Czechoslovak Mathematical Journal, Vol. 69, No. 3, pp. 713-761, 2019
Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective
Xinfeng Liang, Feng Wei, Ajda Fošner
Received November 4, 2017. Published online May 17, 2019.
Abstract: Let $\mathcal{R}$ be a commutative ring, $\mathcal{G}$ be a generalized matrix algebra over $\mathcal{R}$ with weakly loyal bimodule and $\mathcal{Z}(\mathcal{G})$ be the center of $\mathcal{G}$. Suppose that $\mathfrak{q}\colon\mathcal{G}\times\mathcal{G} \rightarrow\mathcal{G}$ is an \hbox{$\mathcal{R}$-bilinear} mapping and that $\mathfrak{T}_{\mathfrak{q}}\colon\mathcal{G}\rightarrow\mathcal{G}$ is a trace of $\mathfrak{q}$. The aim of this article is to describe the form of $\mathfrak{T}_{\mathfrak{q}}$ satisfying the centralizing condition $[\mathfrak{T}_{\mathfrak{q}}(x), x]\in\mathcal{Z(G)}$ (and commuting condition $[\mathfrak{T}_{\mathfrak{q}}(x), x]=0$) for all $x\in\mathcal{G}$. More precisely, we will revisit the question of when the centralizing trace (and commuting trace) $\mathfrak{T}_{\mathfrak{q}}$ has the so-called proper form from a new perspective. Using the aforementioned trace function, we establish sufficient conditions for each Lie-type isomorphism of $\mathcal{G}$ to be almost standard. As applications, centralizing (commuting) traces of bilinear mappings and Lie-type isomorphisms on full matrix algebras and those on upper triangular matrix algebras are totally determined.
Affiliations: Xinfeng Liang, School of Mathematics and Big Data, Anhui University of Science & Technology, Huainan, 232001, P. R. China. e-mail: xfliang@aust.edu.cn; Feng Wei (corresponding author), School of Mathematics and Statistics, Beijing Institute of Technology, 5 South Zhongguancun Street, Haidian District, Beijing, 100081, P. R. China, e-mail: daoshuo@hotmail.com, daoshuowei@gmail.com; Ajda Fošner, Faculty of Management, University of Primorska, Cankarjeva 5, SI-6104 Koper, Slovenia, e-mail: ajda.fosner@fm-kp.si
