A Stone-\v{C}ech Theorem for $C_0(X)$-algebras

For a $C_0(X)$-algebra $A$, we study $C(K)$-algebras $B$ that we regard as compactifications of $A$, generalising the notion of (the algebra of continuous functions on) a compactification of a completely regular space. We show that $A$ admits a Stone-\v{C}ech-type compactification $A^{\beta}$, a $C(\beta X)$-algebra with the property that every bounded continuous section of the C$^*$-bundle associated with $A$ has a unique extension to a continuous section of the bundle associated with $A^{\beta}$. Moreover, $A^{\beta}$ satisfies a maximality property amongst compactifications of $A$ (with respect to appropriately chosen morphisms) analogous to that of $\beta X$. We investigate the structure of the space of points of $\beta X$ for which the fibre algebras of $A^{\beta}$ are non-zero, and partially characterise those $C_0(X)$-algebras $A$ for which this space is precisely $\beta X$.