A Stone-Čech Theorem for $C_0(X)$-algebras

Abstract

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 $A^{\beta }$, a $C(\beta X)$-algebra with the property that every bounded continuous section of the C${}^{\ast }$-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 $\beta X$.

arXiv comment28 pages