Characterizations of additive local Jordan *-derivations by action at idempotents

Let H be a real or complex Hilbert space and B(H) the algebra of all bounded linear operators on H. Recall that a map delta:B(H)-> B(H) is called an inner Jordan & lowast;-derivation if there exists some T is an element of B(H) such that delta(A)=AT-TA & lowast; for all A is an element of B(H). In this paper, it is proved that inner Jordan & lowast;-derivations are the only additive maps delta of B(H) with the property that delta(P)=delta(P)P & lowast;+P delta(P) for all idempotent operators P is an element of B(H) if dim H=infinity, which is satisfied by additive local Jordan & lowast;-derivations. For the finite dimensional case, additional conditions are required for delta to be an inner Jordan & lowast;-derivation. As applications, it is shown that, for any given C,D is an element of B(H), delta satisfies delta(A)B-& lowast;+B delta(A)+delta(B)A(& lowast;)+A delta(B)=D for all A,B is an element of B(H) with AB + BA = C if and only if delta is an inner Jordan &+-derivation and D=delta(C). Also, several known results are generalized.