Norbert Schappacher (Strasbourg), Marie-Françoise Roy (Rennes)
Georges Skandalis (Université Paris Diderot)
Let M be a smooth compact manifold and F a foliation of M. We wish to study leafwise elliptic operators: these are differential operators which differentiate only in the leaf direction - and are elliptic in this direction. When the foliation is trivial (just one leaf), this study uses smoothing operators which are convolution kernels k(x, y) with x, y in M. The corresponding smoothing kernels for a general foliation are kernels k(x,y) with x,y in the same leaf. The convolution algebra they form is Connes’ C*-algebra of the foliation. In this construction, one has to take into account the holonomy of the foliation which tells how the transversal moves along the leaves. Our first task in this talk will be to explain this holonomy phenomenon.
The construction of Connes’ algebra is then very natural, and in fact quite easy. We then use the foliation C*-algebra to analyse a leafwise elliptic operator P. There are in fact two different ways of putting the problem : one can consider the operators P_M and P_L which act respectively on L2 functions (or sections of some bundle) on M or on the generic leaf. Although these two operators are quite different, Connes’ C*-algebra allows to establish selfadjointness results for both of them. Also, with further assumptions (mainly amenability) one proves that P_M and P_L have the same spectrum. Analysis of K-theoretic invariants of the C*-algebra tells us also in some cases what are the possible such spectra.
If time permits, we will extend our study to the case of singular foliations.