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}.
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