Date of Award
Doctor of Philosophy (PhD)
This thesis takes the idea of projective multiresolution analyses and extends it to modules over noncommutative C*-algebras, particularly irrational rotation algebras, by constructing a concrete examples. We provide background information on irrational rotation algebras, Hilbert C*-modules, and Morita equivalence of C*-algebras. We then discuss previous definitions due to J. Packer and M. Rieffel, and modify these definitions to account for our specific situation. Then our first example of a projective multiresolution analysis over Aα is built, using the framework of a Morita equivalence due to M. Rieffel and employing a scaling function due to I. Daubechies. We then use our new PMRA to construct a module frame for a Hilbert module over an irrational rotation algebra, and using the module frame create an explicit isomorphism to a well-known Hilbert module. We create more examples of projective multiresolution analyses over irrational rotation algebras by generalizing our construction of the initial example. We examine direct sums of PMRAs, and then show that a PMRA can be combined with a compatible Morita equivalence structure to create a new projective multiresolution analysis. This construction relies heavily on the use of balanced tensor products, and is employed to create new examples of PMRAs involving Morita equivalences due to M. Rieffel and F. Luef.
Purkis, Benjamin Allen, "Projective Multiresolution Analyses over Irrational Rotation Algebras" (2014). Mathematics Graduate Theses & Dissertations. 33.