Zero Divisors and Linear Independence of Translates
Ahmed Roman ยท VTechWorks (Virginia Tech) ยท 2015
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 ๐ยฒ 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 ๐ถ*-algebras and operator norm ๐ถ*-algebras. For certain classes of groups we address the zero divisor conjecture by providing an isomorphism between the the reduced ๐ถ*-algebra and the operator norm ๐ถ*-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 โ๐บ modules for some groups ๐บ.