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⋊rG 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⋊rG 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⋊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 characterize an equivariant orientation cycle on the triple on A⋊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.