On a Certain Class of *-Algebras of Unbounded Operators

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A *-algebra \mathfrak{A} of linear operators with a common invariant dense domain \mathscr{D} in a Hilbert space is studied relative to the order structure given by the cone \mathfrak{A}+ of positive elements of \mathfrak{A} (in the sense of positive sesquilinear form on \mathscr{D}) and the ρ-topology defined as an inductive limit of the order norm ρA (of the subspace \mathfrak{A}A with A as its order unit) with A∈\mathfrak{A}+. In particular, for those \mathfrak{A} with a countable cofinal sequence Ai in \mathfrak{A}+ such that Ai−1∈\mathfrak{A}, the ρ-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(\mathscr{D}, \mathscr{D}) of continuous sesquilinear forms on \mathscr{D} is shown to be the ultraweak closure of \mathfrak{A}. 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.

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