Pseudo-differential extension for graded nilpotent Lie groups

Abstract

Classical pseudo-differential operators of order zero on a graded nilpotent Lie group $G$ form a ${}^{\ast }$-subalgebra of the bounded operators on $L^2(G)$. We show that its ${\mathrm{C}}^{\ast }$-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 ${\mathbb{R}}_{>0}$-action on a certain ideal in the ${\mathrm{C}}^{\ast }$-algebra of the tangent groupoid of $G$. The action takes the graded structure of $G$ into account. Our construction allows to compute the $\mathrm{K}$-theory of the algebra of symbols.