The image is from the C*-algebras by Example by Kenneth R. Davidson, here F$_2$ is the free product of Z$_2$ and Z$_3$. I have two questions:
1) Why do we have the matrices U$_n$ and V$_n$, I guess it is to make the A$_n$ and B$_n$ unitary. Is it sufficient?
2) As we know, the reduced group C*-algebra C$^*$$_r$(F$_2$) is simple. If we do the same trick to the C$^*$$_r$(F$_2$), which step will break?
1) The point here is to construct finie-dimensional representations. The restrictions $A_n$, $B_n$ are finite-dimensional but not unitaries, so the trick is to use the fact that any contraction is a corner of a unitary. Indeed, given any contraction $T$ n $B(H)$, the operator $$ \begin{bmatrix}T&(I-TT^*)^{1/2}\\ (I-T^*T)^{1/2}&-T^*\end{bmatrix}\in B(H\oplus H) $$ is a unitary. It is a direct computation to check that it is.
2) The problem lies with the assertion that $\pi_n$ is a representation. It works in the universal case because you can lift any representation of $\mathcal F_2$ to a representation of the algebra, but it doesn't work for the reduced case.