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Let $A \subset C$ and $B \subset D$ be unital inclusions of unital$C^*$-algebras. Let ${}_A \mathbf{B}_A (C, A)$ (resp. ${}_B \mathbf{B}_B (D,B)$) be the space of all bounded $A$-bimodule (resp. $B$-bimodule) linear mapsfrom $C$ (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 anisometric isomorphism $f$ of ${}_A \mathbf{B}_A (C, A)$ onto ${}_B \mathbf{B}_B(D, B)$ and we shall study on basic properties about $f$.

Bimodule maps from a unital $C^*$-algebra to its $C^*$-subalgebra and strong Morita equivalence

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