Real spectral triples on crossed products

Abstract

Given a spectral triple on a unital [Formula: see text]-algebra [Formula: see text] and an equicontinuous action of a discrete group [Formula: see text] on [Formula: see text], a spectral triple on the reduced crossed product [Formula: see text]-algebra [Formula: see text] 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 [Formula: see text] is equivariant for an action of [Formula: see text], we show that the triple on [Formula: see text] is equivariant for the dual coaction of [Formula: see text]. If moreover an equivariant real structure [Formula: see text] is given for the triple on [Formula: see text], we give constructions for two inequivalent real structures on the triple [Formula: see text]. We compute the KO-dimension with respect to each real structure in terms of the KO-dimension of [Formula: see text] and show that the first- and the second-order conditions are preserved. Lastly, we characterize an equivariant orientation cycle on the triple on [Formula: see text] coming from an equivariant orientation cycle on the triple on [Formula: see text]. We show, along the paper, that our constructions generalize the respective constructions of the equivariant spectral triple on the noncommutative [Formula: see text]-torus.