What's the application of C*-algebra in topology?

C*-algebras are thought be be non-commutative topological spaces because of Gelfand's theorem that any commutative C*-algebra are isomorphic to C(X) for some locally compact Hausdorff space X. I've been working on C*-algebras and I saw there are many topological methods used in the study of C*-algebras. I am curious about the other direction. Could anyone give me some examples that topology problems can be solved using C*-algebra technics?

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Alain Connes and Henry Moscovici solved the NoviKov conjecture for hyperbolic groups in the following paper. Their method uses a number of advanced techniques of noncommutative geometry beyond basics of $C^*$-algebras.
All papers on the Baum-Connes conjecture (which generalizes the Novikov conjecture) can also be considered as other applications of noncommutative geometry and $C^*$-algebras in topology.