The higher rho invariant defined via Hilsum-Skandalis perturbations equals the invariant defined via Higson-Roe Hilbert Poincare complexes, up to a standard index isomorphism.
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Equivalence of different definitions of higher $\rho$ invariants
The higher rho invariant defined via Hilsum-Skandalis perturbations equals the invariant defined via Higson-Roe Hilbert Poincare complexes, up to a standard index isomorphism.