Find a projection in a type III$_1$ factor

Question

Let $M$ be a type III$_1$ factor and $\omega$ a normal state on $M$. For any $\epsilon>0$, can we find a projection $p$ in $M$ such that $\omega(p)＜\epsilon$?

Answer 1

If $\omega$ is not faithful, there are projections $p$ with $\omega(p)=0$.

When $\omega$ is faithful, start with $\omega(1)=1$. Now halve, $1=p_1+p_2$. Since $\omega(p_1)+\omega(p_2)=1$ and both numbers are positive, one of them is less than or equal $\frac12$. Say, $\omega(p_1)\leq \frac12$. Halving again you get $p_2$ with $\omega(p_2)\leq \frac14$. Repeat as needed.

Normality is not needed.