On a Certain Class of *-Algebras of Unbounded Operators

A *-algebra SI of linear operators with a common invariant dense domain & in a Hilbert space is studied relative to the order structure given by the cone Sl+ of positive elements of SI (in the sense of positive sesquilinear form on &) and the p-topology defined as an inductive limit of the order norm pA (of the subspace SIX with A as its order unit) with A eSl+. In particular, for those SI with a countable cofinal sequence A, in ST+ such that AT1 eSI, the ,0-topology is proved to be order convex, any positive elements in the predual is shown to be a countable sum of vector states, and the bicommutant within the set B(&, &) of continuous sesquilinear forms on 3f is shown to be the ultraweak closure of SI. The structure of the commutant and the bicommutant are explicitly given in terms of their bounded operator elements which are von Neumann algebras and the commutant of each other.