Mysterious central projections in the full group $C^*$-algebra

Question

Let me quote the following theorem about the structure of $C^*(G)$ for property $T$ group (the reference is Higson and Roe "Analitycal K-homology"):

Let $G$ be a property $T$ (discrete) group. then for any class of irreducible unitary representation $\pi$ there is a central projection $P_{\pi}$ with the following properties:
1. In any unitary representation $\rho:G \to B(H)$, $P_{\pi}$ acts as a projection onto the $\pi$-isotypical subspace of $H$.
2. If $\pi$ and $\pi'$ are not equivalent irreducible representations then corresponding projections are orthogonal to each other.
3. $P_{\pi}C^*(G)P_{\pi}$ is isomorphic to $B(H_{\pi})$

I found the following remark: it is impossible to expand $P_{\pi}$ as $\sum_{s \in G}\alpha_ss$ where $\sum_{s \in G}\alpha_s$ is convergent in some suitable sense. It is stated that if such an expansion will be valid then one can recover the coefficients $\alpha_s$ from the action on the regular representation and one can conclude that all $\alpha_s$ are zero. My question is, how to obtain that fact?
Many thanks for any help.
EDIT: I changed the name of the coefficients to $\alpha_s$ (now $\lambda$ can be reserved for a regular representation).

Answer 1

This was too long for a comment, although it might not exactly answer your question.

Suppose $\pi$ is such that the left reg rep does not contain any copy of $\pi$. (E.g if $\pi$ is the trivial representation $G\to \{1\}$.) Then $P_\pi$ lies in the kernel of the natural homomorphism $q:{\rm C}^*(G) \to {\rm C}_r^*(G)$.

For each $s\in G$ let $\theta_s: {\rm C}_r^*(G) \to {\bf C}$ be defined by $\theta_s(a) = \langle a \delta_e, \delta_s\rangle_{\ell^2(G)}$. Note that $\theta_s \vert_{\ell^1(G)}$ just picks out the scalar coefficient labelled by $s$.

Let $\iota: c_{00}(G) \to {\rm C}^*(G)$ be the natural embedding, and note that $q\circ\iota : c_{00}(G) \to {\rm C}_r^*(G)$ coincides with the restriction of the usual left reg. rep. $\lambda: c_{00}(G) \to {\rm C}_r^*(G)$.

Let $(a_n)$ be any sequence in $c_{00}(G)$ such that $\Vert a_n - P_\pi \Vert_{{\rm C}^*(G)} \to 0$. Then for each $s\in G$, $$ (a_n)(s) = \theta_s\circ q (a_n - P_\pi) \to 0 \quad{\rm as}\; n\to\infty $$ This shows that any attempt to approximate $P_\pi$ by applying some kind of noncommutative version of the Fejer kernel will not work.