---
title: Convolution Bimodules of Groupoid Bibundles
url: https://www.emergentmind.com/topics/convolution-bimodules-of-groupoid-bibundles
type: topic
---

# Convolution Bimodules of Groupoid Bibundles

Convolution bimodules of groupoid bibundles are Hilbert \(C^*\)-bimodules obtained from spaces of compactly supported functions on a bibundle \(Z\) carrying commuting left and right groupoid actions. Their defining operations are convolution-type integrals along the groupoid fibers, and when \(Z\) is an equivalence bibundle they implement Morita equivalence of the associated groupoid \(C^*\)-algebras. In the arithmetic setting of noncommutative solenoids, Carrillo Rouse and Guillaume give an explicit realization of this mechanism for solenoidal action groupoids \(S_\alpha\) and \(S_\beta\), thereby turning a geometric equivalence bibundle into an explicit imprimitivity bimodule between noncommutative solenoid algebras [2503.13251].

## 1. Bibundles and the meaning of convolution bimodules

A \(G\)-\(H\) bibundle equivalence, for locally compact groupoids \(G\rightrightarrows G^{(0)}\) and \(H\rightrightarrows H^{(0)}\) with open range maps, is a space \(P\) equipped with a free and proper left \(G\)-action, a free and proper right \(H\)-action, commuting actions, and open moment maps \(l:P\to G^{(0)}\), \(r:P\to H^{(0)}\) inducing homeomorphisms
\[
P/H \xrightarrow{\simeq} G^{(0)},\qquad G\backslash P \xrightarrow{\simeq} H^{(0)}.
\]
Such a bibundle realizes topological Morita equivalence of groupoids, and in the Lie groupoid framework biprincipal bibundles are precisely the data expressing Morita equivalence [2507.03091].

The analytic object attached to an equivalence bibundle is the convolution bimodule. Its underlying pre-Hilbert space is \(C_c(Z)\), the compactly supported continuous functions on the bibundle, and its left and right module structures are defined by convolution against \(C_c(G)\) and \(C_c(H)\). In the standard formulation used in the solenoidal case, the left action and right action are
\[
(g \cdot \varphi)(z)
= \int_G g(\gamma)\,\varphi(\gamma^{-1}\cdot z)\,d\lambda^{r(z)}(\gamma),
\]
\[
(\varphi \cdot h)(z)
= \int_H \varphi(z\cdot\eta)\,h(\eta^{-1})\,d\mu^{s(z)}(\eta),
\]
with analogous convolution-type formulas for the \(C_c(G)\)- and \(C_c(H)\)-valued inner products. The completion of \(C_c(Z)\) is then an imprimitivity bimodule whenever \(Z\) is a \((G,H)\)-equivalence. A common simplification is to treat any space with commuting actions as sufficient; the precise theory requires freeness, properness, and quotient conditions on the moment maps, because these are the hypotheses under which the convolution formulas satisfy the imprimitivity axioms.

## 2. Solenoidal action groupoids and their \(C^*\)-algebras

Fix a prime \(p\), and let \(\mathscr{S}_p\) be the \(p\)-solenoid, realized as
\[
\mathscr{S}_p
=
\left\{
(z_n)_{n\in\mathbb{N}} \in \mathbb{T}^{\mathbb{N}}
:
z_{n+1}^p=z_n\ \forall n
\right\}.
\]
There is an action of \(\mathbb{R}\times\mathbb{Q}_p\) on \(\mathscr{S}_p\) by translations,
\[
\pi : \mathbb{R}\times\mathbb{Q}_p \to \operatorname{Homeo}(\mathscr{S}_p),
\quad
(t,r)\mapsto \bigl(z\mapsto \omega(t)\,\zeta(-r)\,z\bigr),
\]
and \(\mathscr{S}_p\) becomes a homogeneous space for this action. For \(\alpha\in \mathbb{R}\times\mathbb{Q}_p\), the reduced solenoidal groupoid is the transformation groupoid
\[
S_\alpha:=\mathbb{Z}[1/p]\ltimes_\alpha \mathscr{S}_p \rightrightarrows \mathscr{S}_p,
\]
where
\[
q\cdot z=\pi(q\alpha)\,z,\qquad q\in \mathbb{Z}[1/p],\ z\in \mathscr{S}_p.
\]
Its arrows are pairs \((q,z)\), with structure maps
\[
s(q,z)=z,\qquad r(q,z)=q\cdot z,
\]
\[
(q',z')\cdot (q,z)=(q'+q,z)\quad\text{whenever } z'=q\cdot z,
\]
\[
(q,z)^{-1}=(-q,\,q\cdot z).
\]

