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Teoreticheskaya i Matematicheskaya Fizika, 1990, Volume 82, Number 2, Pages 163–177
(Mi tmf5404)
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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.
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
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https://www.mathnet.ru/eng/tmf5404 https://www.mathnet.ru/eng/tmf/v82/i2/p163
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Abstract page: | 347 | Full-text PDF : | 88 | References: | 28 | First page: | 1 |
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