Stability of Some Inequalities in Banach ∗-Algebras

In this paper, we investigate the stability and superstability of a specific class of functional inequalities associated with centrally extended ∗-derivations on Banach ∗-algebras. A CE ∗-derivation δ:R→R is defined as an additive mapping satisfying δ(x+y)−δ(x)−δ(y)∈Z(R) and δ(xy)−δ(x)y∗−xδ(y)∈Z(R) for all x,y∈R, where Z(R) denotes the center of the ring. We consider the functional inequality ∥[a1δ(x1)+a2δ(x2)+a3δ(x3),w]∥ ≤ ∥[δ(a1x1+a2x2+a3x3),w]∥ + Φ(x1,x2,x3,w), where Φ is a perturbing term. By employing the direct method, we establish several theorems concerning the Hyers–Ulam stability of this inequality in the context of unital Banach ∗-algebras. Furthermore, we provide sufficient conditions under which these functional inequalities exhibit superstability. We also explore the implications of our results for linear ∗-derivations in semiprime Banach ∗-algebras with no nonzero central ideals.