Citation
Denby-Wilkes, John Edward (1950) Structure of Commutative Normed Rings. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/tr8p-j588. https://resolver.caltech.edu/CaltechTHESIS:06252025-185103530
Abstract
In a complex commutative normed ring the unit sphere at the origin has a vertex at the unit element. If the ring is finite dimensional, the radical translated to the unit element intersects this sphere only at the unit element.
A finite dimensional ring containing an element of nilpotency degree equal to the dimension or the radical is a direct sum of a ring with a scalar product and a ring with a convolution product. Using this decomposition the conjugate space is made into normed ring, and a duality theory is obtained.
General properties are given or completely continuous and weakly completely continuous elements or various types of rings.
In a star ring, if uniform convergence with respect to the maximal ideals implies weak convergence, then the square or a weakly completely continuous operator is completely continuous. Some of the consequences of this result are: (a) no infinite dimensional ring of this type is reflexive as a Banach space, (b) all weakly completely continuous elements or infinite dimensional indecomposable rings of this type lie in the radical, (c) a new proof or Dunford's theorem that the square of a weakly completely continuous operator from L into L is completely continuous is given.
Item Type: | Thesis (Dissertation (Ph.D.)) |
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Subject Keywords: | (Mathematics and Aeronautics) |
Degree Grantor: | California Institute of Technology |
Division: | Physics, Mathematics and Astronomy |
Major Option: | Mathematics |
Minor Option: | Aeronautics |
Thesis Availability: | Public (worldwide access) |
Research Advisor(s): |
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Thesis Committee: |
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Defense Date: | 1 January 1950 |
Record Number: | CaltechTHESIS:06252025-185103530 |
Persistent URL: | https://resolver.caltech.edu/CaltechTHESIS:06252025-185103530 |
DOI: | 10.7907/tr8p-j588 |
Default Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. |
ID Code: | 17486 |
Collection: | CaltechTHESIS |
Deposited By: | Benjamin Perez |
Deposited On: | 27 Jun 2025 22:33 |
Last Modified: | 27 Jun 2025 23:17 |
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