$G$-invariant functionals on endomorphisms of vector spaces

Question

I am looking for a reference to the following fact.

Let $(V,\rho)$ be a representation of a group $G$ on a finite-dimensional vector space $V$. Let $f$ be a $G$-invariant functional on ${\rm End}(V)$, that is, $f(A) = f(\rho(g^{-1})A\rho(g))$ for any $g\in G$. Then there is $B\in {\rm End}(V)$ such that $f(A) = {\rm tr}(AB)$.

Thank you.

Answer 1

Bilinear pairing $(A,B) \mapsto \mathrm{tr}(AB)$ on $\mathrm{End}(V)$ is non-degenerate, so any functional, invariant or not, admits such a representation. Moreover operator $B$ is uniquely determined. I think the point is that if the functional is invariant, then so is $B$.