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Normalizing rings

Abstract

In this manuscript, we transport the classical Operator Theory on complex Banach spaces to normed modules over absolutely valued rings. In some cases, we are able to extend classical results on complex Banach spaces to normed modules over normed rings. In order to make sure that bounded linear maps on normed modules coincide with the continuous linear maps, it is sufficient that the underlying ring be practical (a topological ring is practical if the invertibles approach zero). In order for other classical results to work on the scope of normed modules, it is sufficient that the underlying ring be normalizing. Normalizing rings is a new class of rings introduced in this manuscript. We provide a characterization of such rings as well as nontrivial examples.

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Acknowledgements

The author would like to deeply thank the referee for the valuable comments and remarks that helped improve the paper. The author has been supported by Research Grant PGC-101514-B-100 awarded by the European Regional Development Fund and the Ministry of Science, Innovation and Universities of Spain.

Author information

Correspondence to Francisco Javier García-Pacheco.

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The authors declare that they have no conflict of interest.

Additional information

Communicated by Krzysztof Jarosz.

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García-Pacheco, F.J., Sáez-Martínez, S. Normalizing rings. Banach J. Math. Anal. (2020). https://doi.org/10.1007/s43037-020-00055-0

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Keywords

  • Normed module
  • Absolutely valued ring
  • Bounded linear operator
  • Closed unit neighborhood of zero
  • Normalizing ring
  • (Left-feasible) neighborhood of zero
  • Feasible dual topology
  • Practical topological ring
  • Hahn–Banach module

Mathematics Subject Classification

  • 46H25
  • 16W80
  • 54H13