The Wodzicki Residue for Pseudo-Differential Operators on Compact Lie Groups

Abstract

Let G be an arbitrary compact Lie group. In this work we apply the method of the analytic continuation of traces in order to compute the Wodzicki residue for a classical pseudo-differential operator on G in terms of its matrix-valued symbol (which is globally defined on the non-commutative phase space $$G\times \widehat {G},$$ with $$\widehat {G}$$ being the unitary dual of G). Our main theorem is complementary to the results in Cardona et al. (Dixmier traces, Wodzicki residues, and determinants on compact Lie groups: the paradigm of the global quantisation. arXiv:2105.14949), where we remove the ellipticity hypothesis when the operators belong to the Hörmander classes on G defined by local coordinate systems.