The space of spectral sections and connections to K-theory

Question

We look at a familiy $D_\alpha$ of Dirac operators over a (compact) base space B. The projection $\Pi^+_\alpha$ onto the positive eigenspaces of $D_\alpha$ is usually not continuous in α. Therefore Melrose and Piazza introduced the concept of a spectral section $P_\alpha$, which is a compact perturbation of $\Pi^+_\alpha$, continuous in α.

Assume, for our space, at least one spectral section exists (and we know all those, which are "very near to $\Pi^+$", in a well-defined sense). What does the "space of all spectral sections" look like?

The difference of the ranges of two such projections can be seen as an element in the K-theory of B. What can be said about this map from spectral sections to K-theory?

Any references or suggestions would be helpful.

Answer 1

First, the existence of a spectral projection implies that the index of the family $(D_\alpha)$ is a trivial element of $K^0(B)$. Next, if you fix a spectral projection $\Pi_0^+ :=\Pi^+_{\alpha_0}$. The space of projections $P$ comensurate with $\Pi_0^+$ (i.e. $P-\Pi_0^+$ is compact) forms a classifying space for $K^0$. See

J. Avron, R. Seiler, B. Simon, The index of a pair of projections, J. Funct. Anal., 120(1994), 220-237.