In the classical theory of operators on a Banach space a beautiful interplay exists between Riesz and Fredholm theory, and the theory of traces and de¬terminants for operator ideals. In this thesis we obtain a complete Riesz de¬composition theorem for Riesz elements in a semi prime Banach algebra and on the other hand extend the existing theory of traces and determinants to a more general setting of Banach algebras. In order to obtain some of these results we use the notion of finite multiplicity of spectral points to give a characterization of the essential spec¬trum for elements in a Banach algebra. As an immediate corollary we obtain the well-known characterization of Riesz elements namely that their non-zero spectral points are isolated and of finite multiplicities. In the final chapter of the thesis we use Plemelj's type formulas to define a determinant on the ideal of finite rank elements and show that it extends continuously to the ideal of nuclear elements. / Thesis (PhD (Mathematics))--University of Pretoria, 2006. / Mathematics and Applied Mathematics / unrestricted

Let A be a von Neumann algebra of linear operators on the Hilbert space H . A linear operator T (resp. a linear bounded. functional ϕ ) on A is said to be normal if for any increasing net [formula omitted] of positive elements in A with least upper bound B , T(B) is the least upper bound of [formula omitted]. Two linear positive functionals ψ1 and ψ2 on A are said to be equivalent if ψ1 (B) = 0 <=> ψ2 (B) = 0 for any positive element B in A. Let ϕ0 be a positive normal linear functional on A . Let S be a semigroup and, {T(s) : s ε S} an antirepresentation of S as normal positive linear contraction operators on A . We find in this thesis equivalent conditions for the existence of a positive normal linear functional ϕ on A which is equivalent to ϕ0 and invariant under the semigroup {T(s) : s ε S} (i.e. ϕ(T(s)B) = ϕ(B) for all B in A and s ε S ). We also extend the concept of weakly-wandering sets, which was first introduced by Hajian-Kakutani, to weakly-wandering projections in A. We give a relation between the non-existence of weakly-wandering projections in A and the existence of positive normal linear functionals on A, invariant with respect to an antirepresentation {T(s) : s ε S} of normal *-homomorphisms on A . Finally we investigate the existence of a complete set of positive normal linear functionals on A which are invariant under the semigroup {T(s) : s ε S}. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis is a study of several spaces of continuous linear functionals on various function spaces with a natural norm inherited from a larger Banach space. The completeness of these normed linear spaces is studied in detail and several necessary and sufficient conditions are obtained in this regard. Since spaces of continuous linear functionals are inherently related to spaces of measures, their measure-theoretic counterparts are also studied. By using these counterparts, several necessary and sufficient conditions are obtained on the separability of these spaces of continuous linear functionals. / Ph. D.

Discusses certain one-parameter subgroups of the low-rank unitary groups called root subgroups. Unitary groups also have representations of Lie type which means they consist of transformations that act as automorphisms of an underlying Lie algebra, in this case the special linear algebra. Exploring this definition of the unitary groups, we find a correlation, via exponentiation, to the basis elements of Lie algebra.

The goal of this paper is to establish the dependency of the topology of a simple Lie group, specifically any of the special linear groups, on its underlying group structure. The intimate relationship between a Lie group's topology and its algebraic structure dictates some necessary topological properties, such as second countability. However, the extent to which a Lie group's topology is an "algebraic phenomenon" is, to date, still not known.

We provide a new relative trace formula approach to the theorem of Waldspurger on toric periods for GL₂, with possible applications to the global Gross-Prasad conjecture for orthogonal groups.

Thesis (M.S.)--Brigham Young University. Dept of Mathematics, 2006. / Includes bibliographical references (p. 30).

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Please refer to the thesis for the abstract. / Thesis (Ph.D.)-University of Natal, Durban, 1998.

The aim of this dissertation is to describe the conjugacy classes and some of the ordinary irreducible characters of the nite general linear group GL(n, q); together with character tables of some of its subgroups. We study the structure of GL(n, q) and some of its important subgroups such as SL(n, q); UT(n, q); SUT(n, q); Z(GL(n, q)); Z(SL(n, q)); GL(n, q)0 ; SL(n, q)0 ; the Weyl group W and parabolic subgroups P : In addition, we also discuss the groups PGL(n, q); PSL(n, q) and the a ne group A (n, q); which are related to GL(n, q): The character tables of GL(2; q); SL(2; q); SUT(2; q) and UT(2; q) are constructed in this dissertation and examples in each case for q = 3 and q = 4 are supplied. A complete description for the conjugacy classes of GL(n, q) is given, where the theories of irreducible polynomials and partitions of i 2 f1; 2; ; ng form the atoms from where each conjugacy class of GL(n, q) is constructed. We give a special attention to some elements of GL(n, q); known as regular semisimple, where we count the number and orders of these elements. As an example we compute the conjugacy classes of GL(3; q): Characters of GL(n, q) appear in two series namely, principal and discrete series characters. The process of the parabolic induction is used to construct a large number of irreducible characters of GL(n, q) from characters of GL(n, q) for m < n: We study some particular characters such as Steinberg characters and cuspidal characters (characters of the discrete series). The latter ones are of particular interest since they form the atoms from where each character of GL(n, q) is constructed. These characters are parameterized in terms of the Galois orbits of non-decomposable characters of F q n: The values of the cuspidal characters on classes of GL(n, q) will be computed. We describe and list the full character table of GL(n, q): There exists a duality between the irreducible characters and conjugacy classes of GL(n, q); that is to each irreducible character, one can associate a conjugacy class of GL(n, q): Some aspects of this duality will be mentioned. / Thesis (M.Sc. (School of Mathematical Sciences)) - University of KwaZulu-Natal, Pietermaritzburg, 2008.