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Vector states on algebras of observables and superselection rules I. Vector states and Hilbert space
Teoretičeskaâ i matematičeskaâ fizika, Tome 4 (1970) no. 2, pp. 171-195 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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     title = {Vector states on algebras of observables and superselection rules {I.~Vector} states and {Hilbert} space},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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V. N. Sushko; S. S. Horuzhy. Vector states on algebras of observables and superselection rules I. Vector states and Hilbert space. Teoretičeskaâ i matematičeskaâ fizika, Tome 4 (1970) no. 2, pp. 171-195. http://geodesic.mathdoc.fr/item/TMF_1970_4_2_a2/

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