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Structure of Commutative Normed Rings

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.))
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):
  • Karlin, Samuel (advisor)
  • Bohnenblust, Henri Frederic (advisor)
Thesis Committee:
  • Unknown, Unknown
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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