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Title: Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective (English)
Author: Liang, Xinfeng
Author: Wei, Feng
Author: Fošner, Ajda
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 69
Issue: 3
Year: 2019
Pages: 713-761
Summary lang: English
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Category: math
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Summary: 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. (English)
Keyword: generalized matrix algebra
Keyword: commuting trace
Keyword: centralizing trace
Keyword: Lie isomorphism
Keyword: Lie triple isomorphism
MSC: 15A78
MSC: 16R60
MSC: 16W10
idZBL: Zbl 07088814
idMR: MR3989276
DOI: 10.21136/CMJ.2019.0507-17
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Date available: 2019-07-24T11:18:08Z
Last updated: 2021-10-04
Stable URL: http://hdl.handle.net/10338.dmlcz/147787
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