The convolution-induced topology onL∞(G)and linearly dependent translates inL1(G)

Abstract

Given a locally compact Hausdorff groupG, we consider onL∞(G)theτc-topology, i.e. the weak topology under all convolution operators induced by functions inL1(G). As a major result we characterize the trigonometric polynomials on a compact group as those functions inL1(G)whose left translates are contained in a finite-dimensional set. From this, we deduce thatτcis different from thew∗-topology onL∞(G)wheneverGis infinite. As another result, we show thatτccoincides with the norm-topology if and only ifGis discrete. The properties ofτcare then studied further and we pay attention to theτc-almost periodic elements ofL∞(G).