Fields of Toeplitz algebras for the principal symbol of regular 2-step nilpotent groups

TL;DR

This paper demonstrates how the C*-algebra of regular 2-step nilpotent Lie groups and related manifolds can be reconstructed using continuous fields of Toeplitz algebras, extending to polycontact and H-type groups.

Contribution

It introduces a method to recover the algebra of principal symbols via Toeplitz fields for 2-step nilpotent groups and polycontact manifolds, including H-type structures.

Findings

C*-algebra of 2-step nilpotent groups can be reconstructed from Toeplitz fields.

Principal symbol algebra is recoverable from Toeplitz fields in polycontact and H-type manifolds.

Extension of the filtered pseudodifferential calculus to these algebraic structures.

Abstract

We show that the C*-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.