روجا حسین زاده

In this Thesis, let A be a -algebra over the complex field C. For A;B 2 A; we define, the new products of A and B by A  B = AB + BA; A ◦ B = AB 􀀀 BA: Then we establish the structure of the nonlinear maps preserving the mixed product (A  B ◦ C) on von Neumann algebras. Also, we assume that be a unital -algebra with the unit I and A contains a nontrivial projection P which satisfies XAP = 0 implies X = 0 and XA(I 􀀀 P) = 0 implies X = 0 Then a map  : A 􀀀! A satisfies (A  B ◦ C) = (A)  B ◦ C + A  (B) ◦ C + A  B ◦ (C) for all A;B;C 2 A if and only if  is an additive -derivation. Then we prove that if A be a factor von Neumann algebra with dimA  2; then a map  : A 􀀀! A satisfies (A  B ◦ C) = (A)  B ◦ C + A  (B) ◦ C + A  B ◦ (C) for all A;B;C 2 A if and only if  is an additive -derivation.