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

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up vote 6 down vote accepted

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.

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Thank you very much. – David May 10 '13 at 18:19

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