Citation
Dodds, Peter Gerard (1969) The Riesz space structure of an Abelian W*algebra. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/MBWY0552. https://resolver.caltech.edu/CaltechTHESIS:02222016142556483
Abstract
Let M be an Abelian W*algebra of operators on a Hilbert space H. Let M_{0} be the set of all linear, closed, densely defined transformations in H which commute with every unitary operator in the commutant M’ of M. A well known result of R. Pallu de Barriere states that if ɸ is a normal positive linear functional on M, then ɸ is of the form T → (Tx, x) for some x in H, where T is in M. An elementary proof of this result is given, using only those properties which are consequences of the fact that ReM is a Dedekind complete Riesz space with plenty of normal integrals. The techniques used lead to a natural construction of the class M_{0}, and an elementary proof is given of the fact that a positive selfadjoint transformation in M_{0} has a unique positive square root in M_{0}. It is then shown that when the algebraic operations are suitably defined, then M_{0} becomes a commutative algebra. If ReM_{0} denotes the set of all selfadjoint elements of M_{0}, then it is proved that ReM_{0} is Dedekind complete, universally complete Riesz spaces which contains ReM as an order dense ideal. A generalization of the result of R. Pallu de la Barriere is obtained for the Riesz space ReM_{0} which characterizes the normal integrals on the order dense ideals of ReM_{0}. It is then shown that ReM_{0} may be identified with the extended order dual of ReM, and that ReM_{0} is perfect in the extended sense.
Some secondary questions related to the Riesz space ReM are also studied. In particular it is shown that ReM is a perfect Riesz space, and that every integral is normal under the assumption that every decomposition of the identity operator has nonmeasurable cardinal. The presence of atoms in ReM is examined briefly, and it is shown that ReM is finite dimensional if and only if every order bounded linear functional on ReM is a normal integral.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Mathematics  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  7 April 1969  
Funders: 
 
Record Number:  CaltechTHESIS:02222016142556483  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:02222016142556483  
DOI:  10.7907/MBWY0552  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  9578  
Collection:  CaltechTHESIS  
Deposited By:  INVALID USER  
Deposited On:  23 Feb 2016 16:25  
Last Modified:  09 Nov 2022 19:20 
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