We build metrized quantum vector bundles, over a generically transcendental quantum torus, from Riemannian metrics, using Rosenberg's Levi-Civita connections for these metrics. We also prove that two metrized quantum vector bundles, corresponding to positive scalar multiples of a Riemannian metric, have distance zero between them with respect to the modular Gromov-Hausdorff propinquity.

Let p be a prime number and set ‫ޚ‬ p = ‫/ޚ‬ p‫.ޚ‬ A ‫ޚ‬ p-sequence is a function S : ‫ޚ‬ → ‫ޚ‬ p. Let be the set {P ∈ ‫[ޒ‬X ] | P(‫)ޚ‬ ⊆ ‫.}ޚ‬ We prove that the set of sequences of the form (P(n) (mod p)) n∈‫ޚ‬ , where P ∈ , is precisely the set of periodic ‫ޚ‬ p-sequences with period equal to a p-power. Given a ‫ޚ‬ p-sequence, we will also determine all P ∈ that correspond to the sequence according to the manner above. MSC2000: 11B83.

In his seminal paper Generalized Fixed Point Algebras and Square-Integrable Group Actions [9], Ralf Meyer showed how to construct generalized fixed-point algebras for C *-dynamical systems via their square-integrable representations on Hilbert C *-modules. His method extends Marc Rieffel's construction of generalized fixed-point algebras from proper group actions in [16]. This dissertation seeks to generalize Meyer's work to construct generalized fixed-point algebras for twisted C *-dynamical systems. To accomplish this, we must introduce some brand-new concepts, the foremost being that of a twisted Hilbert C *-module. A twisted Hilbert C *-module is basically a Hilbert C *-module equipped with a twisted group action that is compatible with the module's right C *-algebra action and its C *-algebra-valued inner product. Twisted Hilbert C *-modules form a category, where morphisms are twisted-equivariant adjointable operators, and we will establish that Meyer's bra-ket operators are morphisms between certain objects in this category. A by-product of our work is a twisted-equivariant version of Kasparov's Stabilization Theorem, which states that every countably generated twisted Hilbert C *-module is isomorphic to an invariant orthogonal summand of the countable direct sum of a standard one if and only if the module is square-integrable. Given a twisted C *-dynamical system, we provide a definition of a relatively continuous subspace of a twisted Hilbert C *-module (inspired by Ruy Exel's paper [5]) and then prescribe a new method of constructing generalized fixed-point algebras that are Morita-Rieffel equivalent to an ideal of the corresponding reduced twisted crossed product. Our construction generalizes that of Meyer and, by extension, that of Rieffel in [16]. Our main result is the description of a classifying category for the class of all Hilbert modules over a reduced twisted crossed product. This implies that every Hilbert module over a d-dimensional non-commutative torus can be constructed from a Hilbert space endowed with a twisted Z d-action and a relatively continuous subspace.

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In this paper, we show that the generalized fixed-point algebra of a proper groupoid dynamical system, under certain assumptions, may be fibered over any locally compact Hausdorff space to which a continuous map exists from the unit space of the underlying groupoid. We will also provide some important examples.