Centralizing additive maps on rank Rblock triangular matrices / Muhammad Hazim Abdul Mutalib

Muhammad Hazim , Abdul Mutalib (2021) Centralizing additive maps on rank Rblock triangular matrices / Muhammad Hazim Abdul Mutalib. Masters thesis, Universiti Malaya.

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      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.

      Item Type: Thesis (Masters)
      Additional Information: Dissertation (M.A.) – Faculty of Science, Universiti Malaya, 2021.
      Uncontrolled Keywords: Centralizing additive maps; Commuting additive maps; Block triangular matrices; Upper triangular matrices; Ranks
      Subjects: Q Science > QA Mathematics
      Divisions: Faculty of Science
      Depositing User: Mr Mohd Safri Tahir
      Date Deposited: 01 Mar 2022 07:26
      Last Modified: 01 Mar 2022 07:26
      URI: http://studentsrepo.um.edu.my/id/eprint/12916

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