I'm following Popa and Anantharaman's "An introduction to $II_1$ factors" and in Proposition 4.1.3 they prove that:
A von Neumann factor $M$ has at most one tracial state, and then such a tracial state is faithful.
In the proof, it seems like they use that if $\tau$ is a tracial state on $M$ then $\tau(p) > 0$ for any projection $p \neq 0$. Is that true, and if so how would one go about showing it?
Because you are in a factor, you can always compare projections. So, given any $p$, you can find projections $p_1,\ldots,p_n,r$, with $p_j\sim p$, $r\prec p$, and $r+\sum_jp_j=1$. If $\tau(p)=0$, then $\tau(p_j)=0$ for all $j$, and also $\tau(r)=0$, since $r\sim r_0\leq p$ and $\tau$ is positive. So you get $\tau(1)=0$, contradicting that $\tau$ is a state.
The construction of the projections is trivial: start with $p_1=p$. If $p\prec 1-p_1$, then you have $p\sim p_2\leq 1-p_1$ for some projection $p_2$; and you keep going. Eventually, you will have $p\succ 1-p_1-\cdots -p_n$, so there exists $r\leq 1-p_1-\cdots-p_n$ with $r\prec p$.