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##
John Phillips - *Spectral flow and index in bounded and unbounded
-summable Fredholm modules: integral formulas*

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

where is the straight line path from

*D*

_{0}to . This is the pairing of [

*u*] in

*K*

_{1}(

*A*) with [

*D*

_{0}] in

*K*

^{1}(

*A*) and can also be interpreted as the spectral flow of the path . A proof of this formula was first outlined by Ezra Getzler.

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
weak--summability) *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 pre-Fredholm module for *A*which is -summable (in a restricted sense) and the *F*_{D} vary
in a certain affine space of bounded self-adjoint operators,
. We show that this map

is suitably smooth. Second, we study the various notions of bounded -summability that arise in this context. In particular, one of our results shows that if (

*H*,

*F*

_{0}) is a pre-Fredholm module which is -summable for

*A*in the strongest sense, then for

*F*in and

*X*in (the tangent space to at

*F*), Tr(

*X e*

^{-|1-F2|-1}) is a closed (and exact) 1-form on the manifold, . The idea to use these differential-geometric notions in this context goes back to I.M. Singer and was also used by Getzler in his proof. The point here is that now the integral

is seen to be independent of path in the space, . Third, we can now reduce to the case of genuine (bounded, -summable) Fredholm modules (

*H*,

*F*

_{0}) (

*i.e.*,

*F*

_{0}

^{2}=1), and paths . In this setting,

*P*=2

*F*

_{0}-1and the formula

is still

*not*a triviality.

As mentioned above, our results are proved in such a way that the type case is included at all stages.

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