Additivity of Jordan maps on Jordan algebras

Let AA and BB be two Jordan algebras. In this paper, we investigate the additivity of maps ϕϕ from AA onto BB that are bijective and satisfyϕ(a∘b)=ϕ(a)∘ϕ(b),ϕ(a∘b)=ϕ(a)∘ϕ(b),for all a,b∈Aa,b∈A. If AA contains an idempotent which satisfies some conditions, then ϕϕ is additive. This result generalizes all results about additivity of Jordan maps in [1,3–9,11].