Classification of traces and associated determinants on odd-class operators in odd dimensions. (English) Zbl 1246.58023

Jiménez, Carolina Neira; Ouedraogo, Marie Françoise

SIGMA, Symmetry Integrability Geom. Methods Appl. 8, Paper 023, 25 p. (2012).

Summary: 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.

MSC:

58J40	Pseudodifferential and Fourier integral operators on manifolds
47C05	Linear operators in algebras

Keywords:

pseudodifferential operators; odd-class; trace; determinant; logarithm; regular Lie group