Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if (T_n) be any sequence of positive linear operators on $C(K)$ satisfying $\|T_n(f)\to f\|_\infty\to 0$ for all $f\in A$ ensures $\|T_n(g)\to g\|_\infty\to 0$ for any $g\in C(K)$.

The famous Korovkin Theorem says $\{1,x,x^2\}$ is a Korovkin set for $C_{\mathbb{R}}[0,1]$.

My question is is it true that $\{1,z,z^2\}$ is Korovkin set for $C_{\mathbb{C}}(\mathbb{T})$?. Here $\mathbb{T}$ is the unit circle in the complex plane.