Analytic index theory and spectral flow in real Hilbert $C^*$-modules

We consider the analytic index and spectral flow of Fredholm operators on Hilbert $C^*$-modules.

Our spaces and algebras are equipped with a real structure, so the analytic index and spectral flow takes value in the real $K$-theory group of a $\sigma$-unital $C^*$-algebra.

We use Van Daele $K$-theory, which allows us to treat the eight real $K$-theory groups and the two complex groups on an equal footing.

We provide a general definition of the analytic index for Clifford anti-linear and skew-adjoint Fredholm operators as well as self-adjoint and odd Fredholm operators.

Our definition of spectral flow and its basic properties are valid for Wahl-continuous paths of Fredholm operators on a real Hilbert $C^*$-module.

We also provide an analytic approach to the spectral flow as a decomposition into a finite sum of relative indices.

Furthermore, we prove a real version of the Robbin-Salamon theorem, relating the spectral flow to a Fredholm index.

Our description of the index and spectral flow relies on various isomorphisms between Kasparov's $KKR$-theory and Van Daele $K$-theory, which we systematically describe in the Appendix.