No, if $\Gamma$ is a group without torsion and with the Infinite conjugacy class property then the algebra $L\Gamma$ is a factor of type $II$ and has only one normal tracial state.
Now, let $x $ in $\Gamma$ different from the identity. The von Neumann algebra $S$ generated by $x$ is isomorphic to $L^\infty(S^1)$, the algebra of measurable functions on the circle. It has many normal tracial states.
