Comparative probability on von Neumann algebras |
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Authors: | Simba A Mutangadura |
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Institution: | Department of Physics, University of Zimbabwe, Box MP 167, Mt. Pleasant, Harare, Zimbabwe - International Centre for Theoretical Physics, Trieste, Italy |
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Abstract: | We continue here the study begun in earlier papers on implementation of comparative probability by states. Let be a von Neumann algebra on a Hilbert space and let denote the projections of . A comparative probability (CP) on (or more correctly on is a preorder on satisfying: -
- with for some .
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- If , then either or .
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- If , and are all in and , , then .
A state on is said to implement a on if for , . In this paper, we examine the conditions for implementability of a CP on a general von Neumann algebra (as opposed to only type I factors). A crucial tool used here, as well as in earlier results, is the interval topology generated on by . A will be termed continuous in a given topology on if the interval topology generated by is weaker than the topology induced on by the given topology. We show that uniform continuity of a comparative probability is necessary and sufficient if the von Neumann algebra has no finite direct summand. For implementation by normal states, weak continuity is sufficient and necessary if the von Neumann algebra has no finite direct summand of type I. We arrive at these results by constructing an appropriate additive measure from the CP. |
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