017), 601–612.
[19] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300.
[20] P. Šemrl, The functional equation of multiplicative derivation is superstable on standard operator algebras, Integral Equations Operator Theory, 18 (1994), 118–122.
[21] I. M. Singer, J. Wermer, Derivations on commutative normed algebras, Math. Ann., 129 (1955), 260–264.
[22] M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math., 128 (1988), 435–460.
[23] S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers, New York-London, (1960).

Downloads

XML export