Let $F$ be a continuous function on the space of $N\times N$ hermitian matrices $\mathcal{H}_N$ taking values in $\mathbb{C}$ which satisfies $F(UMU^*)=F(M)$ for all $M\in \mathcal{H}_N$ and $U\in U(N)$, where the later stands for the unitary group of $\mathbb{C}^N$.
The first examples coming in mind are the Trace, and the Determinant. Do you know is there exists a way to express $F$ in terms of these two functions ? Or maybe under stronger conditions than continuity for $F$ ?