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

Abstract

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

12 pages