I'm currently trying to figure out the following problem, and I'm not sure if the answer is obviously true / false.
Suppose that $A,B$ are Mortia equivalent $C^{*}$-algebras with imprimitivity bimodule $_A\mathsf{X}_B$, and $I \subset A$, $J \subset B$ are closed two-sided ideals of $A,B$ respectively such that $I,J$ are themselves Morita equivalent as $C^{*}$-algebras.
a) Is it possible to find an imprimitivity bimodule $_I\mathsf{Y}_J$ such that there exists some natural embedding $_I\mathsf{Y}_J \hookrightarrow _A\mathsf{X}_B$?
b) If this isn't always possible, is there a nice counter example I can look at?
Thank you.