Bresar, MatejMiers, C. Robert2010-01-072010-01-0719912010-01-07http://hdl.handle.net/1828/2036A map \theta: M->N where M and N are rings is said to preserve commutativity in both directions if the elements a,b \in M commute if and only if \theta(a) and \theta(b) commute. In this paper we show that if M and N are von Neumann algebras with no central summands of type I_1 or I_2 and \theta is a bijective map which preserves commutativity in both directions then \theta(x)=c\phi(x)+f(x) where c is an invertible element in Z_N, the center of N, \phi:M->N is a Jordan isomorphism of M onto N, and f is an additive map of M into Z_N.entechnical reports (mathematics and statistics)Technical ReportDepartment of Mathematics and Statistics