Let $K$ be a pseudocompact Tychonoff space that is, a Hausdorff $T_{3.5}$ space for which every continuous function $\varphi \colon K\to \mathbb{C}$ is bounded. Let $\beta$ be the Stone-Cech extension functor. Then

$$\beta\colon C(K)\to C(\beta K)$$

is an algebra isomorphism.

OK, suppose $K$ is moreover locally compact, so one may consider the C*-algebra $C_0(K)$ of continuous functions which vanish at infinity. My question is how large can be in this case the quotient

$$A:=C(K)/C_0(K)?$$

Can it be infinite dimensional? If so, is there any characterisation of pseudocompactness in terms of relations between $C_0(\beta K)$ and $C(K)$?

PS. By the way, is $\beta$ in this case a *-homomorphism?