Compact Open Sets in the Dual Space of a Wallpaper Group

Abstract

Knowledge of the compact open sets in the dual space of a locally compact group can be used to study projections in the L1-algebra of the group.The wallpaper groups are a class of almost abelian groups which arise as the symmetry groups of wallpaper patterns. We characterize the compact open subsets in the dual space of a wallpaper group G. This is achieved by associating to G a graph that captures the stratified nature of the dual space. We show how this can be applied to the problem of finding projections in L1(G) by constructing a novel projection in the L1-algebra of the wallpaper group, p2.