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 $\pi :M \to N$ is said to be countably additive if $\pi (\sum \nolimits _1^\infty {{e_n}) = \sum \nolimits _1^\infty {\pi ({e_n})} }$, for every sequence $({e_n})$ of orthogonal projections in M. We prove that a $\ast$-homomorphism $\pi :M \to N$ is countably additive if and only if $\pi (e \vee f) = \pi (e) \vee \pi (f)$ for every pair of projections e and f of M. A corollary is that if, in addition, M has no Type ${{\text {I}}_2}$ direct summands, then every lattice morphism from the projections of M into the projections of N is a $\sigma$-lattice morphism.