抄録
The completion with respect to the uniform topology of the maximal Op*-algebra L+(D) on a Fréchet domain D is denoted by \mathscr{L}. It is isomorphic to the second strong dual of the complete injective tensor product D'\bar{⊗}ε\bar{D'} of the strong duals of D and \bar{D}, where D is endowed with the topology generated by the graph norms of operators belonging to L+(D) and \bar{D} denotes the complex conjugate space of D. The predual of \mathscr{L}, i. e., the dual of D'\bar{⊗}ε\bar{D'} is isomorphic to the space \mathscr{N}(\bar{D'}, D) of nuclear operators mapping \bar{D'} into D. These facts, together with the fact that the positive cone of \mathscr{L} is normal with respect to the order topology, are applied to the study of bounded, positive, and continuous linear functionals on \mathscr{L}. It is also shown that D'\bar{⊗}ε\bar{D'} is a barrelled DF-space, that L+(D) is a DF-space, and that the subspace \mathscr{F}⊂L+(D) of finite rank operators is a bornological DF-space. There are given several characterizations of the Montel property of the Fréchet domain D. One of them is the reflexivity of \mathscr{L}.
