Exist matrices such that $f(X) = \sum_{j = j}A_jXB_j$ for all $X \in \text{M}_N(\mathbb{C})$? [closed]

Question

Let $f: \text{M}_n(\mathbb{C}) \to \text{M}_n(\mathbb{C})$ be a $\mathbb{C}$-linear map (not necessarily an algebra homomorphism). Do there exist matrices $A_1, \ldots, A_d \in \text{M}_n(\mathbb{C})$ and $B_1, \ldots, B_d \in \text{M}_n(\mathbb{C})$ such that$$f(X) = \sum_{j = 1}^d A_jXB_j \quad \forall X \in \text{M}_n(\mathbb{C})?$$

Answer 1

Of course. The basis of the space of your linear maps is the set of $n^4$ maps $f_{ijkl}:X\mapsto e_{ij}Xe_{kl}$, where $e_{ij}$ are matrix units. Indeed, $f_{ijkl}$ maps $e_{jk}$ to $e_{il}$ and all other matrix units to 0. So, these maps correspond to matrix units of $n^2\times n^2$ matrices representing your maps, if you choose matrix units as a basis in $M_n(\mathbb{C})$.