Real spectral triples on crossed products

Given a spectral triple on unital $C^{*}$-algebra $A$ and an equicontinuous action of discrete group $G$ $A$, the reduced crossed product $A\rtimes_r G$ was constructed by Hawkins, Skalski, White Zacharias in [On triples products arising from actions, Math. Scand. 113(2) (2013) 262-291], extending construction Belissard, Marcolli Reihani [Dynamical systems metric spaces, preprint (2010), arXiv:1008.4617], using Kasparov to make ansatz for Dirac operator. Supposing that is equivariant $G$, we show dual coaction $G$. If moreover real structure $J$ given give constructions two inequivalent structures $A\rtimes_rG$. We compute KO-dimension with respect each terms first second order conditions are preserved. Lastly, characterise orientation cycle $A\rtimes_rG$ coming $A$. show, along paper, our generalize respective noncommutative $2$-torus.