Let A be a factor von Neumann algebra, with unit I, which contains a nontrivial projection P1, and let : A ! A be a surjective map that satis es one of the two conditions: (A) (P) + (P) (A) = AP + PA and (A) (P) + (P) (A) = AP + PA for all A 2 A and P 2 fP1; I P1g and 2 f1; 1g. Then, we determine the concrete form of .
