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Groupoids, equivalence bibundles and bimodules for noncommutative solenoids

Paulo Carrillo Rouse, Laurent Guillaume

TL;DR

The paper advances a geometric Morita-theory program for noncommutative solenoids by constructing explicit equivalence bibundles for solenoidal groupoids $S_\alpha$ and $S_\beta$ when $\alpha$ and $\beta$ lie in the same $GL_2(\mathbb{Z}[1/p])$-orbit. Building on Connes–Rieffel ideas for noncommutative tori, it introduces solenoidal groupoids $\mathscr{S}_\alpha$ and derives explicit bibundles $P_M$ (including $P_{\alpha,\alpha^{-1}}$) and their moment maps, establishing Morita equivalences between the corresponding groupoids. These equivalences translate into explicit imprimitivity bimodules between the noncommutative solenoids $C^*(S_\alpha)$ and $C^*(S_\beta)$, unifying prior results (LP1–LP3, Lu) within a single groupoid framework and linking to the $K$-theory and dynamical system perspectives. The work thus provides a concrete, scalable method to realize Morita equivalences for a broad class of twisted group C*-algebras arising from solenoids and their groupoid models.

Abstract

Let $p$ be a prime number and $\mathcal{S}_p$ the $p$-solenoid. For $α\in \mathbb{R}\times \mathbb{Q}_p$ we consider in this paper a naturally associated action groupoid $S_α:=\mathbb{Z} [1/p]\ltimes_α\mathcal{S}_p \rightrightarrows \mathcal{S}_p$ whose $C^*-$algebra is a model for the noncommutative solenoid $\mathcal{A}_α^\mathscr{S}$ studied by Latremolière and Packer. Following the geometric ideas of Connes and Rieffel to describe the Morita equivalences of noncommutative torus using the Kronecker foliation on the torus, we give an explicit description of the geometric/topologic equivalence bibundle for groupoids $S_α$ and $S_β$ whenever $α,β\in \mathbb{R}\times \mathbb{Q}_p$ are in the same orbit of the $GL_2(\mathbb{Z}[1/p])$ action by linear fractional transformations. As a corollary, for $α,β\in \mathbb{R}\times \mathbb{Q}_p$ as above we get an explicit description of the imprimitivity bimodules for the associated noncommutative solenoids.

Groupoids, equivalence bibundles and bimodules for noncommutative solenoids

TL;DR

The paper advances a geometric Morita-theory program for noncommutative solenoids by constructing explicit equivalence bibundles for solenoidal groupoids and when and lie in the same -orbit. Building on Connes–Rieffel ideas for noncommutative tori, it introduces solenoidal groupoids and derives explicit bibundles (including ) and their moment maps, establishing Morita equivalences between the corresponding groupoids. These equivalences translate into explicit imprimitivity bimodules between the noncommutative solenoids and , unifying prior results (LP1–LP3, Lu) within a single groupoid framework and linking to the -theory and dynamical system perspectives. The work thus provides a concrete, scalable method to realize Morita equivalences for a broad class of twisted group C*-algebras arising from solenoids and their groupoid models.

Abstract

Let be a prime number and the -solenoid. For we consider in this paper a naturally associated action groupoid whose algebra is a model for the noncommutative solenoid studied by Latremolière and Packer. Following the geometric ideas of Connes and Rieffel to describe the Morita equivalences of noncommutative torus using the Kronecker foliation on the torus, we give an explicit description of the geometric/topologic equivalence bibundle for groupoids and whenever are in the same orbit of the action by linear fractional transformations. As a corollary, for as above we get an explicit description of the imprimitivity bimodules for the associated noncommutative solenoids.

Paper Structure

This paper contains 14 sections, 10 theorems, 31 equations.

Table of Contents

  1. Locally compact groupoids and bibundle equivalences
  2. Solenoids as homogeneous spaces
  3. Exact sequences for solenoids
  4. $\mathscr{S}_{p}$ is an homogeneous space for the actions of ${\mathbb{R}} \times {\mathbb{Z}}_p$ and ${\mathbb{R}} \times {\mathbb Q}_p$
  5. Equivalence bibundles of noncommutative solenoids
  6. Solenoidal groupoids $S_\alpha$ and $\mathscr{S}_\alpha$
  7. $S_\alpha$-$S_{\alpha^{-1}}$ equivalence bibundle
  8. $S_\alpha$-$S_\beta$ equivalence bibundles $P_M$ for $M\in SL_2({\mathbb{Z}}[1/p])$
  9. Extension to $GL_2({\mathbb{Z}}[1/p])$ is trivial up to isomorphism
  10. Equivalence bibundles of noncommutative torus
  11. Equivalence bimodules of noncommutative solenoids
  12. Noncommutative solenoids
  13. The groupoid $C^*$-algebra $C^*(S_\alpha)$ as a noncommutative solenoid
  14. Imprimitivity bimodules and the Equivalence Theorem

Key Result

Proposition 2.1

The maps $\pi$ and $\Delta$ induce the exact sequences of topological groups \xymatrix{ 1 \ar[r] & {\mathbb{Z}} \ar[r]^\delta & {\mathbb{R}} \times {\mathbb{Z}}_p \ar[r]^{\pi} & \mathscr{S}_{p} \ar[r] & 1\\ }\xymatrix{ 1 \ar[r] & {\mathbb{Z}}[1/p] \ar[r]^\delta & {\mathbb{R}} \times {\m

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 1.2
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Example 3.4
  • ...and 6 more