This paper investigates the completion of the maximal Op*-algebra L+ (D) of (possibly) unbounded operators on a dense domain D in a Hilbert space. It is assumed that D is a Frechet space with respect to the graph topology. Let D+ denote the strong dual of D, equipped with the complex conjugate linear structure. It is shown that the completion of L+ (D) (endowed with the uniform topology) is the space of continuous linear operators ??? (D, <7>D+) . This space is studied as an ordered locally convex space with an involution and a partially defined multiplication. A characterization of bounded subsets of D in terms of self-adjoint operators is given. The existence of special factorizations for several kinds of operators is proved. It is shown that the bounded operators are uniformly dense in L+ (D).
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Klaus-Detlef Kürsten, The Completion of the Maximal Op*-Algebra on a Frechet Domain. Publ. Res. Inst. Math. Sci. 22 (1986), no. 1, pp. 151–175DOI 10.2977/PRIMS/1195178378