The convolution-induced topology on L ∞ ( G ) and linearly dependent translates in L 1 ( G )

Given a locally compact Hausdorff group G , we consider on L ∞ ( G ) the τ c -topology, i.e. the weak topology under all convolution operators induced by functions in L 1 ( G ) . As a major result we characterize the trigonometric polynomials on a compact group as those functions in L 1 ( G ) whose left translates are contained in a finite-dimensional set. From this, we deduce that τ c is different from the w ∗ -topology on L ∞ ( G ) whenever G is infinite. As another result, we show that τ c coincides with the norm-topology if and only if G is discrete. The properties of τ c are then studied further and we pay attention to the τ c -almost periodic elements of L ∞ ( G ) .