Convolution Bimodules of Groupoid Bibundles

• Convolution bimodules are Hilbert C*-bimodules defined on spaces of compactly supported functions, encoding groupoid actions via convolution-type integrals.
• The construction employs left and right convolution operations that translate geometric bibundle equivalences into explicit imprimitivity bimodules.
• In noncommutative solenoidal settings, these bimodules concretely realize Morita equivalence through arithmetic structures and GL2(Z[1/p])-orbits.

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 (Rouse et al., 17 Mar 2025).

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 $Z$0 equipped with a free and proper left $Z$1-action, a free and proper right $Z$2-action, commuting actions, and open moment maps $Z$3, $Z$4 inducing homeomorphisms

$Z$5

Such a bibundle realizes topological Morita equivalence of groupoids, and in the Lie groupoid framework biprincipal bibundles are precisely the data expressing Morita equivalence (Farsi et al., 3 Jul 2025).

The analytic object attached to an equivalence bibundle is the convolution bimodule. Its underlying pre-Hilbert space is $Z$6, the compactly supported continuous functions on the bibundle, and its left and right module structures are defined by convolution against $Z$7 and $Z$8. In the standard formulation used in the solenoidal case, the left action and right action are

$Z$9

$Z$0

with analogous convolution-type formulas for the $Z$1- and $Z$2-valued inner products. The completion of $Z$3 is then an imprimitivity bimodule whenever $Z$4 is a $Z$5-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 $Z$6-algebras

Fix a prime $Z$7, and let $Z$8 be the $Z$9-solenoid, realized as

$C^*$0

There is an action of $C^*$1 on $C^*$2 by translations,

$C^*$3

and $C^*$4 becomes a homogeneous space for this action. For $C^*$5, the reduced solenoidal groupoid is the transformation groupoid

$C^*$6

where

$C^*$7

Its arrows are pairs $C^*$8, with structure maps

$C^*$9

$S_\alpha$0

$S_\alpha$1

These groupoids model the noncommutative solenoids of Latremolière and Packer. If $S_\alpha$2, then the associated noncommutative solenoid is the twisted group $S_\alpha$3-algebra

$S_\alpha$4

and Carrillo Rouse and Guillaume identify the groupoid algebra with this solenoid algebra: $S_\alpha$5 The identification is made explicit using generators

$S_\alpha$6

which satisfy the solenoid relations

$S_\alpha$7

This places the study of noncommutative solenoids inside the geometry of transformation groupoids and their bibundles (Rouse et al., 17 Mar 2025).

3. The explicit equivalence bibundle $S_\alpha$8

The central geometric construction arises when $S_\alpha$9 are related by a linear fractional transformation coming from

$S_\beta$0

with

$S_\beta$1

In this situation the bibundle is

$S_\beta$2

Its moment maps are

$S_\beta$3

and the groupoid actions are

$S_\beta$4

for the left $S_\beta$5-action, and

$S_\beta$6

for the right $S_\beta$7-action.

The geometric origin of $S_\beta$8 lies in a larger action groupoid on $S_\beta$9. Two transversals are chosen,

$G$0

and the restricted groupoids satisfy

$G$1

The theorem is that $G$2, with these actions and moment maps, is an equivalence bibundle between $G$3 and $G$4. For $G$5 with determinant $G$6, one writes

$G$7

and the determinant unit only changes $G$8 by an isomorphic parameter, so the relevant Morita classes are controlled by the $G$9-orbit of $H$0 (Rouse et al., 17 Mar 2025).

4. The convolution bimodule $H$1

Once $H$2 is identified as an equivalence bibundle, the associated convolution bimodule is built on

$H$3

Because $H$4 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

$H$5

$H$6

Since $H$7 and $H$8 are discrete in the $H$9-direction, these integrals reduce to countable sums over $G\rightrightarrows G^{(0)}$0. The left action is therefore a twisted convolution built from the translation $G\rightrightarrows G^{(0)}$1, while the right action is governed by the affine translation $G\rightrightarrows G^{(0)}$2.

The inner products are likewise defined by convolution-type integrals over groupoid fibers, producing $G\rightrightarrows G^{(0)}$3- and $G\rightrightarrows G^{(0)}$4-valued pairings. Their compatibility with the module actions is expressed by the usual imprimitivity identities, including

$G\rightrightarrows G^{(0)}$5

The induced norms from the left and right inner products coincide, and the completion is an imprimitivity bimodule

$G\rightrightarrows G^{(0)}$6

implementing

$G\rightrightarrows G^{(0)}$7

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 (Rouse et al., 17 Mar 2025).

5. Morita equivalence of noncommutative solenoids

Combining the identification $G\rightrightarrows G^{(0)}$8 with the bibundle equivalence $G\rightrightarrows G^{(0)}$9, one obtains a concrete Morita equivalence statement for noncommutative solenoids. If

$H\rightrightarrows H^{(0)}$0

then $H\rightrightarrows H^{(0)}$1 completes to a $H\rightrightarrows H^{(0)}$2-$H\rightrightarrows H^{(0)}$3 imprimitivity bimodule $H\rightrightarrows H^{(0)}$4, and therefore the noncommutative solenoids attached to $H\rightrightarrows H^{(0)}$5 and $H\rightrightarrows H^{(0)}$6 are strongly Morita equivalent.

This yields a classification statement: two noncommutative solenoids are strongly Morita equivalent exactly when their parameters lie in the same $H\rightrightarrows H^{(0)}$7-orbit, modulo the determinant units $H\rightrightarrows H^{(0)}$8, which only change the parameter by isomorphic groupoids. The pattern is explicitly compared with the Connes–Rieffel description of Morita equivalence for noncommutative $H\rightrightarrows H^{(0)}$9-tori via $Z$00-orbits. In the solenoidal setting, the ingredients are arithmetic rather than purely real: $Z$01, the $Z$02-solenoid $Z$03, and the mixed parameter space $Z$04.

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 $Z$05, analytically by the convolution bimodule $Z$06, and algebraically by the resulting equivalence of noncommutative solenoid $Z$07-algebras. The bridge between these levels is the equivalence theorem for groupoid bibundles and their convolution completions (Rouse et al., 17 Mar 2025).

6. Broader formulations and extensions

The same pattern extends beyond the specific $Z$08-algebraic solenoidal model. In the bornological framework for Lie groupoids, the complete bornological convolution algebra

$Z$09

and the convolution bimodule

$Z$10

of a groupoid bibundle $Z$11 define a symmetric monoidal weak $Z$12-functor from the $Z$13-category of differentiable stacks to the Morita $Z$14-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 (Aretz et al., 14 Aug 2025).

A bundle-valued extension appears for Fell bundles. If $Z$15 is a Fell bundle over a groupoid $Z$16 and $Z$17 is a Banach bundle over a principal right $Z$18-space $Z$19 with a right $Z$20-action and a $Z$21-valued inner product, then the imprimitivity Fell bundle

$Z$22

is a Fell bundle over the imprimitivity groupoid of $Z$23, and it is the unique Fell bundle equivalent to $Z$24 via $Z$25. This is the Fell-bundle analogue of the compact-operator construction $Z$26 for Hilbert $Z$27-modules, and it recovers constructions such as Kumjian’s stabilization trick (Duwenig, 2023).

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 $Z$28-categorical.