Characterization of Non-Linear Bi-Skew Jordan <i>n</i>-Derivations on Prime ∗-Algebras
Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">A</mi></semantics></math></inline-formula> be a prime *-algebra. A product defined as <inline-fo...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-07-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/8/753 |
| _version_ | 1797585566019092480 |
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| author | Asma Ali Amal S. Alali Mohd Tasleem |
| author_facet | Asma Ali Amal S. Alali Mohd Tasleem |
| author_sort | Asma Ali |
| collection | DOAJ |
| description | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">A</mi></semantics></math></inline-formula> be a prime *-algebra. A product defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>•</mo><mi>V</mi><mo>=</mo><mi>U</mi><msup><mi>V</mi><mo>∗</mo></msup><mo>+</mo><mi>V</mi><msup><mi>U</mi><mo>∗</mo></msup></mrow></semantics></math></inline-formula> for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>,</mo><mi>V</mi><mo>∈</mo><mi mathvariant="fraktur">A</mi></mrow></semantics></math></inline-formula>, is called a bi-skew Jordan product. A map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ξ</mi><mo>:</mo><mi mathvariant="fraktur">A</mi><mo>→</mo><mi mathvariant="fraktur">A</mi></mrow></semantics></math></inline-formula>, defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ξ</mi><mfenced separators="" open="(" close=")"><msub><mi>p</mi><mi>n</mi></msub><mfenced separators="" open="(" close=")"><msub><mi>U</mi><mn>1</mn></msub><mo>,</mo><msub><mi>U</mi><mn>2</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>U</mi><mi>n</mi></msub></mfenced></mfenced><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>p</mi><mi>n</mi></msub><mfenced separators="" open="(" close=")"><msub><mi>U</mi><mn>1</mn></msub><mo>,</mo><msub><mi>U</mi><mn>2</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>U</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>ξ</mi><mrow><mo>(</mo><msub><mi>U</mi><mi>k</mi></msub><mo>)</mo></mrow><mo>,</mo><msub><mi>U</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>U</mi><mi>n</mi></msub></mfenced></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>U</mi><mn>1</mn></msub><mo>,</mo><msub><mi>U</mi><mn>2</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>U</mi><mi>n</mi></msub><mo>∈</mo><mi mathvariant="fraktur">A</mi></mrow></semantics></math></inline-formula>, is called a non-linear bi-skew Jordan <i>n</i>-derivation. In this article, it is shown that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> is an additive ∗-derivation. |
| first_indexed | 2024-03-11T00:07:57Z |
| format | Article |
| id | doaj.art-de9561e4cd714d3fa100c9171f0af0b4 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T00:07:57Z |
| publishDate | 2023-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-de9561e4cd714d3fa100c9171f0af0b42023-11-19T00:14:43ZengMDPI AGAxioms2075-16802023-07-0112875310.3390/axioms12080753Characterization of Non-Linear Bi-Skew Jordan <i>n</i>-Derivations on Prime ∗-AlgebrasAsma Ali0Amal S. Alali1Mohd Tasleem2Department of Mathematics, Aligarh Muslim University, Aligarh 202002, IndiaDepartment of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi ArabiaDepartment of Mathematics, Aligarh Muslim University, Aligarh 202002, IndiaLet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">A</mi></semantics></math></inline-formula> be a prime *-algebra. A product defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>•</mo><mi>V</mi><mo>=</mo><mi>U</mi><msup><mi>V</mi><mo>∗</mo></msup><mo>+</mo><mi>V</mi><msup><mi>U</mi><mo>∗</mo></msup></mrow></semantics></math></inline-formula> for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>,</mo><mi>V</mi><mo>∈</mo><mi mathvariant="fraktur">A</mi></mrow></semantics></math></inline-formula>, is called a bi-skew Jordan product. A map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ξ</mi><mo>:</mo><mi mathvariant="fraktur">A</mi><mo>→</mo><mi mathvariant="fraktur">A</mi></mrow></semantics></math></inline-formula>, defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ξ</mi><mfenced separators="" open="(" close=")"><msub><mi>p</mi><mi>n</mi></msub><mfenced separators="" open="(" close=")"><msub><mi>U</mi><mn>1</mn></msub><mo>,</mo><msub><mi>U</mi><mn>2</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>U</mi><mi>n</mi></msub></mfenced></mfenced><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>p</mi><mi>n</mi></msub><mfenced separators="" open="(" close=")"><msub><mi>U</mi><mn>1</mn></msub><mo>,</mo><msub><mi>U</mi><mn>2</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>U</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>ξ</mi><mrow><mo>(</mo><msub><mi>U</mi><mi>k</mi></msub><mo>)</mo></mrow><mo>,</mo><msub><mi>U</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>U</mi><mi>n</mi></msub></mfenced></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>U</mi><mn>1</mn></msub><mo>,</mo><msub><mi>U</mi><mn>2</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>U</mi><mi>n</mi></msub><mo>∈</mo><mi mathvariant="fraktur">A</mi></mrow></semantics></math></inline-formula>, is called a non-linear bi-skew Jordan <i>n</i>-derivation. In this article, it is shown that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> is an additive ∗-derivation.https://www.mdpi.com/2075-1680/12/8/753∗-derivationbi-skew Jordan n-derivationprime ∗-algebras |
| spellingShingle | Asma Ali Amal S. Alali Mohd Tasleem Characterization of Non-Linear Bi-Skew Jordan <i>n</i>-Derivations on Prime ∗-Algebras Axioms ∗-derivation bi-skew Jordan n-derivation prime ∗-algebras |
| title | Characterization of Non-Linear Bi-Skew Jordan <i>n</i>-Derivations on Prime ∗-Algebras |
| title_full | Characterization of Non-Linear Bi-Skew Jordan <i>n</i>-Derivations on Prime ∗-Algebras |
| title_fullStr | Characterization of Non-Linear Bi-Skew Jordan <i>n</i>-Derivations on Prime ∗-Algebras |
| title_full_unstemmed | Characterization of Non-Linear Bi-Skew Jordan <i>n</i>-Derivations on Prime ∗-Algebras |
| title_short | Characterization of Non-Linear Bi-Skew Jordan <i>n</i>-Derivations on Prime ∗-Algebras |
| title_sort | characterization of non linear bi skew jordan i n i derivations on prime ∗ algebras |
| topic | ∗-derivation bi-skew Jordan n-derivation prime ∗-algebras |
| url | https://www.mdpi.com/2075-1680/12/8/753 |
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