Spectral resolutions in effect algebras

Anna Jenčová; Sylvia Pulmannová

doi:10.22331/q-2022-11-03-849

Spectral resolutions in effect algebras

Anna Jenčová and Sylvia Pulmannová

Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia

Published:	2022-11-03, volume 6, page 849
Eprint:	arXiv:2111.02166v3
Doi:	https://doi.org/10.22331/q-2022-11-03-849
Citation:	Quantum 6, 849 (2022).

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Abstract

Effect algebras were introduced as an abstract algebraic model for Hilbert space effects representing quantum mechanical measurements. We study additional structures on an effect algebra $E$ that enable us to define spectrality and spectral resolutions for elements of $E$ akin to those of self-adjoint operators. These structures, called compression bases, are special families of maps on $E$, analogous to the set of compressions on operator algebras, order unit spaces or unital abelian groups. Elements of a compression base are in one-to-one correspondence with certain elements of $E$, called projections. An effect algebra is called spectral if it has a distinguished compression base with two special properties: the projection cover property (i.e., for every element $a$ in $E$ there is a smallest projection majorizing $a$), and the so-called b-comparability property, which is an analogue of general comparability in operator algebras or unital abelian groups. It is shown that in a spectral archimedean effect algebra $E$, every $a\in E$ admits a unique rational spectral resolution and its properties are studied. If in addition $E$ possesses a separating set of states, then every element $a\in E$ is determined by its spectral resolution. It is also proved that for some types of interval effect algebras (with RDP, archimedean divisible), spectrality of $E$ is equivalent to spectrality of its universal group and the corresponding rational spectral resolutions are the same. In particular, for convex archimedean effect algebras, spectral resolutions in $E$ are in agreement with spectral resolutions in the corresponding order unit space.

► BibTeX data

@article{Jencova2022spectralresolutions,
  doi = {10.22331/q-2022-11-03-849},
  url = {https://doi.org/10.22331/q-2022-11-03-849},
  title = {Spectral resolutions in effect algebras},
  author = {Jen{\v{c}}ov{\'{a}}, Anna and Pulmannov{\'{a}}, Sylvia},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {849},
  month = nov,
  year = {2022}
}

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Cited by

[1] Anna Jenčová and Sylvia Pulmannová, "Spectral order unit spaces and JB-algebras", Journal of Mathematical Analysis and Applications 520 2, 126911 (2023).

[2] Anna Jenčová and Sylvia Pulmannová, "Spectral resolutions in effect algebras", Quantum 6, 849 (2022).

[3] Anna Jenčová and Sylvia Pulmannová, "Spectral order unit spaces and JB-algebras", arXiv:2208.08740, (2022).

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