These groupoids model the noncommutative solenoids of Latremolière and Packer. If \(\pi(\alpha)=a\in \mathscr{S}_p\), then the associated noncommutative solenoid is the twisted group \(C^*\)-algebra
\[
\mathscr{A}_a^\mathscr{S}=C^*(\mathbb{Z}[1/p]^2,\Psi_a),
\]
and Carrillo Rouse and Guillaume identify the groupoid algebra with this solenoid algebra:
\[
C^*(S_\alpha)\cong \mathscr{A}_{\pi(\alpha)}^\mathscr{S}.
\]
The identification is made explicit using generators
\[
U_k=\delta_{1/p^k}\otimes 1_{\mathscr{S}_p},\qquad
V_l=\delta_0\otimes p_l,
\]
which satisfy the solenoid relations
\[
U_kV_l=e^{2i\pi a_{k+l}}V_lU_k,\qquad
U_{k+1}^p=U_k,\qquad
V_{l+1}^p=V_l.
\]
This places the study of noncommutative solenoids inside the geometry of transformation groupoids and their bibundles [2503.13251].

## 3. The explicit equivalence bibundle \(P_M\)

The central geometric construction arises when \(\alpha,\beta\in \mathbb{R}\times\mathbb{Q}_p\) are related by a linear fractional transformation coming from
\[
M=
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}
\in SL_2(\mathbb{Z}[1/p]),\qquad c\neq 0,
\]
with
\[
\beta=M^{-1}\cdot \alpha
=
-\frac{b-d\alpha}{a-c\alpha}.
\]
In this situation the bibundle is
\[
P_M=\mathbb{R}\times\mathbb{Q}_p.
\]
Its moment maps are
\[
\mu(q)=\pi\bigl(q(a-c\alpha)/c\bigr),\qquad
\epsilon(q)=\pi(q/c),
\]
and the groupoid actions are
\[
(n,z)\cdot q = n+q
\quad\text{given } z=\pi\bigl((q+n)(a-c\alpha)/c\bigr),
\]
for the left \(S_\alpha\)-action, and
\[
q\cdot (n,z)= q+n(a-c\alpha)^{-1}
\quad\text{given } z=\pi(q/c),
\]
for the right \(S_\beta\)-action.

The geometric origin of \(P_M\) lies in a larger action groupoid on \(\mathscr{S}_p^2\). Two transversals are chosen,
\[
V=\{(1,y):y\in \mathscr{S}_p\},
\qquad
H_M=\{(\pi(cq),\pi(aq)): q\in \mathbb{R}\times\mathbb{Q}_p\},
\]
and the restricted groupoids satisfy
\[
S_\alpha{}_V^V \simeq S_\alpha,\qquad
S_\alpha{}_{H_M}^{H_M}\simeq S_\beta,\qquad
S_\alpha{}_V^{H_M}\simeq P_M.
\]
The theorem is that \(P_M\), with these actions and moment maps, is an equivalence bibundle between \(S_\alpha\) and \(S_\beta\). For \(M\in GL_2(\mathbb{Z}[1/p])\) with determinant \(\epsilon=\pm p^l\), one writes
\[
\tilde M = M M_\epsilon,\qquad
M_\epsilon=
\begin{pmatrix}
\epsilon & 0\\
0 & 1
\end{pmatrix},
\]
and the determinant unit only changes \(\alpha\) by an isomorphic parameter, so the relevant Morita classes are controlled by the \(GL_2(\mathbb{Z}[1/p])\)-orbit of \(\alpha\) [2503.13251].

## 4. The convolution bimodule \(E_M\)

Once \(P_M\) is identified as an equivalence bibundle, the associated convolution bimodule is built on
\[
C_c(P_M)=C_c(\mathbb{R}\times\mathbb{Q}_p).
\]
Because \(\mathbb{R}\times\mathbb{Q}_p\) is a second countable locally compact Hausdorff group, this is the usual space of compactly supported continuous complex-valued functions. The left and right module structures are obtained from the general bibundle formulas. In the solenoidal case they specialize to
\[
(f\cdot \xi)(q)
=
\int_{S_\alpha}
f(n,z)\,
\xi\bigl((n,z)^{-1}\cdot q\bigr)\,
d\lambda^{\mu(q)}(n,z),
\]
\[
(\xi\cdot g)(q)
=
\int_{S_\beta}
\xi\bigl(q\cdot (n,z)\bigr)\,
g\bigl((n,z)^{-1}\bigr)\,
d\mu^{\epsilon(q)}(n,z).
\]
Since \(S_\alpha\) and \(S_\beta\) are discrete in the \(\mathbb{Z}[1/p]\)-direction, these integrals reduce to countable sums over \(n\in \mathbb{Z}[1/p]\). The left action is therefore a twisted convolution built from the translation \((n,z)\cdot q=n+q\), while the right action is governed by the affine translation \(q\cdot(n,z)=q+n(a-c\alpha)^{-1}\).

