
JOHN PHILLIPS, Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada 
Spectral flow and index in bounded and unbounded summable Fredholm modules: integral formulas 
In joint work with Alan Carey, we study summable Fredholm
modules (H,D_{0}) for Banach algebras, A, and integral
formulas for the pairing of (H,D_{0}) with K_{1}(A). In particular, if
(H, D_{0}) is summable (in Connes' original sense that
for all t>0) then we prove that if u is
a unitary in A^{1} with [u,D_{0}] bounded and
, then
Our proof is quite different from the one indicated by Getzler. Our method is able to handle Connes' new notion of summability (which we dub weaksummability) i.e., for some t>0. We can also handle the various bounded notions of summability (indeed, our method is based on this). Finally our method simultaneously deals with the version of all these results.
Roughly speaking, our proofs involve three steps. First, given (H
,D_{0}) we make a careful analytic study of the map
defined for D in
. The pair
(H,F_{D0}) is a preFredholm module for Awhich is summable (in a restricted sense) and the F_{D} vary
in a certain affine space of bounded selfadjoint operators,
. We show that this map
As mentioned above, our results are proved in such a way that the type case is included at all stages.