Abstract Harmonic Analysis on locally compact right topological groups

Abstract harmonic analysis is well established on compact Hausdorff admissible right topological (CHART) groups. Specifically these groups are one-sided analogues of topological groups, where the elements that multiply continuously on the other side are dense in the group. The analytic theory of such groups was facilitated by strong topological results that lead to the existence of a Haar measure. However, not much is known about whether analogues hold in the non-admissible case, and the locally compact setting has been left untouched. In this thesis our goal is to broaden the scope of the current literature by considering these cases. In particular, we first establish theory on locally compact right topological groups, including a sufficient condition for the existence of a Haar measure. We then consider analogues of various classical function algebras on these groups and discuss their properties. Then we introduce various measure algebra analogues on compact right topological groups and use their properties to characterize the existence of a Haar measure. These are also used to obtain hereditary properties - relating existence of a Haar measure on substructures to that on the group itself. In the process we provide sufficient conditions that do not rely on admissibility. The main challenge in this work lies in the lack of nice algebro-topological properties on right topological groups, which makes classical abstract harmonic analytic techniques unavailable for application. The lack of examples of such groups also impacts empirical evidence to draw inspiration from.