Existence of faithful normal state

Question

Does there always exist faithful normal state on center of von Neumann algebra?

Further, in type II$_{1}$ von Neumann algebras are tracial states coming from the center valued trace?

Answer 1

The first question is the same as asking if every abelian von Neumann algebra has a faithful normal state. The answer is easily no, by taking for instance the example in this answer.

As for all tracial states coming from the center-valued trace, it is not entirely clear to me what you mean. If you have $$ M=\bigoplus_n A_n\otimes N_n, $$ where $A_n$ are abelian and $N_n$ are II$_1$-factors, the center-valued trace is induced by $$ \Phi(\bigoplus_n a_n\otimes x_n)=\bigoplus_n a_n\otimes 1. $$ Tracial states are obtained by taking $f_n\in S(A_n)$ and $\tau_n$ the unique trace on $N_n$, and letting $$ \phi(\bigoplus_n a_n\otimes x_n)=\sum_n f_n(a_n)\tau(x_n). $$ I cannot immediately say in what sense $\phi$ "comes" from $\Phi$.