Let $N$ be a type $II_{1}$-factor with trace $\tau$, and $B$ a von Neumann subalgebra. The existence of the semifinite trace on the Jones basic construction $\langle N, e_{B} \rangle$ is reasonably easy to establish if $B$ is a subfactor of $N$, but appears not to be so easy in general.

Question: What is the shortest known proof of the existence of the trace on the basic construction for a

von Neumann subalgebrainclusion?

Two complete proofs of this appear in the excellent book *Finite von Neumann algebras and Masas* by Sinclair and Smith. I (and others far more adept than I) am curious if any other proofs of this exist in the literature.

**EDIT:** The previous question, as asked, is embarrassingly silly. What seems less than obvious is how to construct the trace and verify that what you have written down is actually a faithful, semifinite trace. If someone could indicate how to do this, I'd very much appreciate it.