Let $X$ and $Y$ be locally compact Hausdorff, consider an algebra homomorphism $\phi:C_0(Y) \rightarrow C_b(X)$. We have a representation, which is an algebra homomorphism, $\pi$ of $C_0(Y)$ on $C_0(X)$ defined by $$\pi(g)(f) := \phi(g) \cdot f$$Assume that this representation is non degenerate i.e. the linear span $\text{span}\{\pi(g)(f):g \in C_0(Y), f \in C_0(X)\}$ is dense in $C_0(X)$. I am trying to find a continuous function $F:X \rightarrow Y$ such that $\phi(g) = g \circ F$.
For the reverse, if $F$ is continuous and define $\phi(g) := g \circ F$, then this representation $\pi$ is non degenerate using Stone-Weierstrass theorem. However I am having trouble showing the reverse.