Gleason-kahane-Żelazko theorem for spectrally bounded algebra
We prove by elementary methods the following generalization of a theorem due to Gleason, Kahane, and Żelazko. Let A be a real algebra with unit 1 such that the spectrum of every element in A is bounded and let φ:A→ℂ be a linear map such that φ(1)=1 and (φ(a))2+(φ(b))2≠0 for all a, b in A satisfying...
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| Format: | Article |
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Wiley
2005-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS.2005.2447 |
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| author | S. H. Kulkarni D. Sukumar |
| author_facet | S. H. Kulkarni D. Sukumar |
| author_sort | S. H. Kulkarni |
| collection | DOAJ |
| description | We prove by elementary methods the following generalization of a theorem due to Gleason, Kahane, and Żelazko. Let A be a real algebra with unit 1 such that the spectrum of every element in A is bounded and let φ:A→ℂ be a linear map such that φ(1)=1 and (φ(a))2+(φ(b))2≠0 for all a, b in A satisfying ab=ba and a2+b2 is invertible. Then φ(ab)=φ(a)φ(b) for all a, b in A. Similar results are proved for real and complex algebras using Ransford's concept of generalized spectrum. With these ideas, a sufficient condition for a linear transformation to be multiplicative is established in terms of generalized spectrum. |
| format | Article |
| id | doaj-art-22c8a96054874bc1b9572e7d8629bbc1 |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2005-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-22c8a96054874bc1b9572e7d8629bbc12025-08-20T02:04:37ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252005-01-012005152447246010.1155/IJMMS.2005.2447Gleason-kahane-Żelazko theorem for spectrally bounded algebraS. H. Kulkarni0D. Sukumar1Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, IndiaDepartment of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, IndiaWe prove by elementary methods the following generalization of a theorem due to Gleason, Kahane, and Żelazko. Let A be a real algebra with unit 1 such that the spectrum of every element in A is bounded and let φ:A→ℂ be a linear map such that φ(1)=1 and (φ(a))2+(φ(b))2≠0 for all a, b in A satisfying ab=ba and a2+b2 is invertible. Then φ(ab)=φ(a)φ(b) for all a, b in A. Similar results are proved for real and complex algebras using Ransford's concept of generalized spectrum. With these ideas, a sufficient condition for a linear transformation to be multiplicative is established in terms of generalized spectrum.http://dx.doi.org/10.1155/IJMMS.2005.2447 |
| spellingShingle | S. H. Kulkarni D. Sukumar Gleason-kahane-Żelazko theorem for spectrally bounded algebra International Journal of Mathematics and Mathematical Sciences |
| title | Gleason-kahane-Żelazko theorem for spectrally bounded algebra |
| title_full | Gleason-kahane-Żelazko theorem for spectrally bounded algebra |
| title_fullStr | Gleason-kahane-Żelazko theorem for spectrally bounded algebra |
| title_full_unstemmed | Gleason-kahane-Żelazko theorem for spectrally bounded algebra |
| title_short | Gleason-kahane-Żelazko theorem for spectrally bounded algebra |
| title_sort | gleason kahane zelazko theorem for spectrally bounded algebra |
| url | http://dx.doi.org/10.1155/IJMMS.2005.2447 |
| work_keys_str_mv | AT shkulkarni gleasonkahanezelazkotheoremforspectrallyboundedalgebra AT dsukumar gleasonkahanezelazkotheoremforspectrallyboundedalgebra |