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TMF, 1990, Volume 82, Number 2, Pages 163–177 (Mi tmf5404)  

This article is cited in 1 scientific paper (total in 1 paper)

$\mathrm{Op}^*$ and $\mathrm C^*$ dynamical systems I. Structural parallels

A. V. Voronin, S. S. Horuzhy


Abstract: The concept of an $\mathrm{Op}^*$ dynamical system is introduced and provides the basis of a systematic study of the problem of describing the vacuum structure of quantum field theory, formulated as a problem of the decomposition of operators and states for an algebra of unbounded operators ($\mathrm{Op}^*$ algebra) with a group of automorphisms. The following result makes it possible to develop a new solution of this problem, namely, it is found (Theorem 1) that for $\mathrm{Op}^*$ algebras Araki's theorem, which states that the commutant of a quasilocal $\mathrm C^*$ algebra with cyclic vacuum is Abelian, is true and can be very easily proved. Introducing the concept of an orthogonal measure on an $\mathrm{Op}^*$ algebra, and generalizing Tomita's theorem on orthogonal measures on $\mathrm C^*$ algebras, we obtain for $\mathrm{Op}^*$ algebras a connection between the spatial decomposition and the decomposition of states. The key Theorem 5 solves the decomposition problem for $\mathrm{Op}^*$ dynamical systems and completely reveals their structural similarity with the wellstudied $\mathrm C^*$ dynamical systems. The physical consequences of this solution are analyzed, and also the properties of Lorentz invariance of an $\mathrm{Op}^*$ system.

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English version:
Theoretical and Mathematical Physics, 1990, 82:2, 113–123

Bibliographic databases:

Document Type: Article
Received: 29.12.1988

Citation: A. V. Voronin, S. S. Horuzhy, “$\mathrm{Op}^*$ and $\mathrm C^*$ dynamical systems I. Structural parallels”, TMF, 82:2 (1990), 163–177; Theoret. and Math. Phys., 82:2 (1990), 113–123

Citation in format AMSBIB
\Bibitem{VorHor90}
\by A.~V.~Voronin, S.~S.~Horuzhy
\paper $\mathrm{Op}^*$ and $\mathrm C^*$ dynamical systems~I.
Structural parallels
\jour TMF
\yr 1990
\vol 82
\issue 2
\pages 163--177
\mathnet{http://mi.mathnet.ru/tmf5404}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1048112}
\zmath{https://zbmath.org/?q=an:0723.47038}
\transl
\jour Theoret. and Math. Phys.
\yr 1990
\vol 82
\issue 2
\pages 113--123
\crossref{https://doi.org/10.1007/BF01079038}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1990DZ44000001}


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    This publication is cited in the following articles:
    1. A. V. Voronin, S. S. Horuzhy, “$\mathrm{Op}^*$ and $\mathrm{C}^*$ dynamical systems. II. Structural differences: Borchers anomaly”, Theoret. and Math. Phys., 82:3 (1990), 225–230  mathnet  crossref  mathscinet  zmath  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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