Let $(M,\tau)$ be a tracial von Neumann algebra and $P,Q$ two von Neumann subalgebras such that there exists an unitary in $u \in M$ with $u^*Pu \subset Q$. Then prove that $H=uL^2(Q)$ is a $\>P-Q$-submodule of $L^2(M)$ with dimension $\dim(H_Q)=1.$
I was reading a proof and I got the above statement inside the proof and I am unable to figure this out. Can you please help me to solve this? Thanks for your time.
Edit: Let $(M,\tau)$ be a tracial von Neumann algebra and $H$ a right $M$-module. Then $\dim(H_M)=(\text{Tr} \otimes \tau)(p),$ where $p$ is any projection in $B(\ell^2(\mathbb N) \overline{\otimes} M$ such that $H$ is isomorphic to $p(\ell^2(\mathbb N) \otimes L^2(M)),$ where $\text{Tr}$ is the usual trace on $B(\ell^2(\mathbb N))_+$.