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.
Similar content being viewed by others
References
Armario, R., García-Pacheco, F.J., Pérez-Fernández, F.J.: On the Krein–Milman property and the Bade property. Linear Algebra Appl. 436(5), 1489–1502 (2012)
Arnautov, V.I., Glavatsky, S.T., Mikhalev, A.V.: Introduction to the Theory of Topological Rings and Modules, Monographs and Textbooks in Pure and Applied Mathematics, vol. 197. Marcel Dekker Inc., New York (1996)
Bourbaki, N.: Topological vector spaces. In: Chapters 1–5: Elements of Mathematics (Berlin). Springer, Berlin (1987) (Translated from the French by H.G. Eggleston and S. Madan)
Cobos-Sánchez, C., García-Pacheco, F.J., Moreno-Pulido, S., Sáez-Martínez, S.: Supporting vectors of continuous linear operators. Ann. Funct. Anal. 8(4), 520–530 (2017)
Diestel, J.: Geometry of Banach Spaces—Selected Topics. Lecture Notes in Mathematics, vol. 485. Springer, Berlin (1975)
Eisele, K.T., Taieb, S.: Weak topologies for modules over rings of bounded random variables. J. Math. Anal. Appl. 421(2), 1334–1357 (2015)
García-Pacheco, F.J.: Complementation of the subspace of \(G\)-invariant vectors. J. Algebra Appl. 16(7), 1750124, 7 (2017)
García-Pacheco, F.J., Murillo-Arcila, M., Miralles-Montolío, A.: The spectral decomposition in unitary algebras (2019) (submitted)
García-Pacheco, F.J., Piniella, P.: Unit neighborhoods in topological rings. Banach J. Math. Anal. 9(4), 234–242 (2015)
García-Pacheco, F.J., Piniella, P.: Linear topologies and sequential compactness in topological modules. Quaest. Math. 40(7), 897–908 (2017)
García-Pacheco, F.J., Piniella, P.: Geometry of balanced and absorbing subsets of topological modules. J. Algebra Appl. 18(6), 1950119, 13 (2019)
García-Pacheco, F.J., Pérez-Fernández, F.J.: Pre-schauder bases in topological vector spaces. Symmetry 11(8), 1026 (2019)
Köthe, G.: Topological Vector Spaces. I. Translated from the German by D. J. H. Garling. Die Grundlehren der mathematischen Wissenschaften, Band 159. Springer, New York (1969)
Megginson, R.E.: An Introduction to Banach Space Theory. Graduate Texts in Mathematics, vol. 183. Springer, New York (1998)
Nowak, P.W.: Group actions on Banach spaces. In: Handbook of Group Actions. Vol. II. Adv. Lect. Math. (ALM), vol. 32, pp. 121–149. Int. Press, Somerville (2015)
Piniella, P.: Existence of non-trivial complex unit neighborhoods. Carpathian J. Math. 33(1), 107–114 (2017)
Sakai, S.: \(C^*\)-algebras and \(W^*\)-algebras. Classics in Mathematics. Springer, Berlin (1998) (reprint of the 1971 edition)
Shulman, T.: On subspaces of invariant vectors. Stud. Math. 236(1), 1–11 (2017)
Warner, S.: Topological Fields. North-Holland Mathematics Studies. Notas de Matemática (Mathematical Notes), vol. 157, p. 126. North-Holland Publishing Co., Amsterdam (1989)
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
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by Krzysztof Jarosz.
Rights and permissions
About this article
Cite this article
García-Pacheco, F.J., Sáez-Martínez, S. Normalizing rings. Banach J. Math. Anal. 14, 1143–1176 (2020). https://doi.org/10.1007/s43037-020-00055-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s43037-020-00055-0
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