Let $M$ be a von Neumann algebra acting on a Hilbert space $H$, and let $\tau$ be a faithful tracial state on $M$.
What is the relation between the GNS representation of $(M,\tau)$ and the original action of $M$ on $H$?
More precisely, let $M_\tau$ denote the Hilbert space obtained by completing $M$ with the inner product $\langle x,y\rangle_\tau=\tau(y^*x)$ (since $\tau$ is faithful, this inner product is nondegenerate), and $M$ acts faithfully (again since $\tau$ is a faithful state) on $M_\tau$ by left multiplication. Let $\varphi_\tau:M\to B(M_\tau)$ denote this representation.
My question is: is $\varphi_\tau(M)$ a von Neumann algebra on $M_\tau$? If so, $\tau$ becomes a vector state under the identification $M\simeq\varphi_\tau(M)$: $\tau(u)=\langle u(1_M),1_M\rangle_\tau$.
In case $M$ admits a cyclic vector $x$ for which $\tau(u)=\langle u(x),x\rangle$, the result is true, but the last comment above becomes useless. In fact, in this case the actions of $M$ on $H$ and on $M_\tau$ are unitarily equivalent (see Murphy, Theorem 5.1.4).