Bimodule maps from a unital $C^{\ast }$-algebra to its $C^{\ast }$-subalgebra and strong Morita equivalence

Let $A\subset C$ and $B\subset D$ be unital inclusions of unital $C^{\ast }$-algebras. Let ${}_A{\mathbf{B}}_A(C,A)$ (resp. ${}_B $\mathbf{B}$_B (D, B)$)bethespaceofallbounded$A$-bimodule(resp\mathrm{.}$B$-bimodule) linear maps from $C$ (resp. $D$) to $A$ (resp. $B$). We suppose that $A\subset C$ and $B \subset D$ are strongly Morita equivalent. We shall show that there is an isometric isomorphism $f$ of ${}_A{\mathbf{B}}_A(C,A)$ onto ${}_B $\mathbf{B}$_B (D, B)$andweshallstudyonbasicpropertiesabout$f$.