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On Some Properties of PBZ*-Lattices

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Abstract

We continue the algebraic investigation of PBZ*-lattices, a notion introduced in Giuntini et al. (Stud. Logica 104, 1145–1177, 2016) in order to obtain insights into the structure of certain algebras of effects of a Hilbert space, lattice-ordered under the spectral ordering.

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Notes

  1. That the spectral ordering is indeed a lattice ordering has been essentially shown by Olson [18] and de Groote [14], who also proved that it coincides with the more familiar ordering of effects induced via the trace functional when both orderings are restricted to the set of projection operators of the same Hilbert space. The same ordering has also been given an algebraic treatment, in a different context, in [10].

  2. We acknowledge here G. Cattaneo’s insightful comments on [12], which we received after the paper itself had gone into its production stage.

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Acknowledgements

R. Giuntini gratefully acknowledges the support of the Regione Autonoma Sardegna within the project “Modeling the uncertainty: quantum theory and imaging processing”, LR 7/8/2007, RAS CRP-59872. A. Ledda gratefully acknowledges the support of the Regione Autonoma della Sardegna within the project ‘Order properties in mathematics and physics’, LR 7/8/2007, CUP F72F16002920002. All authors gratefully acknowledge the support of the Horizon 2020 program of the European Commission: SYSMICS project, Proposal Number: 689176, MSCA-RISE-2015, and the Fondazione Banco di Sardegna within the project ‘Science and its logics, the representation’s dilemma’, CUP F72F16003220002, and the suggestions of two referees.

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  1. University of Cagliari, Cagliari, Italy

    Roberto Giuntini, Antonio Ledda & Francesco Paoli

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  1. Roberto Giuntini
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  2. Antonio Ledda
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  3. Francesco Paoli
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Correspondence to Francesco Paoli.

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Giuntini, R., Ledda, A. & Paoli, F. On Some Properties of PBZ*-Lattices. Int J Theor Phys 56, 3895–3911 (2017). https://doi.org/10.1007/s10773-017-3374-y

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  • DOI: https://doi.org/10.1007/s10773-017-3374-y

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