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 ( x y ) − δ ( x ) y ∗ − x δ ( y ) ∈ Z ( R ) for all x , y ∈ R , where Z ( R ) denotes the center of the ring. We consider the functional inequality ∥ [ a 1 δ ( x 1 ) + a 2 δ ( x 2 ) + a 3 δ ( x 3 ) , w ] ∥ ≤ ∥ [ δ ( a 1 x 1 + a 2 x 2 + a 3 x 3 ) , w ] ∥ + Φ ( x 1 , x 2 , x 3 , 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.