If $ A $ and $ B $ are $ C^{\*} $-algebras, then they are *strongly Morita equivalent* if there exist a $ (B,A) $-bimodule $ E $ and an $ (A,B) $-bimodule $ F $ such that
$$
E \otimes\_{A} F \cong B \quad \text{and} \quad
F \otimes\_{B} E \cong A,
$$
where the isomorphisms are between $ (B,B) $-bimodules and between $ (A,A) $-bimodules respectively. All bimodules are meant to be Hilbert bimodules.


I was wondering:



> 
> **Question.** If two $ C^{\*} $-algebras are strongly Morita equivalent and one knows the bimodules that implement the strong Morita equivalence, then does there exist a morphism between the $ C^{\*} $-algebras that can be defined using the bimodules?
> 
> 
> 


I thank everyone in advance for their help.



Strong Morita Equivalence and Morphisms Between $ C^{*} $-Algebras
Well, here is a remarkable theorem of Brown-Rieffel-Green:


If $A,B$ are $\sigma$-unital and stable, then they are strong Morita equivalent iff they are isomorphic. 


However,if they are not stable, there may not be a morphism, as $\mathbb{C},\mathbb{K}$ are strongly Morita equivalent via the Hilbert space $\mathbb{H}$, but there is no good map for us, nor it is reasonable to expect a map to be constructed via $\mathbb{H}$.


Something trivial can be $A,B$, $B$ stable, $\sigma$-unital, and $B$ stable, then $A$ has a map to $A\otimes \mathbb{K}$, by $a\rightarrow a\otimes p $, where $p$ is a rank one projection,$A\otimes \mathbb{K}$ is isomorphic to $B\otimes \mathbb{K} \cong B$ by the previous theorem.


