Von neumann algebras of observables with non-Abelian commutator algebra and superselection rules

A study is made of representations of the algebra of observables in $\mathscr H$ which are such that every vector functional can be weakly approximated by finite e linear combinations of pure states. It is proved that this assumption is equivalent to $\mathscr H$ being the closure of the linear hull of the set of vectors that represent pure states. A general definition is introduced for superselectton rules and it is shown that the set of superseleetion operators eoineides with the set of selfadjoint operators adjoined to the center of the yon Neurnann algebra of observables. A number of properties of coherent subspaees is established.