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 Semimetrics on the normal states of a W*algebra
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Semimetrics on the normal states of a W*algebra Promislow, S. David
Abstract
In this thesis we investigate certain semimetrics defined on the normal states of a W* algebra and their applications to infinite tensor products. This extends the work of Bures, who defined a metric d on the set of normal states by taking d(μ,v) = inf {; xy; } , where the infimum is taken over all vectors x and y which induce the states μ and v respectively relative to any representation of the algebra as a vonNeumann algebra. He then made use of this metric in obtaining a classification of the various incomplete tensor products of a family of semifinite W* algebras, up to a natural type of equivalence known as product isomorphism. By removing the semifiniteness restriction form Bures' "product formula", which relates the distance under d between two finite product states to the distances between their components, we obtain this tensor product classification for families of arbitrary W* algebras. Moreover we extend the product formula to apply to the case of infinite product states. For any subgroup G of the *automorphism group of a W*algebra, we define the semimetric d(G) on the set of normal states by: d(G) (μ,v) = inf {d(μ(α) ,v (β) : α,β ε G} ; where μ(α).a is defined by μ(α)(A) = μ(α(A)). We show the significance of d(G) in classifying incomplete tensor products up to weak product isomorphism, a natural weakening of the concept of product isomorphism. In the case of tensor products of semifinite factors, we obtain explicit criteria for such a classification by calculating d(G)(μ, v) in terms of the RadonNikodym derivatives of the states. In the course of this calculation we introduce a concept of compatibility which yields some other results about d and d(G) . Two selfadjoint operators S and T are said to be compatible, if given any real numbers α and β , either E(α) ≤ F((β) or F(β) ≤ E(α) ; where {E(λ)} , (F(λ)} , are the spectral resolutions of S,T , respectively. We obtain some miscellaneous results concerning this concept.
Item Metadata
Title 
Semimetrics on the normal states of a W*algebra

Creator  
Publisher 
University of British Columbia

Date Issued 
1970

Description 
In this thesis we investigate certain semimetrics defined on the normal states of a W* algebra and their applications to infinite tensor products.
This extends the work of Bures, who defined a metric d on the set of normal states by taking d(μ,v) = inf {; xy; } , where the infimum is taken over all vectors x and y which induce the states μ and v respectively relative to any representation of the algebra as a vonNeumann algebra. He then made use of this metric in obtaining a classification of the various incomplete tensor products of a family of semifinite W* algebras, up to a natural type of equivalence known as product isomorphism.
By removing the semifiniteness restriction form Bures' "product formula", which relates the distance under d between two finite product states to the distances between their components, we obtain this tensor product classification for families of arbitrary W* algebras. Moreover we extend the product formula to apply to the case of infinite product states.
For any subgroup G of the *automorphism group of a W*algebra, we define the semimetric d(G) on the set of normal states by: d(G) (μ,v) = inf {d(μ(α) ,v (β) : α,β ε G} ; where μ(α).a is defined by μ(α)(A) = μ(α(A)). We show the significance of d(G) in classifying incomplete tensor products up to weak product isomorphism, a natural weakening of the concept of product isomorphism.
In the case of tensor products of semifinite factors, we obtain explicit criteria for such a classification by calculating d(G)(μ, v) in terms of the RadonNikodym derivatives of the states.
In the course of this calculation we introduce a concept of compatibility which yields some other results about d and d(G) . Two selfadjoint operators S and T are said to be compatible, if given any real numbers α and β , either E(α) ≤ F((β) or F(β) ≤ E(α) ; where {E(λ)} , (F(λ)} , are the spectral resolutions of S,T , respectively. We obtain some miscellaneous results concerning this concept.

Genre  
Type  
Language 
eng

Date Available 
20110607

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080513

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.