$\mathbb{C} [G] \longrightarrow \prod_{\rho} \text{End}(V_{\rho})$ an intertwining isomorphism

Question

Consider the vector space $\mathbb C[G]$ of functions $f: G \longrightarrow \mathbb{C}$ where $G$ is a finite group, or equivalently a vector space of all formal linear combinations of elements of $G$ over the complex numbers. How can I prove the map $$\phi : \mathbb{C} [G] \longrightarrow \prod_{\rho} \text{End}(V_{\rho})$$ $$f \mapsto (\rho(f) )_{\rho}$$ where $\rho(f) = \sum_{g \in G} f(g) \rho(g)$ (where $\rho$ runs over all the isomorphism classes of irreducible representations of $G$) is both a ring isomorphism and also an intertwining isomorphism of $G \times G$ where our action, given $T \in \text{End}(V_{\rho})$, is $(g,h)\cdot T := \rho(g) \circ T \circ \rho(h)^{-1}$? We define ring multiplication by $(f_1 \ast f_2)(g) = \sum_{g = x y} f_1(x) f_2(y)$ in $\mathbb{C}[G]$. Does this isomorphism lead to a more natural decomposition of the regular representation $\mathbb{C}[G]$?

I know how to show that each map $\rho : G \longrightarrow \text{GL}(V_{\rho})$ extends to $\phi$ by linearity. I can also see how ring multiplication is respected by $\phi$. I also know that we can show the map is injective because $\dim \mathbb{C}[G] = |G|$, but also $$\dim (\prod_{\rho} \text{End}(V_{\rho}))=\sum_\rho \dim (\text{End}(V_\rho))=\sum_\rho \dim(V^{\ast}_\rho \otimes V_\rho) =$$ $$\sum_\rho \dim(V^{\ast}_\rho) \dim(V_\rho) =\sum_\rho \dim(V_\rho)^2 = |G|$$ by the theorem that the sum of the squares of the degrees of the irreducible representations is equal to the order of the group.

I can't seem to work out how as a linear transformation it intertwines with the action on $G \times G$ described above or how the convolution multiplication defined above is respected by $\phi$ as a ring isomorphism.

Answer 1

$\def\CC{\mathbb C}$Notice that to prove injectivity of the map it is enough to show that for each $f\in\CC G$ there is a $\rho$ such that $\rho(f)\neq0$ in $\operatorname{End}(V_\rho)$.

There is a decomposition $\CC G=\bigoplus_{i=1}^r I_i$ of the algebra $\CC G$ as a direct sum of minimal left ideals. Indeed, $\CC G$ is a representation of $G$ on th left in the obvious way —the left regular representation— so it decomposes as a direct sum of irreducible subrepresentations, and you can easily check that these subrepresentations are in fact left ideals of $\CC G$.

Let $1$ be the unit element in $\CC G$. In view of the above decomposition, there are uniquely determined elements $e_i\in I_i$ for all $i$ such that $1=\sum_{i=1}^re_i$.

Now let $f\in\CC G\neq0$. Since $f=f\cdot 1=f\cdot\sum_ie_i=\sum_if\cdot e_i$, we see that there is an $i_0$ such that $f\cdot e_i\neq0$. This tells us that $f$ acts as non-zero on some irreducible representation of $G$..