Classification of Traces and Associated Determinants on Odd-Class Operators in Odd Dimensions

TL;DR

This paper classifies traces and determinants on odd-class pseudodifferential operators of non-positive order in odd dimensions, extending understanding of their algebraic and geometric structures.

Contribution

It provides a new classification of traces and determinants on odd-class pseudodifferential operators in odd dimensions, expanding prior classifications.

Findings

Classified traces on odd-class pseudodifferential operators.

Established a correspondence between traces and determinants on associated Lie groups.

Discussed potential extensions of determinants beyond neighborhoods of the identity.

Abstract

To supplement the already known classification of traces on classical pseudodifferential operators, we present a classification of traces on the algebras of odd-class pseudodifferential operators of non-positive order acting on smooth functions on a closed odd-dimensional manifold. By means of the one to one correspondence between continuous traces on Lie algebras and determinants on the associated regular Lie groups, we give a classification of determinants on the group associated to the algebra of odd-class pseudodifferential operators with fixed non-positive order. At the end we discuss two possible ways to extend the definition of a determinant outside a neighborhood of the identity on the Lie group associated to the algebra of odd-class pseudodifferential operators of order zero.