Exam

Prove that $P:f\mapsto f_c$ is the orthogonal projection $L_2(G)\rar Z(G)$.

$P$ is a projection; thus we just need to check self-adjointness: \begin{eqnarray*} \la Pf,g\ra &=&\frac1{|G|^2}\sum_{x,y}f(yxy^{-1})\bar g(x) =\frac1{|G|^2}\sum_{x,y}f(yx)\bar g(xy)\\ &=&\frac1{|G|^2}\sum_{x,y}f(x)\bar g(y^{-1}xy) =\frac1{|G|^2}\sum_{x,y}f(x)\bar g(yxy^{-1}) =\la f,Pg\ra \end{eqnarray*} Since $P$ is self-adjoint, it's the orthogonal projection onto its image, which is $Z(G)$.