The inner products are likewise defined by convolution-type integrals over groupoid fibers, producing \(C_c(S_\alpha)\)- and \(C_c(S_\beta)\)-valued pairings. Their compatibility with the module actions is expressed by the usual imprimitivity identities, including
\[
\langle f\cdot\xi,\eta\rangle_{S_\beta}
=
\langle \xi,f^*\cdot\eta\rangle_{S_\beta},
\qquad
\langle \xi,\eta\cdot g\rangle_{S_\alpha}
=
\langle \xi\cdot g^*,\eta\rangle_{S_\alpha}.
\]
The induced norms from the left and right inner products coincide, and the completion is an imprimitivity bimodule
\[
E_M
\]
implementing
\[
C^*(S_\alpha)\sim_{\mathrm{Morita}} C^*(S_\beta).
\]
The term “convolution” is literal: the groupoid algebras are defined by convolution, and the bimodule actions and inner products are built from the same type of integral formulas, now transported through the bibundle [2503.13251].

## 5. Morita equivalence of noncommutative solenoids

Combining the identification \(C^*(S_\alpha)\cong \mathscr{A}_{\pi(\alpha)}^\mathscr{S}\) with the bibundle equivalence \(S_\alpha\simeq S_\beta\), one obtains a concrete Morita equivalence statement for noncommutative solenoids. If
\[
M=
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}
\in GL_2(\mathbb{Z}[1/p]),
\qquad
\beta=M^{-1}\cdot \alpha,
\]
then \(C_c(\mathbb{R}\times\mathbb{Q}_p)\) completes to a \(C^*(S_\alpha)\)-\(C^*(S_\beta)\) imprimitivity bimodule \(E_M\), and therefore the noncommutative solenoids attached to \(\alpha\) and \(\beta\) are strongly Morita equivalent.

This yields a classification statement: two noncommutative solenoids are strongly Morita equivalent exactly when their parameters lie in the same \(GL_2(\mathbb{Z}[1/p])\)-orbit, modulo the determinant units \(\pm p^{\mathbb{Z}}\), which only change the parameter by isomorphic groupoids. The pattern is explicitly compared with the Connes–Rieffel description of Morita equivalence for noncommutative \(2\)-tori via \(GL_2(\mathbb{Z})\)-orbits. In the solenoidal setting, the ingredients are arithmetic rather than purely real: \(\mathbb{Z}[1/p]\), the \(p\)-solenoid \(\mathscr{S}_p\), and the mixed parameter space \(\mathbb{R}\times\mathbb{Q}_p\).

A frequent misconception is that Morita equivalence here is merely an abstract consequence of orbit data. The explicit construction shows otherwise. The equivalence is realized geometrically by the bibundle \(P_M=\mathbb{R}\times\mathbb{Q}_p\), analytically by the convolution bimodule \(C_c(P_M)\), and algebraically by the resulting equivalence of noncommutative solenoid \(C^*\)-algebras. The bridge between these levels is the equivalence theorem for groupoid bibundles and their convolution completions [2503.13251].

## 6. Broader formulations and extensions

The same pattern extends beyond the specific \(C^*\)-algebraic solenoidal model. In the bornological framework for Lie groupoids, the complete bornological convolution algebra
\[
A(G)=C^\infty_{\mathrm c}(G_1)
\]
and the convolution bimodule
\[
M(P)=C^\infty_{\mathrm c}(P)
\]
of a groupoid bibundle \(P\) define a symmetric monoidal weak \(2\)-functor from the \(2\)-category of differentiable stacks to the Morita \(2\)-category of complete bornological algebras. In this setting the algebras are generally non-unital, but they possess one-sided approximate units such that the multiplication operators Mackey converge in the functional bornology of endomorphisms; consequently the convolution algebras are self-induced, the convolution modules are smooth in the sense of R. Meyer, and submersive, proper, transitive actions yield projective convolution modules [2508.11005].

A bundle-valued extension appears for Fell bundles. If \(\mathscr{B}\) is a Fell bundle over a groupoid \(\mathcal{H}\) and \(\mathscr{M}\) is a Banach bundle over a principal right \(\mathcal{H}\)-space \(X\) with a right \(\mathscr{B}\)-action and a \(\mathscr{B}\)-valued inner product, then the imprimitivity Fell bundle
\[
\mathbb{K}_{\mathscr{B}}(\mathscr{M})
=
\mathscr{M}\otimes_{\mathscr{B}}\mathscr{M}^{\mathrm{op}}
\]
is a Fell bundle over the imprimitivity groupoid of \(X\), and it is the unique Fell bundle equivalent to \(\mathscr{B}\) via \(\mathscr{M}\). This is the Fell-bundle analogue of the compact-operator construction \(\mathbb{K}_A(\mathbf X)\) for Hilbert \(C^*\)-modules, and it recovers constructions such as Kumjian’s stabilization trick [2311.15021].

These developments suggest a general principle: groupoid bibundles are geometric carriers of Morita equivalence, and convolution bimodules are their analytic realizations. In the solenoidal case that principle is completely explicit; in bornological and Fell-bundle settings it becomes functorial and \(2\)-categorical.

Source: https://www.emergentmind.com/topics/convolution-bimodules-of-groupoid-bibundles