The convolution-induced topology on L∞(G) and linearly dependent translates in L1(G)

Abstract

Given a locally compact Hausdorff group G, we consider on L∞(G) the τc-topology, i.e. the weak topology under all convolution operators induced by functions in L1(G). As a major result we characterize the trigonometric polynomials on a compact group as those functions in L1(G) whose left translates are contained in a finite-dimensional set. From this, we deduce that τc is different from the w∗-topology on L∞(G) whenever G is infinite. As another result, we show that τc coincides with the norm-topology if and only if G is discrete. The properties of τc are then studied further and we pay attention to the τc-almost periodic elements of L∞(G).

Keywords

• locally compact group
• convolution operator
• topology induced by convolution
• linearly dependent translates
• almost periodic functions.