AIMS Mathematics, 2020, 5(5): 4482-4493. doi: 10.3934/math.2020288

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Homomorphism-derivation functional inequalities in C*-algebras

1 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea
2 Department of Mathematics, Shaanxi University of Science and Technology, Xi’an, Shaanxi, P. R. China

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In this paper, we introduce and solve the following additive-additive $(s,t)$-functional inequality \begin{eqnarray}\label{0.1} && \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))\|, \nonumber \end{eqnarray} where $s$ and $t$ are fixed nonzero complex numbers with $|s| <1$ and $|t| <1$. 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}\label{0.1t} \nonumber && \|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\|, \nonumber \end{eqnarray} where $s$ and $t$ are fixed nonzero complex numbers with $|s| <1$ and $|t| <1$. Furthermore, we investigate $C^*$-ternary derivations and $C^*$-ternary homomorphisms in $C^*$-ternary algebras, associated to the additive-additive $(s,t)$-functional inequality (0.1) under some extra condition.
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