Commuting additive maps on tensor products of matrices.

Let k , n 1 , ... , n k be positive integers such that n i ⩾ 2 for i = 1 , ... , k and let M n i denote the algebra of n i × n i matrices over a field F for i = 1 , ... , k. Let ⨂ i = 1 k M n i be the tensor product of M n 1 , ... , M n k . We obtain a structural characterization of additive maps ψ : ⨂ i = 1 k M n i → ⨂ i = 1 k M n i satisfying ψ ⨂ i = 1 k A i ⨂ i = 1 k A i = ⨂ i = 1 k A i ψ ⨂ i = 1 k A i for all A 1 ∈ S n 1 , ... , A k ∈ S n k , where S n i = E s t (n i) + α E p q (n i) : α ∈ F , 1 ⩽ p , q , s , t ⩽ n i a r e n o t a l l d i s t i n c t i n t e g e r s E p q (n i) and E s t (n i) is the standard matrix unit in M n i for i = 1 , ... , k. In particular, we show that ψ : M n 1 → M n 1 is an additive map commuting on S n 1 if and only if there exist a scalar λ ∈ F and an additive map μ : M n 1 → F such that ψ (A) = λ A + μ (A) I n 1 for all A ∈ M n 1 . As an application, we classify additive maps ψ : ⨂ i = 1 k M n i → ⨂ i = 1 k M n i satisfying ψ (⨂ i = 1 k A i) (⨂ i = 1 k A i) = (⨂ i = 1 k A i) ψ (⨂ i = 1 k A i) for all A 1 ∈ R r 1 n 1 , ... , A k ∈ R r k n k . Here, R r i n i denotes the set of rank r i matrices in M n i and each 1 < r i ⩽ n i is a fixed integer such that r i ≠ n i when n i = 2 and | F | = 2 for i = 1 , ... , k. [ABSTRACT FROM AUTHOR]