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Let R be a commutative ring with identity and let G and H be abelian groups with the group algebras RG and RH isomorphic as R-algebras. In this dissertation we investigate the relationships between G and H. Let inv(R) denote the set of rational prime numbers that are units in R and let G(R) (respectively, H(R)) be the direct sum of the p-components of G (respectively, H) with p ∈ inv(R). It is known that if G(R) or H(R) is nontrivial then it is not necessarily true that G and H are isomorphic. However, if R is an integral domain of characteristic 0 or a finitely generated indecomposable ring of characteristic 0 then G/G(R) ≅ H/H(R). This leads us to make the following definition: We say that R satisfies the Isomorphism Theorem if whenever RG ≅ RH as R-algebras for abelian groups G and H then G/G(R) ≅ H/H(R). Thus integral domains of characteristic 0 and finitely generated indecomposable rings of characteristic 0 satisfy the Isomorphism Theorem. Our first major result shows that indecomposable rings of characteristic 0 (no restrictions on generation) satisfy the Isomorphism theorem. It has been conjectured that all rings R satisfy the Isomorphism Theorem. However, we show that the conjecture may fail if nontrivial idempotents are present in R. This leads us to consider necessary and sufficient conditions for rings to satisfy the Isomorphism Theorem. We say that R is an ND-ring if whenever R is written as a finite product of rings then one of the factors, say Rᵢ, satisfies inv(Rᵢ) = inv(R). We show that every ring satisfying the Isomorphism Theorem is an ND-ring. Moreover, if R is an ND-ring and if inv(R) is not the complement of a single prime we show that R must satisfy the Isomorphism Theorem. This result together with some other fragmentary evidence leads us to conjecture that R satisfies the Isomorphism Theorem if and only if R is an ND-ring. Finally we obtain several equivalent formulations of our conjecture. Among them is the result that every ND-ring satisfies the Isomorphism Theorem if and only if every field of prime characteristic satisfies the Isomorphism Theorem.

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The purpose of this thesis is the study of the Lie algebras of affine vector fields and curvature collineations of space-time, the aim being, in the first case, to obtain upper bounds on the dimension of the Lie algebra of affine vector fields (under the assumption that the space-time is non-flat) as well as to obtain a characterization of such vector fields in terms of other types of symmetries. In the case of curvature collineations the aim was that of characterizing space-times which may admit an infinite-dimensional Lie algebra of curvature collineations as well as to find local characterizations of such vector fields. Chapters 1 and 2 consist of introductory material, in Differential Geometry (Ch.l) and General Relativity (Ch.2). In Chapter 3 we study homothetic vector fields which admit fixed points. The general results of Alekseevsky (a) and Hall (b) are presented, some being deduced by different methods. Some further details and results are also given. Chapter 4 is concerned with space-times that can admit proper affine vector fields. Using the holonomy classification obtained by Hall (c) it is shown that there are essentially two classes to consider. These classes are analysed in detail and upper bounds on the dimension of the Lie algebra of affine vector fields of such space-times are obtained. In both cases local characterizations of affine vector fields are obtained. Chapter 5 is concerned with space-times which may admit proper curvature collineations. Using the results of Halford and McIntosh (d) , Hall and McIntosh (e) and Hall (f) we were able to divide our study into several classes The last two of these classes are formed by those space-times which admit a (1 or 2-dimensional) non-null distribution spanned by vector fields which contract the Riemann tensor to zero. A complete analysis of each class is made and some general results concerning the infinite-dimensionality problem are proved. The chapter ends with some comments in the cases when the distribution mentioned above is null.

<b>Chapter One</b> After the definitions and basic results required for the rest of the thesis, a notion of spectrum for semigroup representations is introduced and some relevant examples given. <b>Chapter Two</b> Any semigroup representation by isometries on a Banach space may be dilated to a group representation on a larger Banach space. A new proof of this result is presented here, and a connection is shown to exist between the dilation and the trajectories of the dual representation. The problem of dilating various types of spaces, including partially ordered spaces, C*-algebras, and reflexive spaces, is discussed, and new dilation theorems are given for dual Banach spaces and von Neumann algebras. <b>Chapter Three</b> In this chapter the spectrum of a representation is examined more closely with the aid of methods from Banach algebra theory. In the case where the representation is by isometries it is shown that the spectrum is non-empty, that it is compact if and only if the representation is norm-continuous, and that any isolated point in the unitary spectrum is an eigenvalue. <b>Chapter Four</b> An analytic characterisation is given of the spectral conditions that imply a representation by isometries is invertible. For representations of Z+_n this con- dition is shown to be equivalent to polynomial convexity. Some topological conditions on the spectrum are also shown to imply invertibility. <b>Chapter Five</b> The ideas of the previous chapters are applied to problems of asymptotic behaviour. Asymptotic stability is described in terms of the behaviour of the dual of a representation. Finally, the case when the unitary spectrum is countable is discussed in detail.

