  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 $\mathbb{R}_{>0}$-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.
