C*-bundles and C₀(X)-algebras

We prove that a C₀(X)-algebra is the section algebra of an upper semi-continuous C*-bundle over X. From this we obtain four corollaries. A C*-algebra A is the section algebra of an upper semi-continuous C*-bundle over Prim ZM(A). If X is a locally compact Hausdorff space and α:Prim A → X is a continuous map with dense range, then A is isomorphic to the section algebra of an upper semi-continuous C*-bundle over X. The induced algebra of an upper semi-continuous C*-bundle is an upper semi-continuous C*-bundle of induced algebras, when the action satisfies suitable condition. We give a necessary and sufficient condition for these bundles to be continuous. With a suitable twisted action, the twisted crossed product of an upper semi-continuous C*-bundle is an upper semi-continuous C*-bundle of twisted crossed products and the twisted crossed product of a continuous C*-bundle by an amenable group is again a continuous C*-bundle.