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

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.