A −→A be a surjective map between some operator algebras such that ψ(A) ◦2 ψ(B) = A◦2 B for all A,B ∈ A, where A◦2 B defined by A◦2 B = (A◦B)◦B and A◦B is Jordan product, i.e. A◦B = AB +BA. In this paper, we determine the concrete form of map ψ on some operator algebras. Such operator algebras include standard operator algebras, properly infinite von Neumann algebras and nest algebras. Particularly, if A is a factor von Neumann algebra that satisfies ψ(A) ◦2 ψ(P) = A ◦2 P for all A ∈ A and idempotents P ∈ A, then there exists nonzero scalar λ with λ3 = 1 such that ψ(A) = λA for all A ∈ A.
