Stability of Functional Inequalities Associated with \( S(A) \)-Extended Jordan \( * \)-Derivations in \( JB^* \)-Algebras

Let A be a unital JB*-algebra with unit 1, Jordan product ∘ and involution \( * \). Since the commutator formed from the Jordan product vanishes, we measure deviations by the Jordan associator {x, w} := (x ∘ w) ∘ w − x ∘ (w ∘ w). Let S(A) consist of those z ∈ A for which {z, a} = 0 for every a ∈ A. A mapping δ : A→ A is called an S(A)-extended Jordan \( * \)-derivation if additivity, the Jordan derivation identity and the involution condition hold modulo S(A). We assume that A ≠ C1, S(A) = C1, and the associator controls the distance to S(A); these conditions hold for Mn(C), n ≥ 2. Under dyadic summability and decay assumptions on the errors in a three-variable functional inequality and in the product and involution conditions, we obtain a unique S(A)-extended Jordan \( * \)-derivation L(x) = limn→∞2nδ(x/2n); an analogous result holds for limn→∞2−nδ(2nx). These limits exist in norm, whereas the quantitative approximation estimates control the associator error and the distance of L(x) − δ(x) to C1. The limit is additive and exactly \( * \)-preserving. An additional one-variable decay condition yields δ = L (superstability), and a unimodular version of the functional inequality yields C-linearity. Matrix examples demonstrate a nonzero limit and the distinction between associator control and full norm control.