Let $\mathcal{A}$ be a unital $C^{*}$-algebra. If $\omega$ is a positive linear functional on $\mathcal{A}$, then we may perform the so-called GNS construction in order to obtain ha Hilbert space $\mathcal{H}_{\omega}$, a representation $\pi_{\omega}$ of $\mathcal{A}$ in $\mathcal{B}(\mathcal{H}_{\omega})$, and a cyclic vector $\xi_{\omega}\in\mathcal{H}_{\omega}$ for $\pi_{\omega}$ such that $$ \omega(a)=\langle\xi_{\omega}|\pi_{\omega}(a)\xi_{\omega}\rangle . $$
We may always decompose $\mathcal{H}_{\omega}$ as the direct sum of the closed subspace $H_{\xi_{\omega}}$ generated by $\xi_{\omega}$ and its orthogonal complemetn $H_{\xi_{\omega}}^{\perp}$. Let me call $H_{\xi_{\omega}}$ the GNS subspace of $\omega$.
Clearly, the orthogonal projection $P$ onto the GNS subspace of $\omega$ is an element of $\mathcal{B}(\mathcal{H}_{\omega})$, and I would like to know if there exists a projection $e\in\mathcal{A}$ (depending on $\omega$, of course) such that $$ \pi_{\omega}=P $$ If it helps, $\mathcal{A}$ may be taken to be a von Neumann algebra, and $\omega$ to be normal.
The projection $e$ cannot exist for $\omega$ faithful when $\mathcal A$ is simple and $\mathcal A\ne\mathbb C$. For instance, it will never exist for a II$_1$-factor, even if $\omega$ is normal. For C$^*$-algebras, you have the added problem that $\mathcal A$ may lack projections.
Write $\hat a$ for the class of $a\in \mathcal A$ in $\mathcal H_\omega$. You have that (recall that $\xi_\omega=\hat 1$) $$\tag1 P\hat a=\langle \hat a,\xi_\omega\rangle\,\xi_\omega=\langle \hat a,\hat 1\rangle\,\hat 1=\omega(a)\,\hat 1. $$ Now you want $P=\pi(e)$ for some projection $e\in \mathcal A$. Then the above becomes $$\tag2 \pi(e)\hat a=\omega(a)\,\hat 1, $$ which is $$\tag3 \widehat {ea}=\omega(a)\,\hat 1. $$ Because $\omega$ is faithful, $ea=\omega(a)1$ for all $a\in A$. This requires $\dim \mathcal A=1$.