Morita equivalence for crossed products by Hilbert C*-bimodules

We introduce the notion of the crossed product A x(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 x B-1 Z. We then present our main result, in which we show that the crossed products A x(X) Z and B x(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 x(A) M similar or equal to M x(B) Y as A - B Hilbert C*-bimodules. We also present ii six-term exact sequence for K-groups of crossed products by Hilbert C*-bimodules.