Classification of traces and associated determinants on odd-class operators in odd dimensions

Carolina Neira Jiménez^a, Marie Françoise Ouedraogo<sup>b

a</sup> Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
^b Départment de Mathématiques, Université de Ouagadougou, 03 BP 7021, Burkina Faso

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.

Keywords: pseudodifferential operators, odd-class, trace, determinant, logarithm, regular Lie group.

MSC: 58J40; 47C05

Received: November 30, 2011; in final form April 11, 2012; Published online April 21, 2012

Language: English

DOI: 10.3842/SIGMA.2012.023

Bibliographic databases:

• 
• 
• 

ArXiv: 1111.6750