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Countably additive homomorphisms between von Neumann algebras

Authors: L. J. Bunce and J. Hamhalter
Journal: Proc. Amer. Math. Soc. 123 (1995), 3437-3441
MSC: Primary 46L50
MathSciNet review: 1285978
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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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Article copyright: © Copyright 1995 American Mathematical Society

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