OPERATORS DETERMINING THE COMPLETE NORM TOPOLOGY OF C(K)

For any uniformly closed subalgebra A of C(K) for a compact Hausdorff space K without isolated points and to x(0) is an element of A, we sho w that every complete norm on A which makes continuous the multiplicat ion by x(0) is equivalent to \\ . \\infinity provided that x(0)(-1)(la mbda) has no interior points whenever lambda lies in C. Actually, thes e assertions are equivalent if A = C(K).