Morita–Rieffel Equivalence and Spectral Theory for Integrable Automorphism Groups of C*-Algebras

Given a C*-dynamical system (A, G, α), we discuss conditions under which subalgebras of the multiplier algebra M(A) consisting of fixed points for α are Morita–Rieffel equivalent to ideals in the crossed product of A by G. In case G is abelian we also develop a spectral theory, giving a necessary and sufficient condition for α to be equivalent to the dual action on the cross-sectional C*-algebra of a Fell bundle. In our main application we show that a proper action of an abelian group on a locally compact space is equivalent to a dual action.