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 ${\mathrm{C}}^{\ast }$-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 ${\mathbb{Z}}_6$.

37 pages