Linear rank preservers of tensor products of rank one matrices

Abstract: Let $n_1,\ldots,n_k $ be integers larger than or equal to 2. We characterize linear maps $\phi: M_{n_1\cdots n_k}\rightarrow M_{n_1\cdots n_k}$ such that $${\mathrm rank}\,(\phi(A_1\otimes \cdots \otimes A_k))=1\quad\hbox{whenever}\quad{\mathrm rank}\, (A_1\otimes \cdots \otimes A_k)=1 \quad \hbox{for all}\quad A_i \in M_{n_i},\, i = 1,\dots,k.$$ Applying this result, we extend two recent results on linear maps that preserving the rank of special classes of matrices.