The construction of the generalized continuous wavelet transform (GCWT) on Hilbert spaces is a special case of the coherent state transform construction, where the coherent state system arises as an orbit of an admissible vector under a strongly continuous unitary representation of a locally compact group. In this thesis we extend this construction to the setting of Hilbert C*-modules. In particular, we define a coherent state transform and a GCWT on Hilbert modules. This construction gives a reconstruction formula and a resolution of the identity formula analogous to those found in the Hilbert space setting. Moreover, the existing theory of standard normalized tight frames in finite countably generated Hilbert modules can be viewed as a discrete case of this construction We also show that the image space of the coherent state transform on Hilbert module is a reproducing kernel Hilbert module. We discuss the kernel and the intertwining property of the group coherent state transform.

The construction of the generalized continuous wavelet transform (GCWT) on Hilbert spaces is a special case of the coherent state transform construction, where the coherent state system arises as an orbit of an admissible vector under a strongly continuous unitary representation of a locally compact group. In this thesis we extend this construction to the setting of Hilbert C*-modules. In particular, we define a coherent state transform and a GCWT on Hilbert modules. This construction gives a reconstruction formula and a resolution of the identity formula analogous to those found in the Hilbert space setting. Moreover, the existing theory of standard normalized tight frames in finite countably generated Hilbert modules can be viewed as a discrete case of this construction We also show that the image space of the coherent state transform on Hilbert module is a reproducing kernel Hilbert module. We discuss the kernel and the intertwining property of the group coherent state transform.

A wavelet is a function which is used to construct a specific type of orthonormal basis. We are interested in using C*-algebras and Hilbert C*-modules to study wavelets. A Hilbert C*-module is a generalisation of a Hilbert space for which the inner product takes its values in a C*-algebra instead of the complex numbers. We study wavelets in an arbitrary Hilbert space and construct some Hilbert C*-modules over a group C*-algebra which will be used to study the properties of wavelets. We study wavelets by constructing Hilbert C*-modules over C*-algebras generated by groups of translations. We shall examine how this construction works in both the Fourier and non-Fourier domains. We also make use of Hilbert C*-modules over the space of essentially bounded functions on tori. We shall use the Hilbert C*-modules mentioned above to study wavelet and scaling filters, the fast wavelet transform, and the cascade algorithm. We shall furthermore use Hilbert C*-modules over matrix C*-algebras to study multiwavelets.

In der vorliegenden Arbeit "Nichtkommutative Blochtheorie" beschäftigen wir uns mit der Spektraltheorie bestimmter Klassen von Hilbertraumoperatoren, den elliptischen Operatoren auf Darstellungsräumen von Hilbert-C*-Moduln. Die auftretenden C*-Algebren kodieren dabei Symmetrieeigenschaften der entsprechenden Operatoren.Für kommutative Symmetrien ist die Blochtheorie ein geeignetes Hilfsmittel. Wir schildern diese Methode zunächst in einem geometrischen Kontext, der allgemein genug ist, um die bekannten Ergebnisse über die Abwesenheit singulärstetigen Spektrums im Hinblick auf physikalische Anwendungen zu erweitern. Wir lassen uns dann durch eine Neuinterpretation der Blochtheorie aus einem nichtkommutativen Blickwinkel inspirieren zur Entwicklung einer nichtkommutativen Blochtheorie. Dabei werden bestimmte Eigenschaften von C*-Algebren verknüpft mit Eigenschaften des Spektrums elliptischer Operatoren. Diese Blochtheorie für Hilbert-C*-Moduln erlaubt es, verschiedene bekannte Resultate aus dem Bereich kommutativer (diskreter und kontinuierlicher) Geometrien mit nichtkommutativen Symmetrien in einem neuen gemeinsamen Rahmen zusammenzufassen, der Raum läßt für Modelle nichtkommutativer Geometrien mit nichtkommutativen Symmetrien. Wichtigstes Beispiel für die behandelte Klasse von Operatoren in der mathematischen Physik sind die Schrödingeroperatoren mit periodischem Magnetfeld und Potential. Wir ordnen sie in den Rahmen kommutativer und nichtkommutativer Blochtheorie ein und wenden die zuvor bereitgestellten Methoden an. / In this doctoral thesis "Nichtkommutative Blochtheorie'' (non-commutative Bloch theory) we investigate the spectral theory of a certain class of operators on Hilbert space: the elliptic operators associated with representations of Hilbert C*-modules. The C*-algebras that arise encode symmetry properties of the corresponding operators. For commutative symmetries Bloch theory is a proper tool. We describe this method in a geometric context which is general enough to extend known results about absence of singular continuous spectrum in view of physical applications. Then --- inspired by a new interpretation of Bloch theory from a non-commutative point of view --- we develop a non-commutative Bloch theory. Here certain properties of C*-algebras get linked to spectral properties of elliptic operators. This Bloch theory for Hilbert \CS-modules allows to unite, in a new common framework, several known results from the field of commutative (discrete and continuous) geometries having non-commutative symmetries; this leaves ample room for models of non-commutative geometries having non-commutative symmetries. In mathematical physics, the most important example for the class of operators considered is given by the Schrödinger operators with periodic magnetic field and potential. We place them into the framework of commutative and non-commutative Bloch theory and apply the methods developed before.