Citation
Talmadge, Richard Bennett (1951) The Representation of Baire Functions. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/7K44T556. https://resolver.caltech.edu/CaltechTHESIS:10062017095007634
Abstract
Gelfand [1]^{1} has shown that a real Banach algebra in which for every element we have x^{2} = x^{2}, is isomorphic and isometric to the ring continuous functions on some compact Hausdorff space. Since he was concerned with an abstract Banach algebra, his representation for this space is necessarily quite complicated; indeed, it is in terms of a space of maximal ideals of the Banach algebra. One would expect, then, that for a particular Banach algebra a simpler characterization of this space would be obtained. It is the purpose of this paper to find such a simpler representation for the collection of Baire functions of class a, for each a ≥, over a topological space S. These collections satisfy the conditions of Gelfand's theorem. Our representation, which is done in terms of lattice, instead of ring, operations, will give the space as a Boolean space associated with a Boolean algebra of subsets of the original space S.
The paper is divided into two parts. In part I, we define the Baire functions of class a and obtain some results connecting them and the Boolean algebra. Part II is concerned with the representation theorem, some of its consequences, and examples to show that the theory is nonvacuous.
1. References to the literature are indicated by numbers in square brackets.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  (Mathematics and Physics) 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Minor Option:  Physics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  1 January 1951 
Record Number:  CaltechTHESIS:10062017095007634 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:10062017095007634 
DOI:  10.7907/7K44T556 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  10493 
Collection:  CaltechTHESIS 
Deposited By:  Benjamin Perez 
Deposited On:  06 Oct 2017 18:03 
Last Modified:  04 May 2023 20:35 
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