Let $A$ be a C*-algebra. Do you confirm the following discussion?
Let us consider a representation $\pi:A\to B(H)$. We denote $M_{\pi}$ by the von Neumann algebra generated by $\pi(A)$. If we denote $V_{\pi}$ by the predual of $M_{\pi}$, then it may be assumed as a closed subspace of $A^*$. Indeed $V_{\pi}$ is an invariant subspace of $A^*$. Therefore there is a unique central projection $e_{\pi}$ in $A^{**}$ (which is also called the support of $\pi$) with $$V_{\pi}=e^{\perp}_{\pi}A^* ~~~\textrm{which implies that}~~~M_{\pi}=e_{\pi}A^{**}$$ To sum up with, for the representation $\pi$, we find the following two formulas:
1- $\pi$ is just the restriction mapping $a\to a_{|_{V_{\pi}}}$
2- $\pi$ is given by $a\to ae_{\pi}$.