Algebras of unbounded operators and vacuum superselection rules in quantum field theory. I. Some properties of Op*-algebras and vector states on them

In connection with the physical problem of describing vacuum superselection rules in quantum field theory, a study is made of some properties of Op* algebras, namely, the structure of their commutants and invariant and reducing subspaces and vector states on such algebras. For this, a formalism is developed that uses intertwining operators of Hermitian representations of a * algebra. The formalism is used to obtain a number of new properties of the commutants of Op* algebras, and a description is given of classes of subspaces the projection operators onto which lie in the strong or weak commutant. A study is made of the correspondence between vector states on the Op* algebra $\mathscr P$ and on its associated yon Neumann algebra $R=({\mathscr P_w}^{'})^{'}$; generalizations are found of the class of self-adjoint Op* algebras for which a detailed investigation of vector states can be made. Classes of weakly regular, strongly regular, and completely regular vectors for which the properties of states on $\mathscr P$ approach closer and closer to states on $R$ are identified and studied.