existence of faithful state in the predual of a von Neumann algebra

Question

Let $M$ be a von Neumann algebra. $M_{*}$ is the predual of $M$.

Does there must exist a faithful state $\psi\in M_{*}$? If such faithful state exists, how to construct it ?

Answer 1

If $M_*$ is separable, just take $\psi=\sum_n2^{-n}\phi_n$, where the $\phi_n$ form a dense subset of the normal state space.

Otherwise such a state might not exist.