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Automatic continuity and C_0(\Omega)-linearity of linear maps between C_0(\Omega)-modules

Chi-Wai Leung Chi-Keung Ng Ngai-Ching Wong

Functional Analysis mathscidoc:1910.43680

arXiv preprint arXiv:1005.4561
Let be a locally compact Hausdorff space. We show that any local -linear map (where" local" is a weaker notion than -linearity) between Banach -modules are" nearly -linear" and" nearly bounded". As an application, a local -linear map between Hilbert -modules is automatically -linear. If, in addition, contains no isolated point, then any -linear map between Hilbert -modules is automatically bounded. Another application is that if a sequence of maps between two Banach spaces" preserve -sequences"(or" preserve ultra- -sequences"), then is bounded for large enough and they have a common bound. Moreover, we will show that if is a bijective" biseparating" linear map from a" full" essential Banach -module into a" full" Hilbert -module (where is another locally compact Hausdorff space), then is" nearly bounded"(in fact, it is automatically bounded if or contains no isolated point) and there exists a homeomorphism such that ( ).
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@inproceedings{chi-waiautomatic,
  title={Automatic continuity and C_0(\Omega)-linearity of linear maps between C_0(\Omega)-modules},
  author={Chi-Wai Leung, Chi-Keung Ng, and Ngai-Ching Wong},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020210432524352209},
  booktitle={arXiv preprint arXiv:1005.4561},
}
Chi-Wai Leung, Chi-Keung Ng, and Ngai-Ching Wong. Automatic continuity and C_0(\Omega)-linearity of linear maps between C_0(\Omega)-modules. In arXiv preprint arXiv:1005.4561. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020210432524352209.
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