SOME MULTIPLICATIVE PRESERVERS ON JORDAN OPERATOR ALGEBRAS

Let A be a unital real standard Jordan operator alge- bra acting on a Hilbert space H of dimension at least 2. We show that every bijection ϕ on A satisfying ϕ(A2 B) = ϕ(A)2 ϕ(B) is of the form ϕ = " where is an automorphism on A and " 2 f1; 1g. As a consequence if A is the real algebra of all self- adjoint operators on a Hilbert space H, then there exists a unitary or conjugate unitary operator U on H such that ϕ(A) = "UAU for all A 2 A