# On Unbounded Positive *-Representationson Fréchet-Domains

### Wolf-Dieter Heinrichs

Technische Universität Dresden, Germany

## Abstract

Let *D* be a Fréchet-domain from Op*-algebra, abbreviated F-domain. The present paper is concerned with the study of positive *-representations of *L*+(*D*), of the Calkin representation of *L*+(*D*) and of bounded sets in ultrapower *Du*. For this the density property plays an important role. It was introduced by S. Heinrich for locally convex spaces in [2].

In the paper [3] we gave several characterizations of the density property of an F-domain *D*. In this work we give a characterizations of continuity of positive *-representations and Calkin representation of *L*+(*D*) by the density property of *D*. This generalizes the well-known result due to K. Schmiidgen, see [12], Further we describe bounded subsets in ultrapower *Du*. If *D* has the density property, then every bounded set *M* c *Du* has a simple structure: For each bounded set *M c Du* there exists a bounded set *N* c *D* with *M* c' *Nu*. S. Heinrich proved an analogous result for bounded ultrapowers on locally convex spaces.

## Cite this article

Wolf-Dieter Heinrichs, On Unbounded Positive *-Representationson Fréchet-Domains. Publ. Res. Inst. Math. Sci. 30 (1994), no. 6, pp. 1123–1138

DOI 10.2977/PRIMS/1195164949