[Question previously asked on Math.SE]

Let $N$ be a von Neumann algebra, and $A$ be a dense $∗$-subalgebra of $N$ (in the ultraweak topology) with $A''=N$. Is it true that:

For any $x∈N^+$, there exists a increasing net $(x_j)$ in $A^+$ such that $x_j \to x$ in the ultraweak topology ?

The case of $A$ being an ideal of $N$ seems known (it is right?) but my question is about the general case.

(Precision: $A$ is not supposed to be normed closed.)