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Much of operator theory hangs its coat on the spectral theorem, but the latter is exclusive to normal operators. Likewise, isometries are well understood via the Wold decomposition. It is von Neumann's inequality that enables a functional calculus for arbitrary contractions on Hilbert spaces. There are essentially two avenues that lead to von Neumann, one being the analytical theory of positive maps, the other marked by geometric dilation theorems. These diverse lines of approach are in fact unified by the inequality. Although our main focus is von Neumann's inequality, for which we provide four different proofs, we shall, however, periodically indulge in some of its intricate cousins.

Thesis (Ph. D.)--University of Missouri-Columbia, 2007. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on October 9, 2007) Vita. Includes bibliographical references.

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

The aim of this thesis is to generalize the classical flow under a function construction to non-abelian W*-algebras. We obtain existence and uniqueness theorems for this generalization. As an application we show that the relationship between a continuous and a discrete decomposition of a properly infinite W*-algebra is that of generalized flow under a function. Since continuous decompositions are known to exist for any properly infinite W*-algebra, this leads to generalizations of Connes' results on discrete decomposition. / Science, Faculty of / Mathematics, Department of / Graduate

For the algebraic group SLl+1(C) we describe a system of positive roots associated to conjugacy classes in its Weyl group Sl+1. Using this we explicitly describe the algebra of regular functions on certain transverse slices to conjugacy classes in SLl+1(C) as a polynomial algebra of invariants. These may be viewed as an algebraic group analogue of certain parabolic invariants that generate the W-algebra in type A found by Brundan and Kleshchev.

Thesis (Ph. D.)--University of Oregon, 2008. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 105-107). Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

Thesis (M.Sc.)(Mathematics)--University of Pretoria, 2004. / Title from opening screen (viewed March 11th, 2005). Includes summary. Includes bibliographical references.

Typescript. Includes vita and abstract. Includes bibliographical references (leaves 112-114) Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

No abstract / Mathematical Sciences / (D. Phil. (Mathematics))