Twisted Convolution Algebras and Applications to Gabor Analysis

Abstract

This thesis concerns several aspects of twisted convolution algebras, with a particular focus on problems arising in Gabor analysis. A significant portion of the thesis is dedicated to the study of Hilbert C -modules known as Heisenberg modules and how they relate to Gabor frame theory. This relation showcases the link between finite Hilbert C -module frames and Gabor frames. Further, the thesis concerns certain properties of twisted convolution algebras of locally compact groups, in particular spectral invariance and C -uniqueness, and we find use for both these properties in Gabor analysis. The problem of C -uniqueness is also considered for the case of twisted convolution algebras of second-countable locally compact Hausdorff étale groupoids