THE CONVOLUTION-INDUCED TOPOLOGY ON Loo (G) AND LINEARLY DEPENDENT TRANSLATES IN L I(G)

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