Crossed product extensions of spectral triples

Abstract

Given a spectral triple $(A,H,D)$ and a $C^{\ast }$-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 $G{⋉}_{\alpha ,\mathrm{red}}\mathbf{A}$ which can be promoted to a modular-type twisted spectral triple within a general procedure exemplified by two cases: the $C^{\ast }$-algebra of the affine group and the conformal group acting on a complete Riemannian spin manifold.