Let $\Omega$ be a domain in $\mathbb{C}^n$. Let $\mathbb{D}$ denote the open unit disc in $\mathbb{C}$. Let $C_b(\Omega)$ and $C_b(\mathbb{D})$ denote the space of all bounded continuous complex valued functions on $\Omega$ and $\mathbb{D}$ respectively.

Let $T:C_b(\Omega)\longrightarrow C_b(\mathbb{D})$ be a positive linear operator which is unital.

Suppose $f\in C_b(\Omega)$ be such that $f(z)\neq 0$ for any $z\in \Omega$. Will it imply that $Tf(y)\neq 0$ for every $y\in\mathbb{D}$?

If not then under what additional conditions will $T$ satisfy this property?