Journal of Operator Theory

Volume 79, Issue 2, Spring 2018 pp. 463-506.

A Stone-Cech theorem for $C_0(X)$-algebras

Authors: David McConnell
Author institution: School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, U.K.

Summary: 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 $\beta X$.

DOI: http://dx.doi.org/10.7900/jot.2017may24.2157
Keywords: $C^*$-algebra, $C_0(X)$-algebra, $C^*$-bundle, Stone-Cech compactification

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