process

A Schwartz type algebra for the tangent groupoid 185see [22]). We can now define a convolution product on Cc 1.G /. Forf; g 2 C c 1.G /we setZ.f g/./ D f. 1 /g./ d s./ ./:G s./This gives a well-defined associative product.Remark 2.3. There is a way to avoid the Haar system when one works with Liegroupoids, using half densities (see Connes’ book [8]).3 Deformation to the normal coneLet M be a C 1 manifold and X M be a C 1 submanifold. We denote by NXMnormal bundle to X in M , i:e:, NXM WD T XM=TX. We define the following settheD M X WD N M X 0 t M .0; 1: (2)The purpose of this section is to recall how to define a C 1 -structure with boundary inDX M . This is more or less classical, for example it was extensively used in [13]. Herewe are only going to do a sketch.Let us first consider the case where M D R n and X D R p f0g (where we identifycanonically X D R p ). We denote by q D n p and by Dp n for DRnRas above. In thispcase we clearly have that Dp n D Rp R q Œ0; 1 (as a set). Consider the bijectiongiven by‰ W R p R q Œ0; 1 ! D n p (3)‰.x; ; t/ D´.x;;0/ if t D 0;.x; t; t/ if t>0;whose the inverse is given explicitly by´‰ 1 .x;;0/ if t D 0;.x;;t/D.x; 1 t ;t/ if t>0:We can consider the C 1 -structure with border on Dp n induced by this bijection.In the general case, let .U;/be a local chart in M and suppose it is an X-slice,so that it satisfies1) W U Š ! U R p R q ;2) if V D U \ X, V D U \ .R p 0/, then we have V D 1 .V /.With this notation we have that D U V Dn pis an open subset. We may define a functionQ W D U V! DU V