Morphisms of extensions of \(C^*\)-algebras: Pushing forward the Busby invariant (Q1960919)

The authors study completions of extension diagrams of \(C^*\)-algebras of certain forms, and describe the solutions in terms of push-outs and pull-backs of certain diagrams. This leads to new results about the \(K\)-theory of amalgamated free products, verifying the Cuntz conjecture in certain cases. They also obtain new results about the extensions of matricial field \(C^*\)-algebras, verifying partially a conjecture of Blackadar and Kirchberg. They show that almost commuting unitaries can be uniformly approximated by commuting unitaries when an index obstruction vanishes.