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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 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?

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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.

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