Trace Expansions and Equivariant Traces on an Algebra of Fourier Integral Operators on $\mathbb R^n$

Abstract

We consider the operator algebra $\mathscr A$ on $\mathscr S(\mathbb R^n)$ generated by the Shubin type pseudodifferential operators, the Heisenberg-Weyl operators and the lifts of the unitary operators on $\mathbb C^n$ to metaplectic operators. With the help of an auxiliary operator in the Shubin calculus, we find trace expansions for these operators in the spirit of Grubb and Seeley. Moreover, we can define a noncommutative residue generalizing that for the Shubin pseudodifferential operators and obtain a class of localized equivariant traces on the algebra.