Relative eta-invariants and C∗-algebra K-theory

We prove a close cousin of a theorem of Weinberger about the homotopy invariance of certain relative eta-invariants by placing the problem in operator K-theory. The main idea is to use a homotopy equivalence h : M′→M to construct a loop of invertible operators whose determinant (in the sense of de la Harpe and Skandalis) is related to eta-invariants. The Baum–Connes conjecture and a technique motivated by the Atiyah–Singer index theorem provides us with the invariance of this determinant under twistings by finite-dimensional unitary representations of Γ.