## Countably additive homomorphisms between von Neumann algebras

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- by L. J. Bunce and J. Hamhalter
- Proc. Amer. Math. Soc.
**123**(1995), 3437-3441 - DOI: https://doi.org/10.1090/S0002-9939-1995-1285978-7
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## 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.

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## Bibliographic Information

- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**123**(1995), 3437-3441 - MSC: Primary 46L50
- DOI: https://doi.org/10.1090/S0002-9939-1995-1285978-7
- MathSciNet review: 1285978