Let A be an arbitrary ∗-algebra with unit I over the real or complex field F that contains a nontrivial idempotent P1 and n ≥ 1 a natural number and φ : A −→ A be a surjective map on A such that φ satisfies condition φ(P) •n−1 φ(P) • φ(A) = P •n−1 P • A, for every A ∈ A and projection P ∈ {P1, I − P1}, where A •n−1 A with repeat n − 1 times A is the Jordan multiple ∗-product. Then φ(A) = φ(I)A for all A ∈ A and φ(I)2 = I.
