Crossed product extensions of spectral triples

TL;DR

This paper develops a method to extend spectral triples to crossed product algebras, enabling the construction of modular-type twisted spectral triples with applications to affine and conformal groups.

Contribution

It introduces a general procedure for extending spectral triples to crossed products, including the promotion to twisted spectral triples, with concrete examples.

Findings

Extension of spectral triples to crossed product $C^*$-algebras.

Construction of modular-type twisted spectral triples.

Applications to affine and conformal groups on Riemannian manifolds.

Abstract

Given a spectral triple $(A,H,D)$ and a $C^*$-dynamical system $(\mathbf{A}, G, \alpha)$ where $A$ is dense in $\mathbf{A}$ and $G$ is a locally compact group, we extend the triple to a triplet $(\mathcal{B},\mathcal{H},\mathcal{D})$ on the crossed product $G \ltimes_{\alpha, \text{red}} \mathbf{A}$ which can be promoted to a modular-type twisted spectral triple within a general procedure exemplified by two cases: the $C^*$-algebra of the affine group and the conformal group acting on a complete Riemannian spin manifold.