# Pseudo-differential extension for graded nilpotent Lie groups

### Eske Ewert

Georg-August-Universität Göttingen, Göttingen, Germany; Leibniz Universität Hannover, Hannover, Germany

## Abstract

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.

## Cite this article

Eske Ewert, Pseudo-differential extension for graded nilpotent Lie groups. DM 28 (2023), no. 6, pp. 1323–1379

DOI 10.4171/DM/940