process

A Schwartz type algebra for the tangent groupoid 197Recall that we have well-defined evaluation morphisms as in (7) and (8). In thecase of the tangent groupoid they are by definition morphisms of algebras. Hence, thealgebra S c .G T / is a field of algebras over the closed interval Œ0; 1, with associatedfiber algebras,S.AG / at t D 0, and Cc 1 .G / for t ¤ 0.It is very interesting to see what this means in the examples given in 3.9.5 Further developmentsLet G G .0/ be a Lie groupoid. In index theory for Lie groupoids the tangent groupoidhas been used to define the analytic index associated to the group, as a morphismK 0 .A G / ! K 0 .Cr .G // (see [19]) or as a KK-element in KK.C 0.A G /; Cr .G //(see [13]), this can be done because one has the following short exact sequence ofC -algebras0 ! Cr .G .0; 1/ ! C r .G T / e 0! C 0 .A G / ! 0; (9)and because of the fact that the K-groups of the algebra Cr .G .0; 1/ vanish (homotopyinvariance). The index defined at the C -level has proven to be very useful (seefor example [9]) but extracting numerical invariants from it, with the existent tools, isvery difficult. In non commutative geometry, and also in classical geometry, the toolsfor obtaining more explicit invariants are more developed for the ’smooth objects’; inour case this means the convolution algebra Cc1 .G /, where we can for example applyChern–Weil–Connes theory. Hence, in some way, the indices defined in K 0 .Cc 1 .G //are more refined objects and for some cases it would be preferable to work with them.Unfortunately these indices are not good enough, since for example they are not homotopyinvariant; in [8] Alain Connes discusses this and also other reasons why it is notenough to keep with the Cc 1-indices. The main reason to construct the algebra S c.G T /is that it gives an intermediate way between the Cc1-level and the C -level and willallow us in [5] to define another analytic index morphism associated to the groupoid,with the advantage that this index will take values in a group that allows to do pairingswith cyclic cocycles and in general to apply Chern–Connes theory to it. The way weare going to define our index is by obtaining first a short exact sequence analogous to(9), that is, a sequence of the following kind0 ! J ! S c .G T / e 0! S.A G // ! 0: (10)The problem here will be that we do not dispose of the advantages of the K-theory forC -algebras, since the algebras we are considering are not of this type (we do not havefor example homotopy invariance).References[1] J. Aastrup, S. T. Melo, , B. Monthubert, E. Schrohe, Boutet de Monvel’s Calculus andGroupoids I, Preprint 2006, arxiv:math.KT/0611336.