A Schwartz type algebra for the Tangent Groupoid

Details A Schwartz type algebra for the Tangent Groupoid A Schwartz type algebra for the Tangent Groupoid

2008-02-25

arxiv.org/abs/0802.3596v1

Mathematics

Differential Geometry

To appear on the "Proceedings of the International Conference on K-theory and Non commutative geometry", VASBI, held at Vallad

Scientific paper

We construct an algebra of smooth functions over the tangent groupoid associated to any Lie groupoid. This algebra is a field of algebras over the closed interval [0, 1] which fiber at zero is the algebra of Schwartz functions over the Lie algebroid, whereas any fiber out of zero is the convolution algebra of the initial groupoid. Our motivation comes from index theory for Lie groupoids. In fact, our construction gives an intermediate algebra between the enveloping C-algebra and the convolution algebra of compactly supported functions of the tangent groupoid; and it will allows us, in a further work, to define other analytic index morphisms as a sort of deformations.

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