Let 𝓐 be a ⋆-algebra, δ : 𝓐 → 𝓐 be a linear map, and z ∈ 𝓐 be fixed. We consider the condition that δ satisfies xδ(y)⋆ + δ(x)y⋆ = δ(z) (x⋆δ(y) + δ(x)⋆y = δ(z)) whenever xy⋆ = z (x⋆y = z), and under several conditions on 𝓐, δ and z we characterize the structure of δ. In particular, we prove that if 𝓐 is a Banach ⋆-algebra, δ is a continuous linear map, and z is a left (right) separating point of 𝓐, then δ is a Jordan derivation. Our proof is based on complex variable techniques. Also, we describe a linear map δ satisfying the above conditions with z = 0 on two classes of ⋆-algebras: zero product determined algebras and standard operator algebras.

Let A be a Banach algebra with unity 1 and M be a unital Banach left A-module. let δ : A → M be a continuous linear map with the property that a, b ∈ A, ab, where z ∈ A. In this article, first we characterize δ for z = 1. Then we consider the case A = M = AlgL, where AlgL is areflexive algebra on a Hilbert space H and z = P is a non-triavial idempotent in A with P (H) ∈ L and describe δ. Finally we apply the main results to CSL-algebras, irreducible CDC algebras and nest algebras on a Hilbert space H.

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