Fields of Toeplitz algebras form the principal symbols of regular 2-step nilpotent groups

Abstract

We show that the $C^{\ast }$-algebra of a regular 2-step nilpotent Lie group can be recovered using continuous fields of Toeplitz algebras and a crossed product. We generalize this result to polycontact manifolds in the sense of van Erp which are endowed with fields of such groups. We also investigate those manifolds with a more rigid structure, namely those modeled on H-type groups. In all those cases, there is a certain pseudodifferential calculus named filtered calculus, we show that the algebra of principal symbols can also be recovered from the field of Toeplitz algebras.