A Note on Twisted Crossed Products and Spectral Triples

TL;DR

This paper extends the construction of spectral triples from group actions on C*-algebras to twisted crossed products, analyzing properties like summability and regularity, with applications to noncommutative coverings.

Contribution

It generalizes existing spectral triple constructions to twisted crossed products with uniformly bounded actions, exploring their fundamental properties and examples.

Findings

Spectral triples can be constructed for twisted crossed products under boundedness conditions.

The resulting triples exhibit properties like summability and regularity.

Noncommutative coverings with finite abelian groups serve as key examples.

Abstract

Starting with a spectral triple on a unital $C^{*}$-algebra $A$ with an action of a discrete group $G$, if the action is uniformly bounded (in a Lipschitz sense) a spectral triple on the reduced crossed product $C^{*}$-algebra $A\rtimes_{r} G$ is constructed in [Hawkins, Skalski, White, Zacharias. Mathematica Scandinavica 2013]. The main instrument is the Kasparov external product. We note that this construction still works for twisted crossed products when the twisted action is uniformly bounded in the appropriate sense. Under suitable assumptions we discuss some basic properties of the resulting triples: summability and regularity. Noncommutative coverings with finite abelian structure group are among the most basic, still interesting, examples of twisted crossed products; we describe their main features.