Let $M$ be a finite von Neumann algebra with $\tau$ a finite faithful trace. Let $N$ be a von Neumann subalgebra of $M$ with trace $\tau|\_N$, obtained by restricting $\tau$ to $N$. If $e\_N$ denotes the Jones projection and $<M,e\_N>$ the basic construction, then we have the strongly-dense \*-subalgebra $span\{M\cup\{xe\_Ny\}|x, y \in M\}\subset <M,e\_N>.$


My question is as follows: I have seen that a trace $\bar{\tau}$ can be defined on $<M,e\_N>$ by first defining $\bar{\tau}(xe\_Ny):=\tau(xy).$ Why is $\bar{\tau}$ a semifinite normal faithful trace?


The closest I've seen to a proof is in a paper by [Austin, Eisner and Tao](http://arxiv.org/pdf/0912.5093v5.pdf) page 25, just after Lemma 3.1. However, the sketch given in the paper is not sufficient for me to fill in the gaps. Is there any other references where such a proof could be found? 



Semifinite Trace on Jones' Basic Construction
A proof can be found in Sinclair and Smith's book "Finite von Neumann algebras and masas" doi:10.1017/CBO9780511666230. I believe Chapter 4 will be what you're looking for.


