Let $X$ be a locally compact Hausdorff space, and let $Y_t$ be a continuous Markov process on $X$ with transition function $P(t, x, \Gamma) := \mathbb{P}_x (Y_t \in \Gamma)$.  Let $T_t$ be the corresponding transition semigroup, i.e. $T_t f (x) = \int_X f(y) P(t,x,dy)$.  Let $C_0$ be the set of continuous functions on $X$ vanishing at infinity, and $C_b$ the bounded continuous functions on $X$.

If $T_t C_0 \subset C_0$, then the process $Y_t$ is **Feller**.  (Since the process is continuous, it follows that $T_t$ is a strongly continuous semigroup on $C_0$.)  

> Suppose we only know that $T_t C_0 \subset C_b$.  (For instance, this happens if the process has a continuous transition density.)  Can we extend the state space $X$ to a larger space $\tilde{X}$ on which a suitably extended version of the process $Y_t$ is Feller?

The example I have in mind is something like Brownian motion on $X = \mathbb{R}^3 \backslash \{0\}$.  If $f \in C_0(X)$, then $T_t f$ need not vanish near 0, so $T_t f \notin C_0$ and the process is not Feller.  However, if we take $\tilde{X} = \mathbb{R}^3$, then Brownian motion on $\mathbb{R}^3$ is Feller.  Moreover, $\tilde{X} \backslash X = \{0\}$ is an exceptional set, so the process started at a point of $X$ doesn't really see the point that we added.

To generalize this, one might consider the $C^*$-subalgebra of $C_b$ generated by $\{T_t C_0 : t \ge 0\}$ and take $\tilde{X}$ to be its spectrum.  However, there are a lot of details that don't seem entirely clear.

This seems like it should be known, so a reference would be much appreciated.