On Unbounded Positive *-Representationson Fréchet-Domains

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.