Explicit Construction of Equivalence Bimodules between Noncommutative Solenoids

Abstract

Let $p\in \mathbb{N}$ be prime, and let $\theta$ be irrational. The authors have previously shown that the noncommutative $p$-solenoid corresponding to the multiplier of the group $\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)^2$ parametrized by $\alpha=(\theta+1, (\theta+1)/p,\cdots, (\theta+1)/p^j,\cdots )$ is strongly Morita equivalent to the noncommutative solenoid on $\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)^2$ coming from the multiplier $\beta= (1-\frac{\theta+1}{\theta},1-\frac{\theta+1}{p\theta}, \cdots, 1-\frac{\theta+1}{p^j\theta}, \cdots )$ . The method used a construction of Rieffel referred to as the "Heisenberg bimodule" in which the two noncommutative solenoid corresponds to two different twisted group algebras associated to dual lattices in $(\mathbb Q_p\times \mathbb R)^2.$ In this paper, we make three additional observations: first, that at each stage, the subalgebra given by the irrational rotation algebra corresponding to $\alpha_{2j}=(\theta+1)/p^{2j}$ is strongly Morita equivalent to the irrational rotation algebra corresponding to the irrational rotation algebra corresponding to $\beta_{2j}= 1-\frac{\theta+1}{p^{2j}\theta}$ by a different construction of Rieffel, secondly, that that Rieffel's Heisenberg module relating the two non commutative solenoids can be constructed as the closure of a nested sequence of function spaces associated to a multiresolution analysis for a $p$-adic wavelet, and finally, at each stage, the equivalence bimodule between $A_{\alpha_{2j}}$ and $A_{\beta_{2j}}$ can be identified with the subequivalence bimodules arising from the $p$-adic MRA. Aside from its instrinsic interest, we believe this construction will guide us in our efforts to show that certain necessary conditions for two noncommutative solenoids to be strongly Morita equivalent are also sufficient.