In this thesis we investigate certain semi-metrics 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 { / x-y / } , 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 von-Neumann algebra. He then made use of this metric in obtaining a classification of the various incomplete tensor products of a family of semi-finite W* -algebras, up to a natural type of equivalence known as product isomorphism.
By removing the semi-finiteness 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 semi-metric 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 semi-finite factors, we obtain explicit criteria for such a classification by calculating d(G)(μ, v) in terms of the Radon-Nikodym 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 self-adjoint 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. / Science, Faculty of / Mathematics, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/35276 |
| Date | January 1970 |
| Creators | Promislow, S. David |
| Publisher | University of British Columbia |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
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