Closure of Cones in Completed Injective Tensor Products

Suppose that E is a complex locally convex space equipped with a continuous involution, ∗ . We consider the cone of finite sums of positive elements( E ⊗ E ) + = {∑ i = 1 n f i ∗ ⊗ f 1 : f 1 ∈ E , 1 ⩽ i ⩽ n , n ∈ ℕ }in the completed injective tensor product E⊗ ˜E . Then if E = E 1 ⊕ E 2 where E 1 is a Fréchet space and E 2 is a complete space whose strong dual is barrelled and such that every continuous linear map from the strong dual into any Banach space has separable range, then the closure of ( E ⊗ E ) + in E⊗ ˜E is exactly the set of convergent infinite sums of positive elements.