Proposition 4

Let W be a weakly closed vector subspace of $L^{\infty}(G)$, invariant under translation and nonzero. Then W contains at least one character of G.

We have seen that $A(W) = V(W^{\circ})$. Since $W\not= 0$, we have $W^{\circ}\not= L^1(G)$, and then $V(W^{\circ})$ is nonempty by th. 1 of II, p. 252.