TITLE: Inductive limit of C*-algebras 
QUESTION [6 upvotes]: It is known that an intertwining (or even approximately intertwining) diagram implies the isomorphism of the limit algebras. Under what conditions the converse holds?

REPLY [7 votes]: 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.