# Existence of faithful normal state

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?

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 $\Phi$.
• Meaning is if $\rho$ is a tracial state in von Neumann algebra(factor), does there exist centervalued trace $\phi$ such that $\phi(A)=\rho(A)I$ Jul 21, 2018 at 18:31