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A Stone-Čech 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-Čech-type compactification $A^β$, a $C(βX)$-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 $A^β$. Moreover, $A^β$ satisfies a maximality property amongst compactifications of $A$ (with respect to appropriately chosen morphisms) analogous to that of $βX$. We investigate the structure of the space of points of $βX$ for which the fibre algebras of $A^β$ are non-zero, and partially characterise those $C_0(X)$-algebras $A$ for which this space is precisely $βX$.