Let $M$ be a von Neumann algebra. Let $\{M_\lambda\}$ be an increasing net of von Neumann sublagebras of $M$, equipped with faithfal normal tracial states $\tau_\lambda$ ($\tau_\lambda=\tau_\gamma$ on $M_\gamma$ when $\lambda >\gamma$). Moreover, $\cup M_\lambda$ is dense in $M$ in the weak operator topology.
Can I say that $M$ can be equipped with a faithful normal tracial state, which coincides with $\tau_\lambda$ on $M_\lambda$?
Extending a trace from a subalgebra to the entire von Neumann algebra
banach-algebrasfunctional-analysisoperator-algebrasvon-neumann-algebras
Best Answer
No. Every AFD type III factor satisfies all the conditions you mentioned, where the $M_\lambda$ can be taken to be $M_{n(\lambda)}(\mathbb C)$