Hilbert modules, rigged modules and stable isomorphism

Every Hilbert module is a module of Morita equivalence between certain $C^*$-algebras $\mathcal A$ and $\mathcal B.$ We present a new subcategory of Hilbert modules, the $\sigma\Delta$-Hilbert modules. Every $\sigma\Delta$-Hilbert module implements a stable isomorphism between $\mathcal A$ and $\mathcal B$. Conversely, if the $C^*$-algebras $\mathcal A$ and $\mathcal B$ are stably isomorphic, there exists a $\sigma\Delta$-Hilbert module which is a module of Morita equivalence between them. We generalize the above theory in the context of rigged modules over nonselfadjoint algebras. We develop a theory of Morita equivalence for rigged modules.