Topological centers and topologically invariant means related to locally compact groups

Abstract

In this thesis, we discuss two separate topics from the theory of harmonic analysis on locally compact groups. The first topic revolves around the topological centers of module actions induced by unitary representations while the second one deals with the set of topologically invariant means associated to an amenable representation. Part I of this thesis is about the topological centers of bilinear maps induced by unitary representations. We give a characterization when the center is minimal in term of a factorization property. We give conditions which guarantee that the center is maximal. Various examples whose topological centers are maximal, minimal nor neither will be given. We also investigate the topological centers related to sub-representations, direct sums and tensor products. In Part II we study of the set of topologically invariant means associated to an amenable representation. We construct topologically invariant means for an amenable representation by two different methods. A lower bound of the cardinality of the set of topologically invariant means will be given.