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.