Tracial and finite von Neumann algebras

Question

A tracial von Neumann algebra $(M,\tau)$ is a von Neumann algebra with a faithful normal tracial state $\tau$ on $M$. That is, $\tau$ is a function from $M \to \mathbb{C}$ such that it is a faithful normal state and $\tau(xy)=\tau(yx)$. I m confused with tracial and finite von Neumann algebras. I could see references saying that a finite von Neumann algebra $M$ has a unique centre valued $Z(M)$ trace. But this need not be scalar valued no? My definition of trace is a positive linear functional $\tau$ satisfying $\tau(xy)=\tau(yx)$. Does a finite von Neumann algebra has a faithful tracial states? That is a scalar valued one?are they unique? I know a finite factor has a unique one.

Answer 1

If you have a faithful tracial state, then the algebra is finite (if $v^*v=1$ then $0\leq \tau(1-vv^*)=\tau(1-v^*v)=0$, so $1$ is finite).

But the converse is not true. An algebra with a faithful state has to be "countably decomposable", it can only admit countably many pairwise orthogonal projections. But there are finite von Neumann algebras that have uncountably many pairwise orthogonal projections. For instance $\ell^\infty[0,1]$.