It is known that an intertwining (or even approximately intertwining) diagram implies the isomorphism of the limit algebras. Under what conditions the converse holds?



Inductive limit of C*-algebras 
The converse holds if the inductive limits involve semiprojective building blocks. That is: suppose
$$\varinjlim (A\_i,\phi\_i^{i'}) \cong \varinjlim (B\_j,\psi\_j^{j'})$$
(here my notation is $\phi\_i^{i'}:A\_i \to A\_{i'}$, etc.), and each $A\_i$ and $B\_j$ are separable and semiprojective, then there exists subsequences $(A\_{i\_k}), (B\_{j\_k})$ and an approximate intertwining between these.


