Linear rank preservers of tensor products of rank one matrices

Let n1,…,nkn1,…,nk be integers larger than or equal to 2. We characterize linear maps ϕ:Mn1⋯nk→Mn1⋯nkϕ:Mn1⋯nk→Mn1⋯nk such thatrank(ϕ(A1⊗⋯⊗Ak))=1wheneverrank(A1⊗⋯⊗Ak)=1 for all Ai∈Mni,i=1,…,k. Applying this result, we extend two recent results on linear maps that preserve the rank of special classes of matrices.