Noncommutative residues, equivariant traces, and trace expansions for an operator algebra on Rn

We consider an algebra A of Fourier integral operators on Rn. It consists of all operators D:S(Rn)→S(Rn) on the Schwartz space S(Rn) that can be written as finite sums D=∑RgTwA, with Shubin type pseudodifferential operators A, Heisenberg-Weyl operators Tw, w∈Cn, and lifts Rg, g∈U(n), of unitary matrices g on Cn to operators Rg in the complex metaplectic group. For D∈A and a suitable auxiliary Shubin pseudodifferential operator H we establish expansions for Tr(D(H−λ)−K) as |λ|→∞ in a sector of C for sufficiently large K and of Tr(De−tH) as t→0+. We also obtain the singularity structure of the meromorphic extension of z↦Tr(DH−z) to C. Moreover, we find a noncommutative residue as a suitable coefficient in these expansions and construct from it a family of localized equivariant traces on the algebra. © 2024 The Authors