Let $M \subset B(\mathcal{H})$ be a von neumann algebra and $\phi$ be a uwo continuous faithful tracial state. I understand that we obtain the GNS $(\mathcal{H}_\phi, \pi_\phi, \xi_\phi)$ with $\xi_\phi$ the cyclic vector can be shown to be separating and a tracial vector for the image of $\pi_\phi$. From the representation $\pi_\phi$, we have an isomorphism $M \cong \pi_\phi(M)$. Defining $m \xi_\phi \rightarrow m^* \xi$, we can extend this to $\mathcal{H}_\phi$(due to $\xi_\phi$ being a tracial cyclic separating vector) denote this as $J$. We can show that $J^2 = I$, $J$ is conjugate linear, and $M' = JMJ$. This essentially shows that our von neumann algebra is in standard form. We denote $M \xi_\phi$ as $L^\infty(M, \phi)$ and $\mathcal{H}_\phi$ as $L^2(M, \phi)$
Let $N$ be a von neumann subalgebra of $M$. We can restrict our $\phi$ to $N$. From GNS again, we can obtain $L^2(N, \phi|_N)$. Maybe because my algebra is quite weak, I am having trouble understanding why $L^2(N, \phi|_N) \subset L^2(M, \phi)$(since it seems the left cosets seem to be different) and to show that $\xi_\phi$ is cyclic vector(the same cyclic vector for $M$) for $N$ (i.e. $N\xi_\phi$ is dense in $L^2(N, \phi|_N)$. Also, is $\mathcal{H}$ isomorphic to $\mathcal{H}_\phi$? Could someone help me understand the all relationships between these spaces. For context, this is to build up to define the condition expectation projection for the closed subspace $L^2(N, \phi|_N)$.
As $\phi$ is faithful, the cosets are singletons. And the inner product is the same, so you are taking the inclusion $N\subset M$ and when you complete $M$ in the $\phi$-topology, the closure of $N$ is the completion of $N$ in the same topology. So $L^2(N,\phi)\subset L^2(M,\phi)$ is an actual inclusion.
The cyclic vector $\xi_\phi$ is $1_M$. As $1_N\in M$ is a projection, $$N\,1_M=N\,(1_N+(1_M-1_N))=N\,1_N,$$ so the same $\xi_\phi$ is cyclic and separating for $N$ in $L^2(N,\phi)$.
As for $H_\phi\simeq H$, this is usually true in the sense that for most common cases they will be infinite-dimensional separable Hilbert spaces. But there is no canonical isomorphism. This can be seen in finite dimension, take $M=M_2(\mathbb C)$, and $\phi$ the normalized trace. Then $M$ acts on $\mathbb C^2$, but $H_\phi=\mathbb C^4$.