Additive functional inequality in C*-algebras

Siriluk Donganont

doi:10.5269/bspm.77682

Siriluk Donganont university of Phayao

Abstract

In this paper, we introduce an additive functional inequality
\begin{eqnarray}\label{1}
\left\| 2 g\left(\frac{\lambda u+ y}{2}\right) - \lambda g(u) - g(y) \right\|\le
\left\| s( g\left(\lambda u+ y\right)- \lambda g(u) - g(y) ) \right\|
\end{eqnarray}
for all $\lambda\in \mathbb{C}$, all unitary elements $u$ in a unital $C^*$-algebra $P$ and all $y\in P$, where $|s|<1$.
Using both the direct method and the fixed point method, we establish the Hyers-Ulam stability of inequality (\ref{1}) in unital $C^*$-algebras.
Furthermore, we apply these results to the study of $C^*$-algebra homomorphisms and $C^*$-algebra derivations in unital $C^*$-algebras.

Downloads

Download data is not yet available.

References

M. Ahmad, A. Zada, Ulam’s stability results of conformable neutral fractional differential equations Ψ-Hilfer fractional integrodifferential equations, Bol. Soc. Parana. Mat. (3) 41 (2023), 1–13.

T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66.

L. Cadariu, V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (2003), no. 1, Art. ID 4.

J. Diaz, 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.

S. Donganont, C. Park, Combined system of additive functional equations in Banach algebras, Open Mathematics 22 (2024), no. 1, 20230177.

P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436.

F. Haddouchi, Existence and Ulam-Hyers stability results for a class of fractional integro-differential equations involving nonlocal fractional integro-differential boundary conditions, Bol. Soc. Parana. Mat. (3) 42 (2024), 1–19.

D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224.

G. Isac, Th. M. Rassias, Stability of ψ-additive mappings: Applications to nonlinear analysis, Int. J. Math. Math. Sci. 19 (1996), 219–228.

S. Jung, D. Popa, M. Th. Rassias, On the stability of the linear functional equation in a single variable on complete metric spaces, J. Global Optim. 59 (2014), 13–16.

R. V. Kadison, J. R. Ringrose, Fundamentals of the Theory of Operator Algebras: Elementary Theory, Academic Press, New York, 1983.

Y. Lee, S. Jung, M. Th. Rassias, Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation, J. Math. Inequal. 12 (2018), 43–61.

D. Mihet¸, V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572.

C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26.

C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397–407.

C. Park, Fixed point method for set-valued functional equations, J. Fixed Point Theory Appl. 19 (2017), 2297–2308.

V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96.

Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.

S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.

Z. Zamani, B. Yousefi, H. Azadi Kenary, Fuzzy Hyers-Ulam-Rassias stability for generalized additive functional equations, Bol. Soc. Parana. Mat. (3) 40 (2022), 1–14.