Additivity of Jordan elementary maps on nest algebras

Let A be a standard subalgebra of a nest algebra on a Hilbert space of dimension greater than one and B an arbitrary algebra. A Jordan elementary map of A×B is a pair (M, M*) where M:A→B and M∗:B→A are two maps satisfyingM(AM∗(B)A)=M(A)BM(A),M∗(BM(A)B)=M∗(B)AM∗(B)forA∈A, B∈B. In this note, it is proved that for a special class of surjective Jordan elementary maps of A×B, every member in it is automatically additive. Also, we construct a counterexample which shows that this result is not necessarily true for all surjective Jordan elementary maps.