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 ${\displaystyle x_0\in A}$, we show that every complete norm on A which makes continuous the multiplication by ${\displaystyle x_0}$ is equivalent to ${\displaystyle \parallel ·{\parallel }_{\infty }}$ provided that ${\displaystyle x_0^{-1}(\lambda )}$ has no interior points whenever λ lies in ℂ. Actually, these assertions are equivalent if A = C(K).