Proposition 45.1.10.label Let $M$ be a factor, then $M$ admits at most one tracial state.
Proof, [Proposition 4.1.3, AP]. Using Proposition 45.1.6, assume without loss of generality that $M$ is diffuse.
Let $p \in \text{Proj}(M)$ with $0 \le p < 1_{M}$. Let $n \in \natp$, then by Corollary 45.1.9, there exist mutually orthogonal equivalent projections $\bracsn{p_j}_{1}^{2^n}\subset \text{Proj}(M)$ such that $\sum_{j = 1}^{2^n}p_{j} = 1$. By the corollary:vna-factor-comparability, there exists a unique $0 \le k \le 2^{n} - 1$ such that
\[\sum_{j = 1}^{k} p_{j} \preceq p \prec \sum_{j = 1}^{k+1}p_{j}\]
As such, for any tracial state $\tau: M \to \complex$,
\[\frac{k}{2^{n}}\le \tau(p) \le \frac{k+1}{2^{n}}\]
so $\tau(p)$ is independent of the choice of $\tau$. By Theorem 43.7.2, $\tau$ is unique.$\square$
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