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Quantum measures and states on Jordan algebras

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Abstract

A problem of Mackey for von Neumann algebras has been settled by the conjunction of the early work of Gleason and the recent advances of Christensen and Yeadon. We show that Mackey's conjecture holds in much greater generality. LetA be a JBW-algebra and letL be the lattice of all projections inA. A quantum measure onL is a countably additive map,m, fromL into the real numbers. Our results imply thatm always has a unique extension to a bounded linear functional onA, provided thatA has no TypeI 2 direct summand.

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Authors and Affiliations

  1. Department of Mathematics, Reading University, Whiteknights Park, Reading, England

    L. J. Bunce & J. D. Maitland Wright

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  1. L. J. Bunce
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  2. J. D. Maitland Wright
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Communicated by H. Araki

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Bunce, L.J., Maitland Wright, J.D. Quantum measures and states on Jordan algebras. Commun.Math. Phys. 98, 187–202 (1985). https://doi.org/10.1007/BF01220507

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  • DOI: https://doi.org/10.1007/BF01220507

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