Equivariant Spectral Flow for Families of Dirac-type Operators

In the setting of a proper, cocompact action by a locally compact, unimodular group $G$ on a Riemannian manifold, we construct equivariant spectral flow of paths of Dirac-type operators. This takes values in the $K$-theory of the group $C^*$-algebra of $G$. In the case where $G$ is the fundamental group of a compact manifold, the summation map maps equivariant spectral flow on the universal cover to classical spectral flow on the base manifold. We obtain "index equals spectral flow" results. In the setting of a smooth path of $G$-invariant Riemannian metrics on a $G$-spin manifold, we show that the equivariant spectral flow of the corresponding path of spin Dirac operators relates delocalised $\eta$-invariants and $\rho$-invariants for different positive scalar curvature metrics to each other.