Additivity of Jordan maps on standard Jordan operator algebras

Abstract

Let A be a standard Jordan operator algebra on a Hilbert space of dimension > 1 and B be an arbitrary Jordan algebra. In this note, we prove that if a bijection ϕ : A → B satisfies ϕ ( a ∘ b ) = ϕ ( a ) ∘ ϕ ( b ) for all a , b ∈ A , where ∘ is the special Jordan product on A, then ϕ is additive.