Real Spectral Triples on Crossed Products

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