Homomorphism-derivation functional inequalities in <em>C</em><sup>*</sup>-algebras

Choonkil Park; XiaoYing Wu; Choonkil Park; XiaoYing Wu

doi:10.3934/math.2020288

AIMS Mathematics

2020, Volume 5, Issue 5: 4482-4493. doi: 10.3934/math.2020288

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Research article

Homomorphism-derivation functional inequalities in C^*-algebras

Choonkil Park ^{1
,
,},
XiaoYing Wu ²

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

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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Abstract

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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