Duality for Maximal Op*-Algebras on Fréchet Domains

Abstract

The completion with respect to the uniform topology of the maximal Op*-algebra L+(D) on a Frechet domain D is denoted by &. It is isomorphic to the second strong dual of the complete injective tensor product D'®tD' of the strong duals of D and D, where D is endowed with the topology generated by the graph norms of operators belonging to L+(D) and D denotes the complex conjugate space of D. The predual of J§?, i. e., the dual of D'<§)aDf is isomorphic to the space ^(D'} D) of nuclear operators mapping D' into D. These facts, together with the fact that the positive cone of 3? is normal with respect to the order topology, are applied to the study of bounded, positive, and continuous linear functionals on 3?, It is also shown that D'®fDr is a barrelled DF-space, that L+(D) is a DF-space, and that the subspace «^cL+(D) of finite rank operators is a bornological DFspace. There are given several characterizations of the Montel property of the Frechet domain D- One of them is the reflexivity of «£?.