En este trabajo examinamos los C*-algebras de operadores Toeplitz sobre la bola unitaria en Cn y en el polidisco unitario en C². Los operadores Toeplitz son ejemplos interesantes de operadores que no son operadores normales y que generan C*-algebras no conmutativas. Además, en los mejores casos de álgebras de operadores Toeplitz (dependiendo de la geometría del dominio) podemos recuperar algunos resultados análogos al teorema espectral módulo operadores compactos. En este contexto, podemos capturar el índice de un operador Fredholm que es un invariante numérico fundamental en Teoría de Operadores / In this work we examine C*-algebras of Toeplitz operators over the unit ball inCn and the unit polydisc in C². Toeplitz operators are interesting examples of non-normal operators that generate non-commutative C*-algebras. Moreover, in the nice cases (depending on the geometry of the domain) of algebras of Toeplitz operators we can recover some analogues of the spectral theorem up to compact operators. In this setting, we can capture the index of a Fredholm operator which is a fundamental numerical invariant in Operator Theory.

Given a C*-algebra $\\A$, an irredundant set in $\\A$ is a subset $\\mathcal$ of $\\A$ such that no $a\\in \\mathcal$ belongs to the C*-subalgebra generated by $\\mathcal\\setminus\\{a\\}$. Every separable C*-algebra has only countable irredundant sets and we ask if every nonseparable C*-algebra has an uncountable irredundant set. For commutative C*-algebras, if $K$ is the Kunen line then $C(K)$ is a consistent example of a nonseparable commutative C*-algebra without uncountable irredundant sets. On the other hand, a result due to S. Todorcevic establishes that it is consistent with ZFC that every nonseparable C*-algebra of the form $C(K)$, for a compact 0-dimensional space $K$, has an uncountable irredundant set. By the method of forcing, we construct a nonseparable and noncommutative scattered C*-algebra $\\A$ without uncountable irredundant sets and with no nonseparable abelian subalgebras. On the other hand, we prove that it is consistent that every C*-subalgebra of $\\B(\\ell_2)$ of density continuum has an irredundant set of size continuum. / Given a C*-algebra $\\A$, an irredundant set in $\\A$ is a subset $\\mathcal$ of $\\A$ such that no $a\\in \\mathcal$ belongs to the C*-subalgebra generated by $\\mathcal\\setminus\\{a\\}$. Every separable C*-algebra has only countable irredundant sets and we ask if every nonseparable C*-algebra has an uncountable irredundant set. For commutative C*-algebras, if $K$ is the Kunen line then $C(K)$ is a consistent example of a nonseparable commutative C*-algebra without uncountable irredundant sets. On the other hand, a result due to S. Todorcevic establishes that it is consistent with ZFC that every nonseparable C*-algebra of the form $C(K)$, for a compact 0-dimensional space $K$, has an uncountable irredundant set. By the method of forcing, we construct a nonseparable and noncommutative scattered C*-algebra $\\A$ without uncountable irredundant sets and with no nonseparable abelian subalgebras. On the other hand, we prove that it is consistent that every C*-subalgebra of $\\B(\\ell_2)$ of density continuum has an irredundant set of size continuum.

The logical system MTL (for Monoidal t-norm Logic) is a formalism of the logic of left-continuous t-norms, which are operations that arise in the study of fuzzy sets and fuzzy logic. The objective is to investigate the important results on MTL and collect them together in a coherent form. The main results considered will be the completeness results for the logic with respect to MTL-algebras, MTL-chains (linearly ordered MTL-algebras) and standard MTL-algebras (left-continuous t-norm algebras). Completeness of MTL with respect to standard MTL-algebras means that MTL is indeed the logic of left-continuous t-norms. The logical system BL (for Basic Logic) is an axiomatic extension of MTL; we will consider the same completeness results for BL; that is we will show that BL is complete with respect to BL-algebras, BL-chains and standard BL-algebras (continuous t-norm algebras). Completeness of BL with respect to standard BL-algebras means that BL is the logic of continuous t-norms.