Existence of a unique normal state on a von Neumann algebra

Question

Let $M$ be a von Neumann algebra. Does there exist a necessary and sufficient condition for $M$ admitting a unique faithful normal state?

Answer 1

The only von Neumann algebra with a unique faithful normal state is $\mathbb C$. In any other von Neumann algebra there will be a non-trivial projection $p$, and given a faithful normal state $\psi$ you can form uncountably many faithful normal states $$ \psi_t(x)=t\,\frac{\psi(pxp)}{\psi(p)}+(1-t)\,\frac{\psi((1-p)x(1-p))}{\psi(1-p)},\qquad t\in[0,1]. $$