Research article

Homomorphism-derivation functional inequalities in C*-algebras

  • Received: 23 January 2020 Accepted: 18 May 2020 Published: 20 May 2020
  • MSC : 47B47, 11E20, 17B40, 39B52, 46L05, 39B72

  • In this paper, we introduce and solve the following additive-additive $ (s, t) $-functional inequality $ \begin{eqnarray} && \left\|g\left(x+y\right) -g(x) -g(y)\right\| + \left\|2 h\left(\frac{x+y}{2}\right) - h(x) - h(y) \right\| \\ && \le \left\| s\left( 2 g\left(\frac{x+y}{2}\right)-g(x)-g(y)\right)\right\|+ \|t ( h(x+y)-h(x)-h(y))\| , \end{eqnarray} $ where $ s $ and $ t $ are fixed nonzero complex numbers with $ |s| \lt 1 $ and $ |t| \lt 1 $. Furthermore, we investigate homomorphisms and derivations in complex Banach algebras and unital $ C^* $-algebras, associated to the additive-additive $ (s, t) $-functional inequality (0.1) under some extra condition. Moreover, we introduce and solve the following additive-additive $ (s, t) $-functional inequality $ \begin{eqnarray} && \|g\left(x+y+z\right) -g(x) -g(y)-g(z)\| +\left\|3h\left(\frac{x+y+z}{3}\right)+ h(x-2y+z) + h(x+y-2z)-3 h(x) \right\| \\ && \le \left\|s\left( 3 g\left(\frac{x+y+z}{3}\right)-g(x)-g(y)-g(z)\right)\right\| \\ && + \left\|t \left( h(x+y+z) + h(x-2y+z) + h(x+y-2z)-3 h(x) \right) \right\| , \end{eqnarray} $ where $ s $ and $ t $ are fixed nonzero complex numbers with $ |s| \lt 1 $ and $ |t| \lt 1 $. Furthermore, we investigate $ C^* $-ternary derivations and $ C^* $-ternary homomorphisms in $ C^* $-ternary algebras, associated to the additive-additive $ (s, t) $-functional inequality (0.2) under some extra condition.

    Citation: Choonkil Park, XiaoYing Wu. Homomorphism-derivation functional inequalities in C*-algebras[J]. AIMS Mathematics, 2020, 5(5): 4482-4493. doi: 10.3934/math.2020288

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  • In this paper, we introduce and solve the following additive-additive $ (s, t) $-functional inequality $ \begin{eqnarray} && \left\|g\left(x+y\right) -g(x) -g(y)\right\| + \left\|2 h\left(\frac{x+y}{2}\right) - h(x) - h(y) \right\| \\ && \le \left\| s\left( 2 g\left(\frac{x+y}{2}\right)-g(x)-g(y)\right)\right\|+ \|t ( h(x+y)-h(x)-h(y))\| , \end{eqnarray} $ where $ s $ and $ t $ are fixed nonzero complex numbers with $ |s| \lt 1 $ and $ |t| \lt 1 $. Furthermore, we investigate homomorphisms and derivations in complex Banach algebras and unital $ C^* $-algebras, associated to the additive-additive $ (s, t) $-functional inequality (0.1) under some extra condition. Moreover, we introduce and solve the following additive-additive $ (s, t) $-functional inequality $ \begin{eqnarray} && \|g\left(x+y+z\right) -g(x) -g(y)-g(z)\| +\left\|3h\left(\frac{x+y+z}{3}\right)+ h(x-2y+z) + h(x+y-2z)-3 h(x) \right\| \\ && \le \left\|s\left( 3 g\left(\frac{x+y+z}{3}\right)-g(x)-g(y)-g(z)\right)\right\| \\ && + \left\|t \left( h(x+y+z) + h(x-2y+z) + h(x+y-2z)-3 h(x) \right) \right\| , \end{eqnarray} $ where $ s $ and $ t $ are fixed nonzero complex numbers with $ |s| \lt 1 $ and $ |t| \lt 1 $. Furthermore, we investigate $ C^* $-ternary derivations and $ C^* $-ternary homomorphisms in $ C^* $-ternary algebras, associated to the additive-additive $ (s, t) $-functional inequality (0.2) under some extra condition.


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    [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66. doi: 10.2969/jmsj/00210064
    [2] J. Bae, I. Chang, Some additive mappings on Banach *-algebras with derivation, J. Nonlinear Sci. Appl., 11 (2018), 335-341.
    [3] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76-86.
    [4] Y. Ding, Ulam-Hyers stability of fractional impulsive differential equations, J. Nonlinear Sci. Appl., 11 (2018), 953-959.
    [5] N. Eghbali, J. M. Rassias, M. Taheri, On the stability of a k-cubic functional equation in intuitionistic fuzzy n-normed spaces, Results Math., 70 (2016), 233-248.
    [6] G. Z. Eskandani, P. Găvruta, Hyers-Ulam-Rassias stability of pexiderized Cauchy functional equation in 2-Banach spaces, J. Nonlinear Sci. Appl., 5 (2012), 459-465.
    [7] P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436. doi: 10.1006/jmaa.1994.1211
    [8] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27 (1941), 222-224. doi: 10.1073/pnas.27.4.222
    [9] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras: Elementary Theory, Academic Press, New York, 1983.
    [10] C. Park, Homomorphisms between Poisson JC*-algebras, B. Braz. Math. Soc., 36 (2005), 79-97. doi: 10.1007/s00574-005-0029-z
    [11] C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal., 9 (2015), 17-26.
    [12] C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal., 9 (2015), 397-407.
    [13] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4 (2003), 91-96.
    [14] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc., 72 (1978), 297-300. doi: 10.1090/S0002-9939-1978-0507327-1
    [15] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113-129. doi: 10.1007/BF02924890
    [16] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publication, New York, 1960.
    [17] Z. Wang, Stability of two types of cubic fuzzy set-valued functional equations, Results Math., 70 (2016), 1-14.
    [18] H. Zettl, A characterization of ternary rings of operators, Adv. Math., 48 (1983), 117-143. doi: 10.1016/0001-8708(83)90083-X
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