the range of trace on projections of II$_1$ factor

Question

Suppose $M$ is a von Neumann algebra factor of type II$_1$ factor,let $P(M)$ be the set of projections in $M$,how to prove that $\{tr (p),p\in P(M)\}=[0,1]$,where tr is a tracial state on $M$?

Answer 1

Clearly $\operatorname{tr}(P(M))\subset[0,1]$. For the other inclusion, first note that $0,1\in\operatorname{tr}(P(M))$. Since $M$ is a II$_1$-factor, for each projection $p\in P(M)$ there exist two equivalent projections $q,r\in P(M)$ such that $p=q+r$. Repeating this halving process ad infinitum, we see that $\{2^{-n}:n\in\mathbb N\}\subset\operatorname{tr}(P(M))$. From this, by adding orthogonal projections constructed above, we see that the collection of dyadic rationals in $[0,1]$, i.e., those rationals of the form $n/2^m$, where $m\in\mathbb N$ and $0\leq n\leq 2^m$, lie in $\operatorname{tr}(P(M))$. Taking now weak limits of sums of orthogonal projections constructed above fills up $[0,1]$.