process

182 P. Carrillo Rousewhere AG is the Lie algebroid of G . The “C -index” ind a is a homotopy invariant ofthe G -pseudodifferential elliptic operators (G -PDO) and has proved to be very useful inmany different situations (see for example [2], [9], [10]). One way to define the aboveindex map is using Connes’ tangent groupoid associated to G as explained by Hilsumand Skandalis in [13] or by Monthubert and Pierrot in [19]. The tangent groupoid is aLie groupoidG T G .0/ Œ0; 1with G T WD AG f0g tG .0; 1, and the groupoid structure is given by the groupoidstructure of AG at t D 0 and by the groupoid structure of G for t ¤ 0. One of the mainfeatures of the tangent groupoid is that its C -algebra C .G T / is a continuous field ofC -algebras over the closed interval Œ0; 1, with associated fiber algebrasC 0 .A G / at t D 0; and C .G / for t ¤ 0.In fact, it gives a C -algebraic quantization of the Poisson manifold A G (in the senseof [14]), and this is the main point why it allows to define the index morphism as asort of ”deformation”. Thus, the tangent groupoid construction has been very useful inindex theory ([2], [10], [13]), but also for other purposes ([14], [21]).Now, to understand the purpose of the present work, let us first say that the indices (inthe sense of Atiyah–Singer–Connes) have not necessarily to be considered as elementsin K 0 .C .G //. Indeed, it is possible to consider indices in K 0 .Cc 1.G //. The C c 1-in-dices are more refined, but they have several inconveniences (see Alain Connes’ bookSection 9:ˇ for a discussion on this matter). Nevertheless this kind of indices have thegreat advantage that one can apply to them the existent tools (such as pairings withcyclic cocycles or Chern–Connes character) in order to obtain numerical invariants.In this work we begin a study of more refined indices. In particular we are lookingfor indices between the Cc1 and the C -levels, trying to keep the advantages of bothapproaches (see [5] for a more complete discussion). In the case of Lie groupoids,this refinement could mean to forget for a moment the powerful tools of the theory ofC -algebras, and instead working in a purely algebraic and geometric level. In thepresent article, we construct an algebra of C 1 functions over G T , denoted by S c .G T /.This algebra is also a field of algebras over the closed interval Œ0; 1, with associatedfiber algebrasS.AG / at t D 0, and Cc 1 .G / for t ¤ 0,where S.AG / is the Schwartz algebra of the Lie algebroid. Furthermore, we will haveC 1 c .G T / S c .G T / C .G T /; (1)as inclusions of algebras. Let us explain in some words why we define an algebra overthe tangent groupoid such that in zero it is Schwartz: The Schwartz algebras have ingeneral the good K-theory groups. For example we are interested in the symbols ofG -PDO, and, more precisely, in their homotopy classes in K-theory, that is, we areinterested in the group K 0 .A G / D K 0 .C 0 .A G //. Here it would not be enough totake the K-theory of Cc1 .AG / (see the example [8], p. 142), however it is enough to