Let $X$ be a complete separable non-compact metric space and let us denote $C_b(X)$ to be the space of all continuous bounded real-value functions on $X$. Consider a functional operator $A$ from the space $C_b(X)$ to itself and let $X^* = X\cup\{\infty\}$ be Alexandroff compactification of $X$. Define: $ C^*_b(X^*) = \{f:X^*\to\mathbb R\text{ s.t. }f|_X\in C_b(X)\} $ to be the space of all extensions of continuous bounded functions from $X$.
Let us extend the operator $A$ as follows: for any $f\in C^*_b(X^*)$ define $ (A^*f)(x) = \begin{cases} \left(Af|_X\right)(x)&\text{ if }x\neq \infty \\ \\ f(\infty)&\text{ if }x = \infty. \end{cases} $ Clearly, $A^*$ maps $ C^*_b(X^*)$ into itself. Does it also map $C_b(X^*)$ into itself?
If it matters, $A$ is a bounded linear operator.