Bi-additive s-Functional Inequalities and Quasi-∗-Multipliers on Banach ∗-Algebras

Abstract

Park introduced and investigated the following bi-additive s-functional inequalities

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle \| f(x{+}y, z{+}w) {+} f(x{+}y, z{-}w) {+} f(x{-}y, z{+}w) {+} f(x{-}y, z{-}w) {-}4f(x,z)\| \\ &\displaystyle \le \left \|s \left(4f\left(\frac{x{+}y}{2}, z{-}w\right) {+} 4f\left(\frac{x{-}y}{2}, z{+}w\right) {-} 4f(x,z ){+} 4 f(y, w)\right)\right\| , \end{array} \end{aligned} $$

(10.1)

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle \left\|4f\left(\frac{x+y}{2}, z-w\right) +4 f\left(\frac{x-y}{2}, z+w\right) -4 f(x,z )+4 f(y, w)\right\| \\ &\displaystyle \qquad \le \|s ( f(x+y, z+w) + f(x+y, z-w) + f(x-y, z+w)\\ &\displaystyle \qquad + f(x-y, z-w) -4f(x,z) )\| , \end{array} \end{aligned} $$

(10.2)

where s is a fixed nonzero complex number with |s| < 1. Using the direct method, we prove the Hyers-Ulam stability of quasi-∗-multipliers on Banach ∗-algebras and unital C ^∗-algebras, associated to the bi-additive s-functional inequalities (10.1) and (10.2).