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Perfect Effect Algebras and Spectral Resolutions of Observables

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We study perfect effect algebras, that is, effect algebras with the Riesz decomposition property where every element belongs either to its radical or to its co-radical. We define perfect effect algebras with principal radical and we show that the category of such effect algebras is categorically equivalent to the category of unital po-groups with interpolation. We introduce an observable on a \(\hbox {Rad}\)-monotone \(\sigma \)-complete perfect effect algebra with principal radical and we show that observables are in a one-to-one correspondence with spectral resolutions of observables.

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Acknowledgements

The authors are very indebted to anonymous referees for their careful reading and suggestions which helped us to improve the presentation of the paper.

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Anatolij Dvurečenskij

  2. Department of Algebra and Geometry, Palacký University, 17. listopadu 12, 771 46, Olomouc, Czech Republic

    Anatolij Dvurečenskij

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  1. Anatolij Dvurečenskij
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Correspondence to Anatolij Dvurečenskij.

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Dedicated to the memory of Paul Busch, an outstanding scholar, pianist and a nice man.

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The paper has been supported by the Grant of the Slovak Research and Development Agency under Contract APVV-16-0073 and the Grant VEGA No. 2/0069/16 SAV.

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Dvurečenskij, A. Perfect Effect Algebras and Spectral Resolutions of Observables. Found Phys 49, 607–628 (2019). https://doi.org/10.1007/s10701-019-00238-2

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