Let $I$ be a both sided $*$- closed ideal of a von Neumann algebra $N$.
Problem: If for each non-zero positive element $n\in N$ there exists a non-zero positive element $\iota\in I$ such that $0\leq \iota\leq n$, then show that $I$ is $s^*$- dense in $N$.
Note that $s^*$- topology is generated by the seminorms $s_{\varphi}(n):=\varphi (n^*n)^{1/2},\,n\in N$ and $s_{\varphi}^*(n):=\varphi(nn^*)^{1/2},\,n\in N$ where $\varphi$ runs over the set of all normal forms on $N$.
I got stuck in this while reading the section on 'The Spatial Derivative' from the book 'Modular Theory in Operator Algebras' by Stratila. The author says that it follows by using the notion of approximate identity in $I$, but I don't get how. Thanks in advance for any help.