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 $\mathbb{Z}_6$.

Disclosure

“visor Dr. Sayan Chakraborty for helpful discussions and valuable suggestions regarding this article. The author also thanks Dhrubajyoti Das for useful discussions on p-adic analysis. The author acknowledges the use of ChatGPT, developed by OpenAI, for assistance with language editing and improving the presentation of parts of this manuscript. The author was supported by the TCG CREST PhD Fellowship. References [AGI17] V. Aiello, D.”

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2026-08-24T00:28:51.963Z

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