The prototype of a near-ring is the set of all self-maps of an additively written (but not necessarily abelian) group under pointwise addition and composition of maps. Moreover, any near-ring with unity can be embedded in a near-ring (with unity) of self-maps of some group. For this reason, a lot of research has been done on near-rings of maps. In 1979, Hofer [16] gave the study of near-rings of maps a topological avour by considering the near- ring of all continuous self-maps of a topological group. In this dissertation we consider some standard constructions of near-rings of maps on a group G and investigate these when G is a topological group and our near-ring consists of continuous maps.

Clifford algebras have been studied for many years and their algebraic properties are well known. In particular, all Clifford algebras have been classified as matrix algebras over one of the three division algebras. But Clifford Algebras are far more interesting than this classification suggests; they provide the algebraic basis for a unified language for physics and mathematics which offers many advantages over current techniques. This language is called geometric algebra - the name originally chosen by Clifford for his algebra - and this thesis is an investigation into the properties and applications of Clifford's geometric algebra. The work falls into three broad categories: - The formal development of geometric algebra has been patchy and a number of important subjects have not yet been treated within its framework. A principle feature of this thesis is the development of a number of new algebraic techniques which serve to broaden the field of applicability of geometric algebra. Of particular interest are an extension of the geometric algebra of spacetime (the spacetime algebra) to incorporate multiparticle quantum states, and the development of a multivector calculus for handling differentiation with respect to a linear function. - A central contention of this thesis is that geometric algebra provides the natural language in which to formulate a wide range of subjects from modern mathematical physics. To support this contention, reformulations of Grassmann calculus, Lie algebra theory, spinor algebra and Lagrangian field theory are developed. In each case it is argued that the geometric algebra formulation is computationally more efficient than standard approaches, and that it provides many novel insights. - The ultimate goal of a reformulation is to point the way to new mathematics and physics, and three promising directions are developed. The first is a new approach to relativistic multiparticle quantum mechanics. The second deals with classical models for quantum spin-I/2. The third details an approach to gravity based on gauge fields acting in a fiat spacetime. The Dirac equation forms the basis of this gauge theory, and the resultant theory is shown to differ from general relativity in a number of its features and predictions.

In this thesis we consider the group of fractional linear transformations of a variable x over an algebraically closed field k. The purpose of the thesis is to determine all finite subgroups of this group whose orders are not divisible by the characteristic of k. / Science, Faculty of / Mathematics, Department of / Graduate

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

M.Sc. / The aim of this dissertation will be an investigation into a classical result which asserts the uniqueness of the norm topology on a semi-simple Banach algebra. For a commutative semi-simple Banach algebra, say A, it is relatively simple matter, with the aid of the Closed Graph Theorem, to show that all Banach algebra norms on A must be equivalent. The same result for non-commutative Banach algebras was conjectured by I. Kaplansky in the 1950’s and solved more then a decade later, in 1967, by B E Johnson. However, Johnson’s proof was difficult and relied heavily on representation theory. As a result, the problem remained unsolved for the more difficult situation of Jordan Banach algebras. Fifteen years later in 1982, B. Aupetit succeeded in proving Johnson’s result, using a subharmonic method that was independent of algebra representations. Moreover he could, using these techniques, also settle the problem in the Jordan Banach algebra case. A while later, in 1989, T. Ransford provided a shorter algebraic proof of Johnson’s result using the well-known spectral radius formula. This dissertation will be a comparative study of the three different approaches on the problem for Banach algebras.

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

In this thesis we present two important closely related examples of Bratteli diagrams: the Pascal triangle and the Eulerian Bratteli diagram. The former is well-known and related to binomial coefficients. The latter, which is the main object of the thesis, is related to the Eulerian numbers. Bratteli diagrams were introduced in 1972 by Ola Bratteli in his study of approximately finite dimensional (AF) C*-algebras. In 1976, George Arthur Elliott associated to an AF C*-algebra or to a corresponding Bratteli diagram an ordered group, he called dimension group. In the first part of the thesis we study the space of infinite paths of the Eulerian diagram, and we realize it as a projective limit of finite permutation groups. In the second part, we study the state space of the dimension group associated to the Eulerian Bratteli diagram. It is a compact convex set and we describe its extremal points. Finally, we use this description to give a necessary and sufficient condition for an element of this dimension group to be positive.

The theory of function algebras has been an active field of research over the past two decades and its coming of age has been heralded by the appearance within the last twelve months of three textbooks devoted entirely to them, namely the books by Browder, Leibowitz and Gamelin. One of the attractive features of the theory of function algebras is that it draws on diverse specialities like the theory of Banach algebras, harmonic analysis and the theory of analytic functions of several complex variables. The last mentioned theory has led to some of the most powerful results in the theory of function algebras. Not surprisingly, many of these results, for example Rossi's local maximum modulus principle theorem 2.24, were first proved for finitely generated and then extended to arbitrary function algebras. This observation, together with the fact that there has been no systematic study of finitely generated function algebras, led to the writing of this thesis. We have made use of some of the results of the theory of analytic functions of several complex variables, though we have not specifically used the methods thereof. What we have looked for is ways in which the functions of finitely generated function algebras behave like analytic functions and then tried to see if arbitrarily generated function algebras behave in a similar way.

It is the purpose of this paper to discuss some important properties of unitary matrices and particularly to discuss the real unitary, or real orthogonal, matrices. The main results stated in this paper may be found in the works listed in the bibliography, but it is believed that the organization of the paper is such that the material will be more readily understandable and useful to the reader in the form in which it is here given. / Typescript. / "June, 1953." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: T. L. Wade, Professor Directing Paper. / Includes bibliographical references (leaf 65).

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