Let $B$ be a C*-algebra. I want to prove that $\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules if and only if $B$ is stable, that is, $\mathcal{K} \otimes B$ is isomorphic to $B$ as C*-algebras.
Assuming that $\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules, we have that $\mathcal{K}_B(\mathcal{H}_B) \simeq \mathcal{K}_B(B)$ and, hence, with no troubles I have proved that: $$\mathcal{K} \otimes B \simeq \mathcal{K} \otimes \mathcal{K}_B(B) \simeq \mathcal{K}_B(\ell^2 \otimes B) \simeq \mathcal{K}_B(\mathcal{H}_B) \simeq \mathcal{K}_B(B) \simeq B$$
Can anyone give me any hint on the converse argument? Thank you!