For a fixed prime $p$, a noncommutative solenoid as defined by Frederic Latremoliere and Judith PackerÂ is a twisted group C*-algebra ${C}^{*}\phantom{\rule{0.167em}{0ex}}\left(\mathbb{Z}[\frac{1}{p}]\times \mathbb{Z}[\frac{1}{p}],\sigma \right)$, where $\mathbb{Z}[\frac{1}{p}]:=\left\{\frac{j}{{p}^{k}}\in \mathbb{Q}:j\in \mathbb{Z},k\in \mathbb{N}\right\}$ is an additive discrete group and $\sigma$ is a $\mathbb{T}-$valued group $2$-cocycle (multiplier) on $\mathbb{Z}[\frac{1}{p}]\times \mathbb{Z}[\frac{1}{p}]$. In this talk, we first review the classification of all NC solenoids in terms of their defining multipliers using $K-$theory. From there, we discuss two constructions for projective modules over the irrational NC solenoids and how they are related: one by constructing the Heisenberg equivalence bimodule of Rieffel utilizing the $p-$adic numbers, and the other by writing the NC solenoids as direct limits of noncommutative tori.

Noncommutative solenoids and their finitely generated projective modules