Zero Divisors and Linear Independence of Translates

Abstract

In this thesis we discuss linear dependence of translations which is intimately related to the zero divisor conjecture. We also discuss the square integrable representations of the generalized Wyle-Heisenberg group in $n^2$ dimensions and its relations with Gabor's question from Gabor Analysis in the light of the time-frequency equation. We study the zero divisor conjecture in relation to the reduced $C^*$-algebras and operator norm $C^*$-algebras. For certain classes of groups we address the zero divisor conjecture by providing an isomorphism between the the reduced $C^*$-algebra and the operator norm $C^*$-algebra. We also provide an isomorphism between the algebra of weak closure and the von Neumann algebra under mild conditions. Finally, we prove some theorems about the injectivity of some spaces as $mathbb{C}G$ modules for some groups $G$.