Metric Spectral Geometry of $\mathbb{Z}^d$-Crossed Products beyond Equicontinuity

Abstract

We construct linearly twisted spectral triples on $C^\ast$-crossed products $\mathfrak{A}\rtimes_α\mathbb{Z}^d$, starting from a spectral triple $(\mathfrak{A},\mathcal{H},D)$ and an action $α$ admitting an adapted weight. The weight compensates for the growth of the base Dirac operator under the dynamics while preserving the canonical differential structure in the $\mathbb{Z}^d$-directions. The construction takes place directly on the original crossed product and does not require the action to be equicontinuous. We prove that, whenever the base spectral triple is metric, the resulting twisted spectral triple is $κ$-metric for every $κ>0$: its associated Monge--Kantorovich metric metrizes the weak* topology on the state space of the crossed product. Our framework applies, in particular, to every action preserving the domain of the Lipschitz seminorm of the base triple. In this case, an exponential weight may be chosen so that the induced metric geometry restricts exactly to the original metric geometry on the base algebra. In the equicontinuous case, the weight may be chosen constant, the twist disappears, and the standard crossed-product spectral triple is recovered. We also establish a general finite-mode criterion for convergence in the spectral propinquity. As an application, we prove joint continuity, with respect to the deformation parameter and suitable fluctuations of the flat metric, for the twisted spectral triples associated with crossed products of quantum $2$-tori by fixed automorphisms in $\operatorname{SL}(2,\mathbb{Z})$, including Anosov automorphisms.