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 $C^{\ast }$-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 $C^{\ast }$-algebra of the tangent groupoid of $G$. The action takes the graded structure of $G$ into account. Our construction allows to compute the $K$-theory of the algebra of symbols.

40 pages; parts of the paper were reorganized; Proposition 8.4 is a strengthening of result 10.5 in [V1], where only a bijection was shown

References in corpus (1)

• Spectral Continuity for Aperiodic Quantum Systems I. General Theory

Cited by in corpus (1)

• Shubin calculi for actions of graded Lie groups