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Semi-metrics on the normal states of a W*-algebra Promislow, S. David


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.

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