In this thesis an algebraic structure, called real calculus, is used as a way to represent noncommutative manifolds in an algebraic setting. Several classical geometric concepts are defined for real calculi, such as metrics and affine connections, and real calculus homomorphisms are introduced. These homomorphisms are then used to define embeddings of real calculi representing manifolds, anda notion of minimal embedding is introduced. The motivating example of the thesis is the noncommutative torus as embedded into a localization of the noncommutative 3-sphere, where it is shown that the noncommutative torus is a minimal embedding of the noncommutative 3-sphere for certain perturbations of the standard metric.

Noncommutative geometry generalizes many geometric results from such fields as differential geometry and algebraic geometry to a context where commutativity cannot be assumed. Unfortunately there are few concrete non-trivial examples of noncommutative objects. The aim of this thesis is to construct a noncommutative surface <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathcal%7BC%7D_%5Chbar" /> which will be a generalization of the well known surface called the catenoid. This surface will be constructed using the Diamond lemma, derivations will be constructed over <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathcal%7BC%7D_%5Chbar" /> and a general localization will be provided using the Ore condition.

A formulação de uma teoria da relatividade geral em espaços não-comutativos tem sido investigada usando-se diferentes abordagens atualmente. Neste contexto, é feita uma revisão do formalismo que descreve a não-comutatividade do espaço-tempo. A seguir, são investigadas as interações entre os campos do gráviton e o bóson escalar, formuladas em um espaço comutativo mostrando-se um método alternativo para reescrever a ação desta interação. Finalmente, utilizando-se este método, obtém-se o termo dominante da ação de interação entre os campos do gráviton e escalar em espaços não-comutativos. Deste modo, somos levados a considerar transformações de coordenadas restritas, com parâmetro infinitesimal "~ 'SOBRE' W "POT.'mü'"(x) tal que " " IND.'mü'"~ 'SOBRE' W "POT.'mü'" = 0. Temos assim a versão não-comutativa da gravidade unimodular / The formulation of a general relativity theory on noncommutative spaces has been investigated using many different approaches. In this context, a revision of the formalism that describes noncommutative spacetime is made. Next, the interactions between the fields of graviton and scalar is investigated in a commutative space showing an alternative method to rewrite the action of this interaction. Finally, using this method, one obtains the dominant term of the action of graviton interacting with a scalar particle in a noncomutative space. In this way, we are led to consider restricted transformations with infinitesimal parameter w&#956;(x), such that, &#8706;&#956; ˜ w&#956; = 0. Thus we have the noncommutative version of unimodular gravity.

Wong, Chun Yin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 133-135). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

by Lo Chui-sim. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 91-93). / Abstract --- p.i / Introcution --- p.1 / Chapter 1 --- Preliminaries --- p.3 / Chapter 1.1 --- Preliminaries on von-Neumann algebra --- p.3 / Chapter 1.2 --- Modular theory --- p.6 / Chapter 2 --- Abstract Lp Spaces --- p.10 / Chapter 2.1 --- "Preliminaries on dual action, dual weights and extended positive part" --- p.10 / Chapter 2.2 --- Abstract LP spaces associated with von-Neumann algebras --- p.20 / Chapter 2.3 --- "LP(M) is a Banach space for p E [1, ∞ ]" --- p.25 / Chapter 2.4 --- Independence of the choice of ψ --- p.32 / Chapter 3 --- Spatial Lp Spaces --- p.34 / Chapter 3.1 --- Definition and elementary properties of spatial derivative --- p.35 / Chapter 3.2 --- Modular properties of spatial derivatives --- p.47 / Chapter 3.3 --- Spatial Lp spaces --- p.51 / Chapter 4 --- LP Spaces constructed by using complex interpolation method --- p.60 / Chapter 4.1 --- The complex interpolation space --- p.60 / Chapter 4.2 --- LP space with respect to a faithful normal state --- p.71 / Chapter 4.3 --- LP spaces with respect to a normal faithful semifinite weight . . --- p.78 / Chapter 4.4 --- Equivalence to spatial LP spaces --- p.87 / Bibliography --- p.91

My research studies algebras of holomorphic functions from $d$-tuples of $n\times n$- matrices, $M_n(\bC)^d$, to $M_n(\bC)$. In particular, I study the holomorphic functions that can be approximated by \emph{polynomial matrix concomitants}, that is polynomial maps from $M_n(\bC)^d$ to $M_n(\bC)$ that satisfy the relationship \[ f(g^{-1}\fz g) = g^{-1}f(\fz)g \] for every $\fz \in M_n(\bC)^d$ and $g\in GL_n(\bC)$. In a sense, these are the polynomial maps that “remember” the structure of the $d$-tuple $\fz$. My first result is that these holomorphic matrix concomitants can be identified with holomorphic cross sections of certain matrix bundles. A holomorphic matrix bundle is a fibred space in which every fibre is $M_n(\bC)$ and the fibres are glued together in such a way that the total space has a holomorphic structure. Once the identification between holomorphic cross sections and holomorphic concomitants is established, the structure of the matrix bundle is used to endow the algebra of continuous cross sections with a $C^*$-algebra structure. Then we study the subalgebra of cross sections that can be approximated by polynomial concomitants. By identifying the matrix concomitants with cross sections, we are able to prove interesting results about these algebras.

The subject of the thesis is the study of operator functions in the setting of symmetric operator spaces. In this latter setting, it is of great importance to analyze the properties of so-called operator functions A --> f(A), where the variable A is a self-adjoint operator and f is a complex-valued Borel function on the real line. The thesis study the question of differentiability of this type of operator functions. The latter question is intimately related to the study of commutators. Text not only extends existing results to the setting of unbounded self-adjoint linear operators, but it is also shown that this can be obtained via a unified approach utilizing the left regular representation of von Neumann algebras.

"Submitted September 2002 ... Amended September 2004." / Bibliography: p. 152-157. / xvi, 157 p. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, School of Mathematical Sciences, Discipline of Pure Mathematics, 2004

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The goal of this thesis is to examine the actions of finite symmetry groups on aperiodic tilings. To an aperiodic tiling with finite local complexity arising from a primitive substitution rule one can associate a metric space, transformation groupoids, and C*-algebras. Finite symmetry groups of the tiling act on each of these objects and we investigate appropriate constructions on each, namely the orbit space, semidirect product groupoids, and crossed product C*-algebras respectively. Of particular interest are the crossed product C*-algebras; we derive important structure results about them and compute their K-theory.