A STONE-CECH THEOREM FOR C0(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-Cech-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.