Spectral resolutions in effect algebras

• URL: http://arxiv.org/abs/2111.02166v3
• Date: Wed, 26 Oct 2022 09:26:24 GMT
• Title: Spectral resolutions in effect algebras
• Authors: Anna Jen\v{c}ov\'a and Sylvia Pulmannov\'a
• Abstract summary: Effect algebras were introduced as an abstract 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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