For $N>4$, let $A=C([0,N])$ and $1_A=\mathbf{1}_{[0,N]}$ (the identity on $A$). Consider $\operatorname{id}:=\operatorname{id}_{[0,N]}=(t\mapsto t)$ (not the identity on $A$ but $f(t)=t$) and the functional $\varphi:=\operatorname{ev}_{N-2}$, $f\mapsto f(N-2)$. I am interested in the spectral projection $\mathbf{1}_{\{N-2\}}(\operatorname{id})$ and the GNS representation $(\mathsf{H}_\varphi,\pi_\varphi,\xi_\varphi)$. I guess I'll start with the GNS representation as that seems more straightforward.
The nullspace of $\varphi$, $N_\varphi$ consists of those functions that vanish at $N-2$. We quotient to get the (one-dimensional?) space of functions $A/N_\varphi$ which agree at $N-2$. We have an inner product: $$\langle f+N_\varphi,g+N_\varphi\rangle=\overline{f(N-2)}g(N-2).$$ We take the (trivial?) Hilbert space completion $\overline{A/N_\varphi}=\mathsf{H}_\varphi$. The representation $\pi_\varphi:A\rightarrow B(\mathsf{H}_\varphi)$ is: $$\pi_\varphi(f)(g+N_\varphi)=fg+N_\varphi.$$ The cyclic vector is $\xi_\varphi=1_A+N_\varphi$. The importance of this vector is that it is an eigenvector of eigenvalue $N-2$ for $\pi_\varphi(\operatorname{id})$ as: $$\pi_\varphi(\operatorname{id})(1_A+N_\varphi)=\operatorname{id}+N_\varphi=(N-2)(1_A+N_\varphi),$$ as $\operatorname{id}(N-2)=N-2=(N-2)\varphi(1_A)=\varphi((N-2)1_A)$. So in this representation $N-2$ is an eigenvalue.
Now onto $\mathbf{1}_{N-2}(\operatorname{id})$. I am wondering is it possible to describe this concretely. When I try and play around with $L^\infty([0,N])$ and $L^2([0,N])$ and multipliers and such I cannot get it to work, and the best I think I can do is something like: $$\mathbf{1}_{(N-2-\varepsilon,N-2+\varepsilon)}(\operatorname{id})(f)(t)=\begin{cases}f(t)& \text{ if }|t-(N-2)|<\varepsilon \\ 0 & \text{otherwise}\end{cases}$$ I don't think it should be zero though in the generality I am interested in. I think I should be looking at the enveloping von Neumann algebra $\pi_U(A)''$, and then my understanding is that $\mathbf{1}_{\{N-2\}}(\operatorname{id})$ is the projection onto the kernel of $\operatorname{id}-(N-2)1_A$, and if I have the big $\mathsf{H}_U$ then I at least have $\xi_\varphi\in\mathsf{H}_U$ and so we should have something like: $$\mathbf{1}_{\{N-2\}}(\pi_U(\operatorname{id}))\xi_{\varphi}=\xi_\varphi,\qquad (*),$$ which might be written as $\mathbf{1}_{\{N-2\}}(\operatorname{id})\xi_{\varphi}=\xi_\varphi$, and so $\mathbf{1}_{\{N-2\}}(\operatorname{id})\neq 0$.
My questions I guess are:
- Does this all check out? In particular do I have to be more careful with the quotient map in (*)
- If $\mathbf{1}_{\{N-2\}}(\operatorname{id})$ is indeed non-zero in $\pi_U(A)''\subset B(\mathsf{H}_U)$, is there any hope of describing it concretely?
What I am hoping to do eventually is condition states $\omega$ (on what algebra now I am not sure, $A$ or normal extensions to $\pi_U(A)''$) by $p_{N-2}:=\mathbf{1}_{\{N-2\}}(\operatorname{id})$: $$\omega\mapsto \frac{\omega(p_{N-2}\cdot p_{N-2})}{\omega(p_{N-2})},$$
so that, for example, we could condition $(\operatorname{ev}_{N-3}+\operatorname{ev}_{N-2})/2$ to $\varphi=\operatorname{ev}_{N-2}$.
Let $A=C_0(X)$ where $X$ is a Hausdorff space. Then the pure states are the evaluation maps, which are also the characters. As such $A/N_\varphi$ is one-dimensional for every pure state $\varphi$.
So while your GNS construction works it is pretty much overkill, in particular every non-zero element of $A/N_\varphi$ is a cyclic vector and $\pi(f)$ is always proportional to the identity (ie has an eigenvalue). The spectral projection of the point $\{N-2\}$ on the representation $A/N_{ev_{N-2}}$ is given by $\Bbb1$, the identity operator.
If you want to find the spectral projection in the enveloping von Neumann algebra it goes like this:
One construction of $A''$ is as the bi-commutant of the image of $A$ in the representation
$$\bigoplus_{\varphi} A/N_{\varphi}$$ where $\varphi$ sums over all pure states / irreducible representations. In this case these are the characters so you are summing over $$\bigoplus_{x\in X} \Bbb C = \ell^2(X)$$ where the action of $f\in C_0(X)$ one the summand corresponding to $x\in X$ is given by multiplication with $f(x)$. The bi-commutant of this image is $\ell^\infty(X)$.
The spectral projection of a set $S\subseteq X$ in $A''$ is then given by pointwise multiplication with the characterstic function $\chi_S$ of $S$ on $\ell^2(X)$.