We introduce the notion of the crossed product A X Z of a C *algebra A by a Hilbert C * -bimodule X. It is shown that given a C * -algebra B which carries a semi-saturated action of the circle group (in the sense that B is generated by the spectral subspaces B 0 and B 1 ), then B is isomorphic to the crossed product B 0 B 1 Z. We then present our main result, in which we show that the crossed products A X Z and B Y Z are strongly Morita equivalent to each other, provided that A and B are strongly Morita equivalent under an imprimitivity bimodule M satisfying X ⊗ A M M ⊗ B Y as A − B Hilbert C * -bimodules. We also present a six-term exact sequence for K-groups of crossed products by Hilbert C * -bimodules.

A study of Hilbert C*-bimodules over commutative C*-algebras is carried out and used to establish a sufficient condition for two quantum Heisenberg manifolds to be isomorphic.

Given a correspondence X over a C * -algebra A, we construct a C * -algebra A X ∞ and a Hilbert C * -bimodule X∞ over A X ∞ such that the augmented Cuntz-Pimsner C * -algebras ÕX and the crossed product A X ∞ ⋊X∞ are isomorphic. This construction enables us to establish a condition for two augmented Cuntz-Pimsner C * -algebras to be Morita equivalent.

Abstract. With each Fell bundle over a discrete group G we associate a partial action of G on the spectrum of the unit fiber. We discuss the ideal structure of the corresponding full and reduced cross-sectional C * -algebras in terms of the dynamics of this partial action.

Let G and H be two locally compact groups acting on a C*-algebra A by commuting actions λ and σ. We construct an action on A × λ G out of σ and a unitary 2-cocycle u. For A commutative, and free and proper actions λ and σ, we show that if the roles of λ and σ are reversed, and u is replaced by u * , then the corresponding generalized fixed-point algebras, in the sense of Rieffel, are strong-Morita equivalent. We apply this result to the computation of the K-theory of quantum Heisenberg manifolds. 1991 MR Classification: Primary 46L55; Secondary 46L80, 46L87.