# The Completion of the Maximal Op*-Algebra on a Frechet Domain

### Klaus-Detlef Kürsten

Universität Leipzig, Germany

## Abstract

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*).

## Cite this article

Klaus-Detlef Kürsten, The Completion of the Maximal Op*-Algebra on a Frechet Domain. Publ. Res. Inst. Math. Sci. 22 (1986), no. 1, pp. 151–175

DOI 10.2977/PRIMS/1195178378