Rank nonincreasing linear maps preserving the determinant of tensor product of matrices

Let l,m1,m2,…ml≥2l,m1,m2,…ml≥2 be positive integers. We describe some linear maps ϕ:Mm1…ml(F)→Mm1…ml(F)ϕ:Mm1…ml(F)→Mm1…ml(F) satisfyingdet⁡(ϕ(A1⊗…⊗Al))=det⁡(A1⊗…⊗Al),det⁡(ϕ(A1⊗…⊗Al))=det⁡(A1⊗…⊗Al), for all Ak∈Mmk(F)Ak∈Mmk(F), k=1,…,lk=1,…,l.