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"): 



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> > 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:  
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> >  1. In any unitary representation $\rho:G \to B(H)$, $P\_{\pi}$ acts as a projection onto the $\pi$-isotypical subspace of $H$.  
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> >  2. If $\pi$ and $\pi'$ are not equivalent irreducible representations then corresponding projections are orthogonal to each other.  
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> >  3. $P\_{\pi}C^\*(G)P\_{\pi}$ is isomorphic to $B(H\_{\pi})$ 
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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).



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


