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 TMF, 1970, Volume 4, Number 2, Pages 171–195 (Mi tmf4142)

Vector states on algebras of observables and superselection rules I. Vector states and Hilbert space

V. N. Sushko, S. S. Horuzhy

Abstract: A detailed investigation is made of vector states on an arbitrary reducible $W^*$-algebra of observables $R$. The properties of vector states (purity, subordination, etc.) are reformulated and studied in terms of their “preimages”, i.e., the sets of vectors in the Hilbert space $\mathscr H$ corresponding to one and the same vector state. The properties of preimages of pure vector states are described exhaustively. A special class of quantum theories is studied for which $\mathscr H$ coincides with the closure $\mathscr H$ of the linear hull of the set of all vectors representing pure states. It is proved that a theory belongs to this class if and only if $R$ is a direct sum of type I factors. The structure of $R$ and $\mathscr H$ is analyzed exhaustively for this class of theories, i.e., different representations of $\mathscr H$ are given; the number of pure vector states and the number of subspaces that are irreducible under $R$ are determined. The connection between the results of the present paper and the formalism of the abstract algebraic approach is established.

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English version:
Theoretical and Mathematical Physics, 1970, 4:2, 758–774

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Citation: V. N. Sushko, S. S. Horuzhy, “Vector states on algebras of observables and superselection rules I. Vector states and Hilbert space”, TMF, 4:2 (1970), 171–195; Theoret. and Math. Phys., 4:2 (1970), 758–774

Citation in format AMSBIB
\Bibitem{SusHor70} \by V.~N.~Sushko, S.~S.~Horuzhy \paper Vector states on algebras of observables and superselection rules I.~Vector states and Hilbert space \jour TMF \yr 1970 \vol 4 \issue 2 \pages 171--195 \mathnet{http://mi.mathnet.ru/tmf4142} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=464976} \transl \jour Theoret. and Math. Phys. \yr 1970 \vol 4 \issue 2 \pages 758--774 \crossref{https://doi.org/10.1007/BF01066486} 

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This publication is cited in the following articles:
1. V. N. Sushko, S. S. Horuzhy, “Vector states on algebras of observables and superselection rules II. Algebraic theory of superselection rules”, Theoret. and Math. Phys., 4:3 (1970), 877–889
2. S. S. Horuzhy, “Fields and local observables in an axiomatic algebraic theory with superselection rules”, Theoret. and Math. Phys., 5:1 (1970), 942–952
3. V. N. Sushko, S. S. Horuzhy, “Properties of $H$ images of vector states on algebras of observables”, Theoret. and Math. Phys., 8:3 (1971), 862–864
4. A. V. Bulinski, “On the classes of $C^*$ algebras that satisfy the Haag–Kastler axioms”, Theoret. and Math. Phys., 8:3 (1971), 865–869
5. V. N. Sushko, S. S. Horuzhy, “Local and asymptotic structure of quantum systems with superselection rules”, Theoret. and Math. Phys., 13:3 (1972), 1147–1160
6. V. N. Sushko, S. S. Horuzhy, “$H$-images of vector states and causal properties of local algebras”, Theoret. and Math. Phys., 15:2 (1973), 460–466
7. S. G. Kharatyan, “Von neumann algebras of observables with non-Abelian commutator algebra and superselection rules”, Theoret. and Math. Phys., 14:3 (1973), 227–232
8. V. M. Maksimov, “Macroscopic observables in algebraic statistical physics”, Theoret. and Math. Phys., 20:1 (1974), 632–638
9. Yu. M. Zinoviev, V. N. Sushko, “Physical symmetries in a theory of local observables of the $P$-class”, Theoret. and Math. Phys., 18:1 (1974), 9–18
10. S. S. Horuzhy, “Superposition principle in Algebraic quantum theory”, Theoret. and Math. Phys., 23:2 (1975), 413–421
11. K. Yu. Dadashyan, “Causality properties in nonrelativistic algebraic models”, Theoret. and Math. Phys., 31:3 (1977), 492–496
12. K. Yu. Dadashyan, S. S. Horuzhy, “Algebras of observables of the free Dirac field”, Theoret. and Math. Phys., 36:2 (1978), 665–675
13. A. V. Voronin, V. N. Sushko, S. S. Horuzhy, “Algebras of unbounded operators and vacuum superselection rules in quantum field theory. I. Some properties of Op*-algebras and vector states on them”, Theoret. and Math. Phys., 59:1 (1984), 335–350
14. A. V. Voronin, “Discrete vacuum superselection rule in Wightman theory with essentially self-adjoint field operators”, Theoret. and Math. Phys., 66:1 (1986), 8–19
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