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

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文摘

Let l,m₁,m₂,…m_l≥2

l, m_{1}, m_{2}, \dots m_{l} \geq 2

be positive integers. We describe some linear maps ϕ:M_{m₁…m_l}(F)→M_{m₁…m_l}(F)

ϕ : M_{m_{1} \dots m_{l}} (F) \to M_{m_{1} \dots m_{l}} (F)

satisfying

det⁡(ϕ(A₁⊗…⊗A_l))=det⁡(A₁⊗…⊗A_l),

\det (ϕ (A_{1} \otimes \dots \otimes A_{l})) = \det (A_{1} \otimes \dots \otimes A_{l}),

for all A_k∈M_{m_k}(F)

A_{k} \in M_{m_{k}} (F)

, k=1,…,l

k = 1, \dots, l

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