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