Weakly parametric pseudodifferential calculus for twisted $C^{\ast }$-dynamical systems

Abstract

For a twisted $C^{\ast }$-dynamical system $(\mathcal{A},{\mathbb{R}}^n,\alpha ,e)$ over a unital $C^{\ast }$-algebra, we establish a weakly parametric pseudodifferential calculus analogously to the celebrated weakly parametric calculus due to Grubb and Seeley [Invent. Math. 121 (1995), 481–529]. If the $C^{\ast }$-algebra $\mathcal{A}$ has an $\alpha$-invariant trace, then we prove an expansion of the resolvent trace (with respect to the dual trace on multipliers) for suitable pseudodifferential multipliers. The question whether the expansion holds true as a Hilbert space trace expansion in concrete GNS spaces for $\mathcal{A}$ will be addressed in a future publication.