strictly semifinite weights restrict to a von Neumann subalgebra

Question

Suppose $\omega$ is a strictly semi-finite weight on a von Neumann algebra $M$ and $N$ is the von Neumann subalgebra of $M$.

Is $\omega|_N$ a strictly semifinite weight on $N$?

Answer 1

No. Say $\omega$ is the canonical trace on $M = B(l^2)$ and $N \subset M$ is a type III factor.