# Strong Morita Equivalence and Morphisms Between $C^{*}$-Algebras

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.

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.
You cannot says anything about morphisms between $A$ and $B$ in general, but from the two bi-modules you can construct a third algebra $C$, such that both $A$ and $B$ embeds into $C$ with the embeddings inducing Morita equivalence between $A$ and $C$ and $B$ and $C$ compatible with the equivalence between $A$ and $B$. The algebra $C$ is constructed as an algebra of $2\times2$ matrices, with diagonals coefficient in $A$ and $B$ and non diagonal coefficient in the two bi-modules.