Local~*-derivations On Rings With Involution

Let R be a ring with an involution~*,which will be called a~*-ring.An additive map?:R?R is called a~*-derivation if ?(ab)= ?(a)b~*+ a?(b)holds for all a,b?R;is called a Jordan~*-derivation if ?(a2)=?(a)a~*+a?(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:(?)ARPi={0}(?)A = 0(i = 1,2),where P2 = I-P1;(?)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 ?:R?R is an additive map.Then ? satisfies ?(AB + BA)= ?(A)B~*+A?(B)+ ?(B)A~*+B?(A)for A,B ?R with AB=G if and only if ? 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 ?:B(H)B(H)is an additive map.If G = 0 or dim ker G>1,then ? satisfies ?(ST)= ?(S)T~*+ S?(T)for any S,T ? B(H)with ST=G if and only if?(S)=0 holds for all S ?B(H).