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Journal of Operator Theory

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

A Stone-Cech theorem for C0(X)-algebras

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

Summary:  For a C0(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 Aβ, 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 Aβ. Moreover, Aβ 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 Aβ are non-zero, and partially characterise those C0(X)-algebras A for which this space is precisely βX.

DOI: http://dx.doi.org/10.7900/jot.2017may24.2157
Keywords: C-algebra, C0(X)-algebra, C-bundle, Stone-Cech compactification

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