Poisson C*-algebra derivations in Poisson C*-algebras

• Autorzy: Wang Yongqiao , Park Choonkil , Chang Yuan

Identyfikatory

DOI

10.1515/dema-2024-0053

• Języki publikacji: EN

Abstrakty

EN

In this study, we introduce the following additive functional equation: 𝑔(𝜆𝑢+𝑣+2𝑦)=𝜆𝑔(𝑢)+𝑔(𝑣)+2𝑔(𝑦) for all 𝜆 ∈ℂ, all unitary elements 𝑢,𝑣 in a unital Poisson C*-algebra P, and all 𝑦 ∈𝑃. Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of the aforementioned additive functional equation in unital Poisson C*-algebras. Furthermore, we apply to study Poisson C*-algebra homomorphisms and Poisson C*-algebra derivations in unital Poisson C*-algebras.

Słowa kluczowe

EN

Hyers-Ulam stability; fixed point method; additive functional equation; Poisson C* -algebra derivation in Poisson C*-algebra; Poisson C*-algebra homomorphism in Poisson C*-algebra

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2024
• Tom: Vol. 57, nr 1
• Strony: art. no. 20240053
• Opis fizyczny: Bibliogr. 30 poz.

Twórcy

autor

Wang Yongqiao

• School of Science, Dalian Maritime University, Dalian 116026, P. R. China

autor

Park Choonkil

• Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

autor

Chang Yuan

• School of Mathematics, Dongbei University of Finance and Economics, Dalian, 116025, P. R. China

Bibliografia

• [1] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publications, New York, 1960.
• [2] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224, DOI: http://dx.doi.org/10.1093/jahist/jav119.
• [3] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66, DOI: http://dx.doi.org/10.2969/jmsj/00210064.
• [4] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300, DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0507327-1.
• [5] Th. M. Rassias, Problem 16; 2, Report of the 27th International Symposium on Functional Equations, Aequationes Mathematics, vol. 39, 1990, pp. 292–293; 309.
• [6] Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. 14 (1991), 431–434, DOI: http://dx.doi.org/10.1155/S016117129100056X.
• [7] Th. M. Rassias and P. Šemrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325–338, DOI: http://dx.doi.org/10.1006/jmaa.1993.1070.
• [8] P. Gǎvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436, DOI: https://doi.org/10.1006/jmaa.1994.1211.
• [9] O. H. Ezzat, Functional equations related to higher derivations in semiprime rings, Open Math. 19 (2021), 1359–1365, DOI: https://doi.org/10.1515/math-2021-0123.
• [10] S. Jung, D. Popa, and M. Th. Rassias, On the stability of the linear functional equation in a single variable on complete metric groups, J. Global Optim. 59 (2014), 165–171, DOI: https://doi.org/10.1007/s10898-013-0083-9.
• [11] S. Karthikeyan, C. Park, P. Palani, and T. R. K. Kumar, Stability of an additive-quartic functional equation in modular spaces, J. Math. Comput. Sci. 26 (2022), no. 1, 22–40, DOI: http://dx.doi.org/10.22436/jmcs.026.01.04.
• [12] G. Kim and Y. Lee, Stability of an additive-quadratic-quartic functional equation, Demonstr. Math. 53 (2020), 1–7, DOI: https://doi.org/10.1515/dema-2020-0001.
• [13] Y. Lee, S. Jung, and M. Th. Rassias, Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation, J. Math. Inequal. 12 (2018), 43–61, DOI: http://dx.doi.org/10.7153/jmi-2018-12-04.
• [14] A. L. Olutimo, A. Bilesanmi, and I. D. Omoko, Stability and boundedness analysis for a system of two nonlinear delay differential equations, J. Nonlinear Sci. Appl. 16 (2023), no. 2, 90–98, DOI: http://dx.doi.org/10.22436/jnsa.016.02.02.
• [15] S. Paokanta, C. Park, N. Jun-on, and R. Superatulatorn, An additive-cubic functional equation in a Banach space, J. Math. Comput. Sci. 33 (2024), no. 3, 264–274, DOI: http://dx.doi.org/10.22436/jmcs.033.03.05.
• [16] C. Park, K. Tamilvana, G. Balasubramanian, B. Noori, and A. Najati, On a functional equation that has the quadratic-multiplicative property, Open Math. 18 (2020), 837–845, DOI: https://doi.org/10.1515/math-2020-0032.
• [17] A. Thanyacharoen and W. Sintunavarat, On new stability results for composite functional equations in quasi-β-normed spaces, Demonstr. Math. 54 (2021), 68–84, DOI: https://doi.org/10.1515/dema-2021-0002.
• [18] A. Turab, N. Rosli, W. Ali, and J. J. Nieto, The existence and uniqueness of solutions to a functional equation arising in psychological learning theory, Demonstr. Math. 56 (2023), 20220231, DOI: https://doi.org/10.1515/dema-2022-0231.
• [19] L. Cădariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (2003), no. 1, 4, DOI: http://eudml.org/doc/123714.
• [20] J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309, DOI: https://doi.org/10.1090/S0002-9904-1968-11933-0.
• [21] G. Isac and Th. M. Rassias, Stability of ψ-additive mappings: Applications to nonlinear analysis, Int. J. Math. Math. Sci. 19 (1996), 219–228, DOI: https://doi.org/10.1155/s0161171296000324.
• [22] C. Park, Fixed point method for set-valued functional equations, J. Fixed Point Theory Appl. 19 (2017), 2297–2308, DOI: https://doi.org/10.1007/s11784-017-0418-0.
• [23] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96.
• [24] C. Bai, R. Bai, L. Guo, and Y. Wu, Transposed Poisson algebras, Novikov-Poisson algebras and 3-Lie algebras, J. Algebra. 632 (2023), 535–566, DOI: https://doi.org/10.1016/j.jalgebra.2023.06.006.
• [25] Y. Cho, R. Saadati, and J. Vahidi, Approximation of homomorphisms and derivations on non-Archimedean Lie C*-algebras via fixed point method, Discrete Dyn. Nat. Soc. 2012 (2012), 373904, DOI: https://doi.org/10.1155/2012/373904.
• [26] S. Donganont, C. Park, and Th. M. Rassias, An additive functional inequality in Poisson C*-algebras, Nonlinear Analysis, Geometry and Differential Equations, Springer, Cham (in press).
• [27] J. Lee, C. Park, and Th. M. Rassias, C*-Bi-ternary derivations in C*-algebra-ternary algebras, Functional Equations and Ulam’s Problem, Springer, Cham.
• [28] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras: Elementary Theory, Academic Press, New York, 1983.
• [29] C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26, DOI: http://dx.doi.org/10.7153/jmi-09-02.
• [30] D. Miheţ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572, DOI: http://dx.doi.org/10.1016/j.jmaa.2008.01.100.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2026).