Multiplicative Isomorphisms Between Von Neumann Algebras

Let M,N be von Neumann algebras without central summands of type I1. For a scalar ξ, a map Φ: M→N is called a ξ-Lie multiplicative isomorphism if Φ is bijective and Φ(AB-ξBA)=Φ(A)Φ(B)-ξΦ(B)Φ(A) for all A,B∈M It is shown that a map Φ:M→N is a ξ-Lie multiplicative isomorphism (where ξ∈C and ξ≠1) if and only if one of the following statements holds (1)ξ=0, Φ is a ring isomorphism;(2) ξ=-1,there exist central projections P∈M, Q∈N such that Φ=Φ1(?)Φ2, where Φ1=Φ|PM:PM→QN is a ring isomorphism, Φ2=Φ|(IM-P)M:(IM-P)M→(IN-Q)N is a ring anti-isomorphism;(3) ξ≠0,-1, there exist central projections P∈M,Q∈N such that Φ=Φ1(?)Φ2, where Φ1: PM→QN is a ring isomorphism with Φ1(ξA1)=ξΦ1(A1) for all A1∈PM;-ξΦ2:(IM-P)M→(IN-Q)N is a ring anti-isomorphism with Φ2(ξA2)=1/ξΦ2(A2) for all A2∈(IM-P)M. Here IM,IN denote the units of M and N. Besides, the Jordan semi-triple multiplicative isomorphism and the skew Jordan semi-triple multiplicative isomorphism are alsocharacterized.