On Realization of Partially Ordered Abelian Groups

J 2013

On Realization of Partially Ordered Abelian Groups

PASEKA, Jan, Ivan CHAJDA a Lei QUIANG

Základní údaje

Originální název

On Realization of Partially Ordered Abelian Groups

Autoři

PASEKA, Jan (203 Česká republika, garant, domácí), Ivan CHAJDA (203 Česká republika) a Lei QUIANG (156 Čína)

Vydání

International Journal of Theoretical Physics, Springer, 2013, 0020-7748

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10101 Pure mathematics

Stát vydavatele

Spojené státy

Utajení

není předmětem státního či obchodního tajemství

Odkazy

URL

Impakt faktor

Impact factor: 1.186

Kód RIV

RIV/00216224:14310/13:00070761

Organizační jednotka

Přírodovědecká fakulta

DOI

http://dx.doi.org/10.1007/s10773-012-1426-x

UT WoS

000318373700031

Klíčová slova anglicky

Non-classical logics; Orthomodular lattice; Effect algebras; Generalized effect algebras; States; Generalized states; Operators on Hilbert spaces

Štítky

AKR, rivok

Příznaky

Mezinárodní význam, Recenzováno

Změněno: 3. 1. 2014 17:58, prof. RNDr. Jan Paseka, CSc.

Anotace

V originále

The paper is devoted to algebraic structures connected with the logic of quantum mechanics. Since every (generalized) effect algebra with an order determining set of (generalized) states can be represented by means of an abelian partially ordered group and events in quantum mechanics can be described by positive operators in a suitable Hilbert space, we are focused in a representation of partially ordered abelian groups by means of sets of suitable linear operators. We show that there is a set of points separating R-maps on a given partially ordered abelian group G if and only if there is an injective non-trivial homomorphism of G to the symmetric operators on a dense set in a complex Hilbert space H which is equivalent to an existence of an injective non-trivial homomorphism of G into a certain power of R. A similar characterization is derived for an order determining set of R-maps and symmetric operators on a dense set in a complex Hilbert space H . We also characterize effect algebras with an order determining set of states as interval operator effect algebras in groups of self-adjoint bounded linear operators.

Návaznosti

EE2.3.20.0051, projekt VaV

Název: Algebraické metody v kvantové logice

Citovat

PASEKA, Jan, Ivan CHAJDA a Lei QUIANG. On Realization of Partially Ordered Abelian Groups. International Journal of Theoretical Physics. Springer, 2013, roč. 52, č. 6, s. 2028-2037. ISSN 0020-7748. Dostupné z: https://dx.doi.org/10.1007/s10773-012-1426-x.

@article{1138657,
   author = {Paseka, Jan and Chajda, Ivan and Quiang, Lei},
   article_number = {6},
   doi = {http://dx.doi.org/10.1007/s10773-012-1426-x},
   keywords = {Non-classical logics; Orthomodular lattice; Effect algebras; Generalized effect algebras; States; Generalized states; Operators on Hilbert spaces},
   language = {eng},
   issn = {0020-7748},
   journal = {International Journal of Theoretical Physics},
   title = {On Realization of Partially Ordered Abelian Groups},
   url = {http://link.springer.com/article/10.1007/s10773-012-1426-x},
   volume = {52},
   year = {2013}
}

TY  - JOUR
ID  - 1138657
AU  - Paseka, Jan - Chajda, Ivan - Quiang, Lei
PY  - 2013
TI  - On Realization of Partially Ordered Abelian Groups
JF  - International Journal of Theoretical Physics
VL  - 52
IS  - 6
SP  - 2028-2037
EP  - 2028-2037
PB  - Springer
SN  - 00207748
KW  - Non-classical logics
KW  - Orthomodular lattice
KW  - Effect algebras
KW  - Generalized effect algebras
KW  - States
KW  - Generalized states
KW  - Operators on Hilbert spaces
UR  - http://link.springer.com/article/10.1007/s10773-012-1426-x
L2  - http://link.springer.com/article/10.1007/s10773-012-1426-x
N2  - The paper is devoted to algebraic structures connected with the logic of quantum mechanics. Since every (generalized) effect algebra with an order determining set of (generalized) states can be represented by means of an abelian partially ordered group and events in quantum mechanics can be described by positive operators in a suitable Hilbert space, we are focused in a representation of partially ordered abelian groups by means of sets of suitable linear operators. We show that there is a set of points separating R-maps on a given partially ordered abelian group G if and only if there is an injective non-trivial homomorphism of G to the symmetric operators on a dense set in a complex Hilbert space H which is equivalent to an existence of an injective non-trivial homomorphism of G into a certain power of R. A similar characterization is derived for an order determining set of R-maps and symmetric operators on a dense set in a complex Hilbert space H . We also characterize effect algebras with an order determining set of states as interval operator effect algebras in groups of self-adjoint bounded linear operators.
ER  -

PASEKA, Jan, Ivan CHAJDA a Lei QUIANG. On Realization of Partially Ordered Abelian Groups. \textit{International Journal of Theoretical Physics}. Springer, 2013, roč.~52, č.~6, s.~2028-2037. ISSN~0020-7748. Dostupné z: https://dx.doi.org/10.1007/s10773-012-1426-x.

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