Morita equivalence for crossed products by Hilbert $C^*$-bimodules

We introduce the notion of the crossed product $A \rtimes _X{\Bbb{Z}}$ 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 $B_0 \rtimes _{B_1}{\Bbb{Z}}$. We then present our main result, in which we show that the crossed products $A \rtimes _X{\Bbb{Z}}$ and $B \rtimes _Y{\Bbb{Z}}$ are strongly Morita equivalent to each other, provided that $A$ and $B$ are strongly Morita equivalent under an imprimitivity bimodule $M$ satisfying $X\otimes _A M \simeq M\otimes _B Y$ as $A-B$ Hilbert $C^*$-bimodules. We also present a six-term exact sequence for $K$-groups of crossed products by Hilbert $C^*$-bimodules.