Coefficient spaces arising from locally compact groups

Abstract

This thesis studies two disjoint topics involving coefficient spaces and algebras associated to locally compact groups. First, Chapter 3 investigates the connection between amenability and compactness conditions on locally compact groups and the homology of the Fourier algebra when viewed as a completely contractive Banach algebra. We provide characterizations of relative 1-projectivity, 1-flatness, and 1-biflatness of the Fourier algebra. These allow us to deduce a new hereditary property for an amenability condition, namely that inner amenability passes to closed subgroups. Our techniques also allow us to show that inner amenability coincides with Property (W) and to settle a conjecture regarding the cb-multiplier completion of the Fourier algebra. Our second theme is coefficient spaces arising from $L^p$-representations of locally compact groups. Chapter 4 is motivated by a question of Kaliszewski, Landstad, and Quigg regarding whether two coefficient space constructions coincide. We are able to provide a positive answer in special cases, in particular for the group $SL(2,\mathbb{R})$. We establish several results regarding the non-separability of algebras related to the $L^p$-Fourier algebras, and characterize when these algebras have a bounded approximate identity.