Local~*-derivations On Rings With Involution

Let R be a ring with an involution~*,which will be called a~*-ring.An additive mapδ:R→R is called a~*-derivation if δ(ab)= δ(a)b~*+ aδ(b)holds for all a,b∈R;is called a Jordan~*-derivation if δ(a2)=δ(a)a~*+aδ(a)holds for all a∈R.Since the question of whether each quasi-quadratic functional is generated by some sesquilinear functional is intimately connected with the structure of Jordan~*-derivations,the study of Jordan~*-derivations becomes more active in operator algebra and operator theory.In this paper,we mainly discuss the relationship beteween additive local Jordan~*-derivations and~*-derivations on prime~*-rings.The following are the main results.1.Let R be a 2-torsion free unital~*-ring with a nontrivial symmetric idempotent P1,and satisfies the following conditions:(ⅰ)ARPi={0}(?)A = 0(i = 1,2),where P2 = I-P1;(ⅱ)for any A ∈R,there exists some integer n such that nP1-P1AP1 is invertible in P1TRP1.Assume that G ∈ R is any fixed point with G = P1GP1 and δ:R→R is an additive map.Then δ satisfies δ(AB + BA)= δ(A)B~*+Aδ(B)+ δ(B)A~*+Bδ(A)for A,B ∈R with AB=G if and only if δ is a Jordan~*-derivation.2.Let H be a Hilbert space over the real or complex field IF with dim H>1.Assume that G E B(H)and δ:B(H)B(H)is an additive map.If G = 0 or dim ker G>1,then δ satisfies δ(ST)= δ(S)T~*+ Sδ(T)for any S,T ∈ B(H)with ST=G if and only ifδ(S)=0 holds for all S ∈B(H).