A Note on Twisted Crossed Products and Spectral Triples

Abstract

Starting with a spectral triple on a unital $C^{\ast }$-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 $C^{\ast }$-algebra $A{⋊}_{r}G$ 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.

33 pages, various corrections and improvements

References in corpus (1)

• Lifting spectral triples to noncommutative principal bundles

Cited by in corpus (2)

• Realizations of free actions via their fixed point algebras
• Non-Abelian extensions of groupoids and their groupoid rings