Proposition 45.1.13.label Let $M$ be a $\vnII_{1}$ factor with tracial normal state $\tau: M \to \complex$, then the mapping $p \mapsto \tau(p)$ induces a bijection from the equivalence classes of $\text{Proj}(M)$ to $[0, 1]$.
Proof, [Proposition 4.1.6, AP]. (Injective): Let $p, q \in \text{Proj}(M)$ with $\tau(p) = \tau(q)$. By the comparability theorem, either $p \prec q$, $p \sim q$, or $q \prec p$. Since $\tau$ is faithful, $p \prec q$ implies that $\tau(p) < \tau(q)$, and $q \prec p$ implies that $\tau(q) < \tau(p)$. Therefore the only possibility is $p \sim q$.
(Surjective): By Lemma 45.1.8, there exists a family $\bracsn{p_r|r \in \mathbb{D} \cap [0, 1]}\subset \text{Proj}(M)$ such that:
- (i)
For each dyadic rational number $r \in \mathbb{D} \cap [0, 1]$, $\tau(p_{r}) = r$.
- (ii)
For each $r, s \in \mathbb{D} \cap [0, 1]$ with $r \le s$, $p_{r} \le p_{s}$.
For each $t \in [0, 1]$, let $p_{t} = \sup\bracsn{p_r|r \in \mathbb{D} \cap [0, 1], r \le t}$, then since $\tau$ is normal,
$\square$
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