I have a question about the Fourier transfomation on a finite non-comutative group. I hope that it is a known fact in the Representation Theory but I cannot find it written explicitly in textbooks.

Let $G$ be a finite non-abelian group, $\hat{?}:L_2(G)\to \hat L_2(\hat G)$ be its Fourier transformation and $\check{?}:\hat L_2(\hat G)\to L_2(G)$ be the inverse Fourier transformation. The space $L_2(G)$ carries the standard norm induced by the inner product $\langle x,y\rangle=\sum_{g\in G}x(g)\cdot\overline{y(g)}$. The norm on $\hat L_2(\hat G)$ is normalized so that the Fourier transformations $\hat?$ is an isometry.

Let $P:\hat L_2(\hat G)\to \hat L_2(\hat G)$ be the function assigning to a sequence of matrices $(M_\alpha)_{\alpha\in\hat G}$ in $\hat L_2(\hat G)$ the sequence $(\frac 1{\dim(\alpha)}Tr(M_\alpha)\cdot\mathrm{Id}_{\alpha})_{\alpha\in\hat G}$ of normalized traces of those matrices multiplied by the identity matrices. So, $P$ is a linear projection of $\hat L_2(\hat G)$ onto its $|\hat G|$-dimensional subspace.

Question 1.What is the norm of a sequence $(M_\alpha)_{\alpha\in \hat G}\in P(\hat L_2(\hat G))$ in the Hilbert space $\hat L_2(\hat G)$? Is it equal to $(\frac1{|\hat G|}\sum_{\alpha\in\hat G}|\frac1{\dim(\alpha)}Tr(M_\alpha)|^2)^{1/2}$?

I am interested in describing the projector $A=\check{?}\circ P\circ \hat{?}:L_2(G)\to L_2(G)$. It should assign to each function $f\in L_2(G)$ some class function on $G$.

Question 2.Is $A$ the averaging of $f$ over conjugacy classes? If yes, where can I find a reference to this fact?

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