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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 X (D, D+) . This space is studied as an ordered locally convex space with an involution and a partially defined multiplication. A characterizati on 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

The completion of the maximal {${\rm Op}\sp \ast$}-algebra on a Fréchet domain