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\(KK\)-equivalence for amalgamated free product \(C^*\)-algebras. (English) Zbl 1404.46047

Summary: We prove that any reduced amalgamated free product \(C^\ast\)-algebra is \(KK\)-equivalent to the corresponding full amalgamated free product \(C^\ast\)-algebra. The main ingredient of its proof is Julg-Valette’s geometric construction of Fredholm modules with Connes’s view for representation theory of operator algebras.

MSC:

46L05 General theory of \(C^*\)-algebras
19K35 Kasparov theory (\(KK\)-theory)