Pseudo-differential extension for graded nilpotent Lie groups

Classical pseudo-differential operators of order zero on a graded nilpotent Lie group G form a *-subalgebra of the bounded operators on L-2(G). We show that its C*-closure is an extension of a noncommutative algebra of principal symbols by compact operators. As a new approach, we use the generalized fixed point algebra of an R->0-action on a certain ideal in the C*-algebra of the tangent groupoid of G. The action takes the graded structure of G into account. Our construction allows to compute the K-theory of the algebra of symbols.