Continuous spectral decompositions of Abelian group actions on C∗-algebras

Let G be a locally compact Abelian group. Following Ruy Exel, we view Fell bundles over the Pontrjagin dual group of G as continuous spectral decompositions of G-actions on C∗-algebras.We classify such spectral decompositions using certain dense subspaces related to Marc Rieffel’s theory of square-integrability. There is a unique continuous spectral decomposition if the group acts properly on the primitive ideal space of the C∗-algebra. But there are also examples of group actions without or with several inequivalent spectral decompositions. © 2007 Elsevier Inc. All rights reserved.