Convergence of inductive sequences of spectral triples for the spectral propinquity

In the context of metric geometry, we introduce a new necessary and sufficient condition for the convergence of an inductive sequence of quantum compact metric spaces for the Gromov-Hausdorff propinquity, which is a noncommutative analogue of the Gromov-Hausdorff distance for compact metric spaces. This condition is easy to verify in many examples, such as quantum compact metric spaces associated to AF algebras or certain twisted convolution C*-algebras of discrete inductive limit groups. Our condition also implies the convergence of an inductive sequence of spectral triples in the sense of the spectral propinquity, a generalization of the Gromov-Hausdorff propinquity on quantum compact metric spaces to the space of metric spectral triples. In particular we show the convergence of the state spaces of the underlying C*-algebras as quantum compact metric spaces, and also the convergence of the quantum dynamics induced by the Dirac operators in the spectral triples. We apply these results to new classes of inductive limit of even spectral triples on noncommutative solenoids and Bunce-Deddens C*-algebras. Our construction, which involves length functions with bounded doubling, adds geometric information and highlights the structure of these twisted C*-algebras as inductive limits.