Countably additive homomorphisms between von Neumann algebras

Abstract

Let M and N be von Neumann algebras where M has no abelian direct summand. A ∗ \ast -homomorphism π : M → N \pi :M \to N is said to be countably additive if π ( ∑ 1 ∞ e n ) = ∑ 1 ∞ π ( e n ) \pi (\sum \nolimits _1^\infty {{e_n}) = \sum \nolimits _1^\infty {\pi ({e_n})} } , for every sequence ( e n ) ({e_n}) of orthogonal projections in M. We prove that a ∗ \ast -homomorphism π : M → N \pi :M \to N is countably additive if and only if π ( e ∨ f ) = π ( e ) ∨ π ( f ) \pi (e \vee f) = \pi (e) \vee \pi (f) for every pair of projections e and f of M. A corollary is that if, in addition, M has no Type I 2 {{\text {I}}_2} direct summands, then every lattice morphism from the projections of M into the projections of N is a σ \sigma -lattice morphism.