ADDITIVITY OF JORDAN MULTIPLICATIVE MAPS ON JORDAN OPERATOR ALGEBRAS

Abstract

Let $H$ be a Hilbert space and $\mathcal{N}$ a nest in $H$. Denote by $S^a(H)$ the Jordan ring of all self-adjoint operators on $H$ and $\mathrm{Alg}\mathcal{N}$ the nest algebra associated to $\mathcal{N}$. We show that a bijective map $\Phi : S^a(H) \to S^a(H)$ satisfying (1) $\Phi(ABA) = \Phi(A) \Phi(B) \Phi(A)$ for every pair of $A,B$, or (2) $\Phi(AB + BA) = \Phi(A) \Phi(B) + \Phi(B) \Phi(A)$ for every pair of $A,B$, or (3) $\Phi(\frac{1}{2} (AB + BA)) = \frac{1}{2} (\Phi(A) \Phi(B) + \Phi(B) \Phi(A))$ for every pair of $A,B$ must be additive, that is, a Jordan ring isomorphism. We also show that if a bijective map $\Phi : \mathrm{Alg}\mathcal{N} \to \mathrm{Alg}\mathcal{N}$ satisfies the Jordan multiplicativity of the form (2) or (3), then $\Phi$ must be a Jordan isomorphism. Moreover, such Jordan multiplicative maps are characterized completely.