process

A Schwartz type algebra for the tangent groupoid 193The vector space S.E/ is an associative algebra with the product given as follows:for f; g 2 S.E/,weputZ.f g/./ D f. /g./ d p./ ./; (6)E p./where is a smooth Haar system of the Lie groupoid E X. A classical Fourierargument can be applied to show that the algebra S.E/ is isomorphic to .S.E /; /(punctual product). In particular this implies that K 0 .S.E// Š K 0 .E /.In the case we are interested in, we have a couple .M; X/ and a vector bundleassociated to it, that is, the normal bundle over X, NXM . The reason why we gave thelast definition is because we get evaluation linear mapse 0 W S c .D M X / ! S.N M X/; (7)ande t W S c .D M X / ! C 1 c.M / (8)for t ¤ 0. Consequently, we have that the vector space S c .DX M / is a field of vectorspaces over the closed interval Œ0; 1 whose fiber spaces are S.NXM / at t D 0 andCc 1 .M / for t ¤ 0.Let us finish this subsection by giving the examples of spaces S c .DX M / correspondingto the DCN manifolds in Examples 3.5.Examples 4.7. 1. For X D;, we have that S c .D; M / Š C c 1 .M .0; 1/.2. For X M an open subset we have that S c .DX M / Š C c 1 .W / where W M Œ0; 1 is the open subset consisting of the union of X Œ0; 1 and M .0; 1.4.2 Schwartz type algebra for the tangent groupoid. In this section we define analgebra structure on S c .G T /. We start by defining a function m c W S c .D G .2// !G .0/S c .D G / by the following formulas:G .0/For F 2 S c .D G .2//,weletG .0/ Zm c .F /.x; ; 0/ D F.x; ; ; 0/ d x ./T x G xandZm c .F /.; t/ D F. ı ı 1 ;ı;t/tq d s./ .ı/:G s./If we canonically identify D G .2/with .G T / .2/ , the map above is nothing else than theG .0/integration along the fibers of m T W .G T / .2/ ! G T . We have the following proposition:Proposition 4.8. m c W S c ..G T / .2/ / ! S c .G T / is a well-defined linear map.