ACTA issues

Centralizing additive maps on rank $r$ block triangular matrices

W. L. Chooi, M. H. A. Mutalib, L. Y. Tan

Acta Sci. Math. (Szeged) 87:1-2(2021), 75-106

Abstract. Let $\mathbb F $ be a field and let $k,n_1,\ldots ,n_k$ be positive integers with $n_1+\cdots +n_k=n\geqslant 2$. We denote by ${\cal T}_{n_1,\ldots ,n_k}$ a block triangular matrix algebra over $\mathbb F $ with unity $I_n$ and center $Z({\cal T}_{n_1,\ldots ,n_k})$. Fixing an integer $1<r\leq n$ with $r\neq n$ when $\left |\mathbb F \right |=2$, we prove that an additive map $\psi \colon {\cal T}_{n_1,\ldots ,n_k}\rightarrow {\cal T}_{n_1,\ldots ,n_k}$ satisfies $ \psi (A)A-A\psi (A)\in Z({\cal T}_{n_1,\ldots ,n_k})$ for every rank $r$ matrices $A\in {\cal T}_{n_1,\ldots ,n_k}$ if and only if there exist an additive map $\mu \colon {\cal T}_{n_1,\ldots ,n_k}\rightarrow \mathbb F $ and scalars $\lambda ,\alpha \in \mathbb F $, in which $\alpha \neq 0$ only if $r=n$, $n_1=n_k=1$ and $\left |\mathbb F \right |=3$, such that $ \psi (A)=\lambda A+\mu (A)I_n+\alpha (a_{11}+a_{nn})E_{1n} $ for all $A=(a_{ij})\in {\cal T}_{n_1,\ldots ,n_k}$, where $E_{ij}\in {\cal T}_{n_1,\ldots ,n_k}$ is the matrix unit whose $(i,j)$th entry is one and zero elsewhere. Using this result, a complete structural characterization of commuting additive maps on rank $s>1$ upper triangular matrices over an arbitrary field is addressed.

DOI: 10.14232/actasm-020-586-y

AMS Subject Classification (1991): 15A03, 15A04, 16R60

Keyword(s): centralizing map, commuting map, block triangular matrix, rank, functional identity

received 6.8.2020, revised 10.8.2020, accepted 13.8.2020. (Registered under 86/2020.)