Commuting additive maps on tensor products of matrices

• Main Authors: Chooi, Wai Leong, Wong, Jian Yong
• Format: Article
• Published: Taylor & Francis Ltd 2022
• Subjects: QA Mathematics
• Online Access: http://eprints.um.edu.my/46308/
• Institution: Universiti Malaya

Let k, n(1),..., n(k) be positive integers such that n(i) >= 2 for i = 1,..., k and let M-ni denote the algebra of n(i) x n(i) matrices over a field F for i = 1,..., k. Let circle times(i=1)k M-ni be the tensor product of M-n1,..., M-nk. We obtain a structural characterization of additive maps psi : circle times(i=1)k M-ni -> circle times(i=1)k M-ni satisfying psi(circle times(k)(i=1) A(i)) (circle times(k)(i=1) A(i)) = (circle times(k)(i=1) Ai) psi (circle times(k)(i=1) A(j)) for all A(1) is an element of S-n1,..., A(k) is an element of S-nk, where S-ni = {E-st((ni)) + alpha E-pq((ni)) : alpha is an element of F, 1 <= p, q, s, t <= n(i) are not all distinct integers} and E-st((ni)) is the standard matrix unit in M-ni for i = 1,..., k. In particular, we show that psi : M-n1 -> M-n1 is an additive map commuting on S-n1 if and only if there exist a scalar lambda is an element of F and an additive map mu : M-n1 -> F such that psi(A) = lambda A + mu(A)I-n1 for all A is an element of M-n1. As an application, we classify additive maps psi : circle times(i=1)k M-ni -> circle times(i=1)k M-ni satisfying psi(circle times(i=1)k A(i))(circle times(i=1)k A(i)) = (circle times(i=1)k A(i))psi(circle times(i=1)k A(i)) for all A(1) is an element of R-r1(n1),..., A(k) is an element of R-rk(nk) . Here, R-ri(ni) denotes the set of rank r(i) matrices in M-ni and each 1 < r(i) <= n(i) is a fixed integer such that r(i) not equal n(i) when n(i) = 2 and vertical bar F vertical bar = 2 for i = 1,..., k.