Heisenberg-Weyl Representations and Morita equivalence for crossed products of Noncommutative solenoids

Abstract

We study strong Morita equivalence for crossed products of noncommutative solenoids by cyclic subgroups of $\mathrm{SL}_2(\mathbb Z[1/p])$. For a large class of parameters, we construct a multiplier on $\mathbb{Z}[1/p]^2$ which is invariant under the natural action of $\mathrm{SL}_2(\mathbb{Z}[1/p])$ and cohomologous to the usual multiplier defining the solenoid. This invariant representative allows us to describe the corresponding crossed products as twisted group $\mathrm{C}^*$-algebras. We also show that the induced action of $\mathrm{SL}_2(\mathbb{Z})$ on the noncommutative solenoid is compatible with the classical Watatani action on the rotation algebras in the inductive-limit system. We then develop a Heisenberg--Weyl framework on $\mathrm{L}^2(\mathbb{Q}_p\times\mathbb{R})$ adapted to these invariant multipliers. Using explicit unitary operators implementing the generators of $\mathrm{SL}_2(\mathbb{Z}[1/p])$, we extend the Heisenberg equivalence bimodule to the crossed-product setting. As a consequence, we obtain strong Morita equivalences for crossed products by infinite cyclic subgroups and by the finite cyclic subgroups $\mathbb{Z}_2,\mathbb{Z}_3,\mathbb{Z}_4$ and $\mathbb{Z}_6$.