Centralizing additive maps on rank Rblock triangular matrices / Muhammad Hazim Abdul Mutalib - Muhammad Hazim , Abdul Mutalib (2021)
In this dissertation, we study centralizing additive maps on block triangular matrix algebras. The main focus of this research is to classify centralizing additive maps on rank r block triangular matrices over an arbitrary field. Let k, n1, nk be positive integers with n1 + · · · + nk = n ! 2. Let Tn1,...,nk be the n1, . . . ,nk block triangular matrix algebra over a field F with center Z(Tn1,...,nk) and unity In. We first obtain a characterization of centralizing additive maps on Tn1,...,nk . Then, by using this preliminary result together with the classification of rank factorization of block triangular matrices, we characterize centralizing additive maps : Tn1,...,nk !Tn1,...,nk on rank r block triangular matrices, i.e, additive maps satisfying A (A) − (A)A 2 Z(Tn1,...,nk) for all rank r matrices A 2 Tn1,...,nk , where r is a fixed integer 1 < r  n such that r 6= n when F is the Galois field of two elements, and we prove these additive maps are of the form (A) = "A + μ(A)In + ↵(a11 + ann)E1n for all A = (aij) 2 Tn1,...,nk , where μ : Tn1,...,nk ! F is an additive map, ", ↵ 2 F are scalars in which ↵ 6= 0 only if r = n, n1 = nk = 1 and |F| = 3, and E1n 2 Tn1,...,nk is the standard matrix unit whose (1, n)th entry is one and zero elsewhere. Using this result, together with the recent works on commuting additive maps on upper triangular matrices, we give a complete description of commuting additive maps on rank r > 1 upper triangular matrices.