(This is a crosspost of a question on MathStackExchange which did not receive any answer.)

Let $E$ be a locally compact metric space, let $C_0(E)$ be the set of real-valued continuous functions of $E$ vanishing at infinity endowed with the sup norm $|\cdot|_{\infty}$, let $C_b(E)$ be the set of real-valued continuous and bounded functions of $E$ endowed with the sup norm. Let $(P_t)_{t \geq 0}$ be a Feller semi-group. Depending on sources (see this discussion for example), a Feller semi-group maps (i) $C_0(E) \to C_0(E)$ or (ii) $C_b(E) \to C_b(E)$. My question is : does (i) imply (ii) ? If the space $E$ is compact, it is true as $C_b(E) = C_0(E)$. I am interested in the case where $E$ is not compact.

I have tried all the tricks in my book and I cannot prove it. However, I cannot find a counter-example as well. The reverse is not true, take the example from the already mentionned discussion $$P_tf(x) = f\left(\frac{1}{\sqrt{2t + x^{-2}}}\right)$$ associated to the deterministic process $d_t x(t) = - x^3$. We have $P_t : C_b(\mathbb{R}_+) \to C_b(\mathbb{R}_+)$ but not $P_t : C_0(\mathbb{R}_+) \to C_0(\mathbb{R}_+)$.

My questions stems from the fact that (ii) is sometimes called weak-Feller (for example in page 8-9 of Markov Chains on Metric Space, Benaim et al.) which to me suggest that (i) would imply (ii). With (ii) comes continuity of $x \mapsto P_t(x,\cdot)$ with respect to the weak convergence topology of probability measure which is a useful property, and I would like to know if it actually also apply to (i).