No description available.

Anderson and Putnam, and Kellendonk discovered methods of defining a C*- algebra on a noncommutative space associated with a tiling. The method employed was to use Renault's theory of groupoid C*-algebras of an equivalence relation on the tiling metric space. C*-algebras of a tiling have two purposes, on one hand they reveal information about the long range order of the tiling and on the other hand they provide interesting examples of C*-algebras. However, the two constructions do not include tilings such as the pinwheel tiling, with tiles appearing in an infinite number of orientations. We rectify this deficiency, with many interesting results appearing in the process.

The investigation of this paper is introduced by describing an important collection of morphisms between two topological groupoids. Characterizations of the different types of morphisms of this collection will be formulated in order to facilitate the construction and the classication of the various morphisms considered. In addition, the relationships existing among the various members of this collection will be determined.

No description available.

The definition of a groupoid is presented as well as examples of common structures from which a groupoid can be formed. Haar measure existence and uniqueness theorems for topological groups are used for the construction of Haar systems on groupoids. Some Haar systems are presented in addition to an example of a groupoid which admits no Haar system.

O objetivo desta tese é oferecer um arcabouço que permita a modelagem de simetrias em fibrados suaves, que possuam um bom comportamento local. Para tanto, usa-se ferramentas de grupoides de Lie e correlatas, com a finalidade de reduzir, quando possível, simetrias dadas pela ação de um grupo diferenciável, possivelmente de dimensão infinita, sobre um fibrado suave, a problemas em dimensão finita. Uma definição de invariância de uma forma diferencial, definida no espaço total de um fibrado suave, sob a ação de um grupoide de Lie, é apresentada e desenvolvida. A seguir, discute-se estas ferramentas no contexto da formulação lagrangiana da teoria clássica de campos com o objetivo de descrever, simultaneamente, simetrias internas e no espaço-tempo, de maneira unificada. Obtém-se então, nesta linguagem, alguns objetos de estudo centrais da teoria, como os teoremas de Noether e, no caso das teorias de calibre, os teoremas de acoplamento mínimo e Utiyama. Por fim, discute-se brevemente o caso de simetrias a menos de elementos de contato e divergências totais. / The aim of this thesis is to provide a framework that allows the modelling of symmetries in smooth fibre bundles which have good local behaviour. For that, we use Lie groupoids and related tools in order to reduce, whenever possible, symmetries given by the action of a possibly infinite dimensional differentiable group on a smooth fibre bundle to finite dimensional problems. We give a definition of invariance of a differential form, defined on the total space of a fibre bundle, by the action of a Lie groupoid. Then, we discuss these tools in the case of a Lagrangian classical field theory to describe internal and space-time symmetries simultaneously, in a unified way. With this language, we get some central objects of the theory such as Noether\'s theorems and, in the case of gauge theories, the minimal coupling and Utiyama\'s theorems. Lastly, we briefly discuss the case of symmetries up to contact elements and a total divergence.

O objetivo desta tese é oferecer um arcabouço que permita a modelagem de simetrias em fibrados suaves, que possuam um bom comportamento local. Para tanto, usa-se ferramentas de grupoides de Lie e correlatas, com a finalidade de reduzir, quando possível, simetrias dadas pela ação de um grupo diferenciável, possivelmente de dimensão infinita, sobre um fibrado suave, a problemas em dimensão finita. Uma definição de invariância de uma forma diferencial, definida no espaço total de um fibrado suave, sob a ação de um grupoide de Lie, é apresentada e desenvolvida. A seguir, discute-se estas ferramentas no contexto da formulação lagrangiana da teoria clássica de campos com o objetivo de descrever, simultaneamente, simetrias internas e no espaço-tempo, de maneira unificada. Obtém-se então, nesta linguagem, alguns objetos de estudo centrais da teoria, como os teoremas de Noether e, no caso das teorias de calibre, os teoremas de acoplamento mínimo e Utiyama. Por fim, discute-se brevemente o caso de simetrias a menos de elementos de contato e divergências totais. / The aim of this thesis is to provide a framework that allows the modelling of symmetries in smooth fibre bundles which have good local behaviour. For that, we use Lie groupoids and related tools in order to reduce, whenever possible, symmetries given by the action of a possibly infinite dimensional differentiable group on a smooth fibre bundle to finite dimensional problems. We give a definition of invariance of a differential form, defined on the total space of a fibre bundle, by the action of a Lie groupoid. Then, we discuss these tools in the case of a Lagrangian classical field theory to describe internal and space-time symmetries simultaneously, in a unified way. With this language, we get some central objects of the theory such as Noether\'s theorems and, in the case of gauge theories, the minimal coupling and Utiyama\'s theorems. Lastly, we briefly discuss the case of symmetries up to contact elements and a total divergence.

W. D. Munn proved that a finite dimensional representation of an inverse semigroup is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this thesis will be to give new analytic proof that every finite dimensional representation of a compact inverse semigroup is equivalent to a ⋆-representation. The second goal is to parameterize all finite dimensional irreducible representations of a compact inverse semigroup in terms of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new and simpler proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Our last theorem is the following: every norm continuous irreducible ∗-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.

W. D. Munn proved that a finite dimensional representation of an inverse semigroup is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this thesis will be to give new analytic proof that every finite dimensional representation of a compact inverse semigroup is equivalent to a ⋆-representation. The second goal is to parameterize all finite dimensional irreducible representations of a compact inverse semigroup in terms of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new and simpler proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Our last theorem is the following: every norm continuous irreducible ∗-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.

W. D. Munn proved that a finite dimensional representation of an inverse semigroup is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this thesis will be to give new analytic proof that every finite dimensional representation of a compact inverse semigroup is equivalent to a ⋆-representation. The second goal is to parameterize all finite dimensional irreducible representations of a compact inverse semigroup in terms of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new and simpler proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Our last theorem is the following: every norm continuous irreducible ∗-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.