© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Mathematica Slovaca

Spectral automorphisms in CB-effect algebras

Dan Caragheorgheopol

Published Dec 1, 2012• 0 citations

Abstract

Abstract Spectral automorphisms have been introduced in [IVANOV, A.—CARAGHEORGHEOPOL, D.: Spectral automorphisms in quantum logics, Internat. J. Theoret. Phys. 49 (2010), 3146–3152]_in an attempt to construct, in the abstract framework of orthomodular lattices, an analogue of the spectral theory in Hilbert spaces. We generalize spectral automorphisms to the framework of effect algebras with compression bases and study their properties. Characterizations of spectral automorphisms as well as necessary conditions for an automorphism to be spectral are given. An example of a spectral automorphism on the standard effect algebra of a finite-dimensional Hilbert space is discussed and the consequences of spectrality of an automorphism for the unitary Hilbert space operator that generates it are shown. The last section is devoted to spectral families of automorphisms and their properties, culminating with the formulation and proof of a Stone type theorem (in the sense of Stone’s theorem on strongly continuous one-parameter unitary groups — see, e.g. [REED, M.#x2014;SIMON, B.: Methods of Modern Mathematical Physics, Vol. I, Acad. Press, New York, 1975]) for a group of spectral automorphisms.

DOI: 10.2478/s12175-012-0072-3

Related papers

Spectral resolutions in effect algebras

Anna Jenčová, Sylvia Pulmannová

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∈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∈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.

arXiv:2111.02166•Quantum•Nov 3, 2022•4 citations

Characterizations of Spectral Automorphisms and a Stone-Type Theorem in Orthomodular Lattices

Dan Caragheorgheopol, Josef Tkadlec

International Journal of Theoretical Physics•Mar 15, 2011•2 citations

Automorphisms of Hilbert space effect algebras

Peter Šemrl

Let H be a Hilbert space and E ( H ) the effect algebra on H . A bijective map is called an ortho-order automorphism of E ( H ) if for every we have and . The classical theorem of Ludwig states that every such ϕ is of the form , , for some unitary or antiunitary operator U . It is also known that each bijective map on E ( H ) preserving order and coexistency in both directions is of the same form. Can we improve these two theorems by relaxing the bijectivity assumption and/or replacing the above preserving properties by the weaker assumptions of preserving above relations in one direction only and still get the same conclusion? For both characterizations of automorphisms of effect algebras we will prove the optimal versions and give counterexamples showing the optimality of the obtained results.

Journal of Physics A Mathematical and Theoretical•Apr 17, 2015•7 citations

Continuous coexistency preservers on effect algebras *

Michiya Mori, Peter Šemrl

Abstract Let H be a finite-dimensional Hilbert space, dim H ⩾ 2. We prove that every continuous coexistency preserving map on the effect algebra E ( H ) is either a standard automorphism of E ( H ), or a standard automorphism of E ( H ) composed with the orthocomplementation. We present examples showing the optimality of the result.

Journal of Physics A Mathematical and Theoretical•Nov 18, 2020•1 citation

Symmetries of Hilbert space effect algebras

Peter Šemrl

Let H be a Hilbert space and E(H) the effect algebra on H, that is, E(H) is the set of all self-adjoint operators A: H→H satisfying 0⩽A⩽I. The effect algebra can be equipped with several operations and relations that are important in mathematical foundations of quantum mechanics. Automorphisms with respect to these operations or relations are called symmetries. We present a new method that can be used to describe the general form of such maps. The main idea is to reduce this kind of problem to the study of adjacency-preserving maps. The efficiency of this approach is illustrated by reproving some known results as well as by obtaining some new theorems.

Journal of the London Mathematical Society•Jul 1, 2013•13 citations

⊥-order automorphisms of Hilbert space effect algebras: The two-dimensional case

Lajos Molnár, Zsolt Páles

It is well known that the ⊥-order automorphisms of the effect algebra of a Hilbert space of dimension not less than three are implemented by unitary or antiunitary operators. The aim of this paper is to show that the same assertion also holds true in the two-dimensional case.

Journal of Mathematical Physics•Apr 1, 2001•19 citations

Abstract Spectral automorphisms have been introduced in [IVANOV, A.—CARAGHEORGHEOPOL, D.: Spectral automorphisms in quantum logics, Internat. J. Theoret. Phys. 49 (2010), 3146–3152]_in an attempt to construct, in the abstract framework of orthomodular lattices, an analogue of the spectral theory in Hilbert spaces. We generalize spectral automorphisms to the framework of effect algebras with compression bases and study their properties. Characterizations of spectral automorphisms as well as necessary conditions for an automorphism to be spectral are given. An example of a spectral automorphism on the standard effect algebra of a finite-dimensional Hilbert space is discussed and the consequences of spectrality of an automorphism for the unitary Hilbert space operator that generates it are shown. The last section is devoted to spectral families of automorphisms and their properties, culminating with the formulation and proof of a Stone type theorem (in the sense of Stone’s theorem on strongly continuous one-parameter unitary groups — see, e.g. [REED, M.#x2014;SIMON, B.: Methods of Modern Mathematical Physics, Vol. I, Acad. Press, New York, 1975]) for a group of spectral automorphisms.

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∈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∈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.

Let H be a Hilbert space and E ( H ) the effect algebra on H . A bijective map is called an ortho-order automorphism of E ( H ) if for every we have and . The classical theorem of Ludwig states that every such ϕ is of the form , , for some unitary or antiunitary operator U . It is also known that each bijective map on E ( H ) preserving order and coexistency in both directions is of the same form. Can we improve these two theorems by relaxing the bijectivity assumption and/or replacing the above preserving properties by the weaker assumptions of preserving above relations in one direction only and still get the same conclusion? For both characterizations of automorphisms of effect algebras we will prove the optimal versions and give counterexamples showing the optimality of the obtained results.

Let H be a Hilbert space and E(H) the effect algebra on H, that is, E(H) is the set of all self-adjoint operators A: H→H satisfying 0⩽A⩽I. The effect algebra can be equipped with several operations and relations that are important in mathematical foundations of quantum mechanics. Automorphisms with respect to these operations or relations are called symmetries. We present a new method that can be used to describe the general form of such maps. The main idea is to reduce this kind of problem to the study of adjacency-preserving maps. The efficiency of this approach is illustrated by reproving some known results as well as by obtaining some new theorems.