Proposition 45.1.11.label Let $(M, \tau)$ be a tracial von Neumann algebra, then $M$ is a factor if and only if $\tau$ is the unique normal faithful tracial state on $M$.
Proof, [Proposition 4.1.4, AP]. If $M$ is a factor, then $\tau$ is unique by Proposition 45.1.10. Suppose that $M$ is not a factor, then $M$ admits a non-trivial central projection $z \in Z(M)$. In which case, for any $\alpha \in (0, 1)$,
\[\tilde \tau(x) = \frac{\alpha}{\tau(z)}\tau(xz) + \frac{1 - \alpha}{1 - \tau(z)}\tau(x(1 - z))\]
is also a normal faithful tracial state. In particular,
\[\tilde \tau(z) = \frac{\alpha}{\tau(z)}\tau(z) + \frac{1 - \alpha}{1 - \tau(z)}\tau(z(1 - z)) = \alpha\]
so if $\alpha \ne \tau(z)$, then $\tilde \tau \ne \tau$.$\square$
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