Lie isomorphisms of factors

Introduction.By a von Neumann algebra M we mean a weakly closed, selfadjoint algebra of operators (containing the identity /) on a complex Hubert space //.In this paper we consider a mapping $ between von Neumann algebras M and N which is one-one, onto, «-linear, and which preserves Lie brackets of operators (that is <f>[A, B] = [<j>iA), <¿(P)] where [X, Y] = XY-YX).Such mappings will be called Lie «-isomorphisms.We prove, as our main result, that if Af is a factor ( = center trivial von Neumann algebra), then <f>=S+\ where 9 is either a ♦-isomorphism, or the negative of a *-anti-isomorphism and A is a *-linear map from M into the center of N which annihilates brackets of operators in M.This was proved by L. Hua [6] in the case that M i = N) is a factor of type /n (n>2).Subsequently Hua's result was generalized in an algebraic sense by W. S. Martindale [8], [9], to the case where M and N are simple rings with M containing two nonzero idempotents whose sum is the identity.The algebraic techniques of these papers, however, are not sufficient in our setting since von Neumann factors are not, in general, simple.In all that follows <f>: M -> N is a Lie «-isomorphism between the von Neumann algebras M and N.2. Preliminary results.Lemma 1.Let X, YeM.Then XY= YX iff <j>iX)<j>iY) = <j>iY)j>iX).ProofAs an easy consequence of this definition we have Theorem 1.If M0 is a normal von Neumann subalgebra of M, then <l>iM0) is a normal von Neumann subalgebra of N having the same linear dimension. Corollary.If M is a factor, then so is N.