Weakly Parametric Pseudodifferential Calculus for Twisted $C^*$-dynamical Systems

TL;DR

This paper develops a weakly parametric pseudodifferential calculus for twisted $C^*$-dynamical systems, extending classical methods to noncommutative settings and analyzing resolvent trace expansions under certain invariance conditions.

Contribution

It introduces a novel weakly parametric pseudodifferential calculus for twisted $C^*$-dynamical systems and proves resolvent trace expansion results when an invariant trace exists.

Findings

Established a pseudodifferential calculus for twisted $C^*$-dynamical systems.

Proved resolvent trace expansion under invariant trace conditions.

Addressed future questions on Hilbert space trace expansions.

Abstract

For a twisted $C^*$-dynamical system $(\mathscr{A},\mathbb{R}^n,\alpha,e)$ over a unital $C^*$-algebra we establish a weakly parametric pseudodifferential calculus analogously to the celebrated weakly parametric calculus due to Grubb and Seeley. If the $C^*$-algebra $\mathscr{A}$ 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 $\mathscr{A}$ will be addressed in a future publication.