Part I Selected generalities about groupoids

2 Groupoids and Haar systems

We use Dana Williams’s book [Reference Williams36] as our basic reference about groupoids. We supplement it with Jean Renault’s thesis [Reference Renault26] and his treatise on amenable groupoids, written with Claire Anantharaman-Delaroche [Reference Anantharaman-Delaroche and Renault1]. A generic second countable, locally compact, Hausdorff groupoid is denoted by G. Its unit space is denoted $G^{(0)}$ and r and s denote the range and source maps, mapping G to $G^{(0)}$ . It is part of the definition of a ‘locally compact groupoid’ that these maps are continuous and open.

The important role of ‘open-ness’ is revealed through ‘duality’.

Remark 2.2. The map $\pi :Y\to X$ makes $C_c(Y)$ a right module over $C_c(X)$ via the formula

$$ \begin{align*} \phi\cdot\psi(y):= \phi(y)\psi\circ \pi(y), \quad \phi\in C_c(Y),\;\psi\in C_c(X), \;y\in Y. \end{align*} $$

Also, one may view $C(X)$ as a right module over itself induced by the identity map on X. So, from the perspective of these modules, a continuous $\pi $ -system, $\alpha $ , defines a $C_c(X)$ -module map from $C_c(Y)_{C_c(X)}$ to $C_c(X)_{C_c(X)}$ that maps nonnegative functions on Y to nonnegative functions on X. Conversely, the Riesz representation theorem implies that every such module map defines a continuous $\pi $ -system. Further, it is a consequence of what we like to call the Williams–Blanchard theorem [Reference Williams36, Corollary B.18] that full continuous $\pi $ -systems exist if and only if $\pi $ is an open map.

A key feature of groupoid theory is the superabundance of fibered products. Suppose X, Y, and Z are three sets, and $f:X\to Z$ and $g:Y\to Z$ are two maps, then the fibered product of X and Y determined by f and g is

$$ \begin{align*} X {_f \ast_g} Y:=X\ast Y:= \{(x,y)\in X\times Y \mid f(x)=g(y)\}. \end{align*} $$

Right actions of groupoids are defined similarly. The corresponding moment map is denoted by $\sigma $ .

Definition 2.4. Let G act continuously on X and on Y, and let $\pi :Y\to X$ be an invariant continuous map, that is, $\pi $ intertwines the two actions— $\pi (\gamma \cdot x)=\gamma \cdot \pi (x)$ , then a continuous $\pi $ -system of measures $\alpha = \{\alpha ^x\}_{x\in X}$ is invariant (or equivariant) for the actions in the case where the map $\alpha $ viewed as a map from $C_c(Y)$ to $C_c(X)$ is invariant, that is,

$$ \begin{align*} \int f(\gamma\cdot x)\,d\alpha^{s(\gamma)}(x) = \int f(x)\,d\alpha^{r(\gamma)}(x). \end{align*} $$

If $X=Y=G$ , while $\pi $ is the identity map, and if $\rho $ is the range map, r, then a full invariant system of measures is a Haar system on G. Typically, we denote Haar systems by $\lambda $ and write also $\lambda = \{\lambda ^u\}_{u\in G^0}$ . Together with a Haar system $\lambda $ , the space $C_c(G)$ , consisting of continuous complex-valued functions on G endowed with the inductive limit topology, becomes a locally convex topological $*$ -algebra, called the convolution algebra of G and $\lambda $ , and denoted $C_c(G,\lambda )$ . The product is defined by the formula

$$ \begin{align*} f*g(\gamma):=\int_G f(\alpha)g(\alpha^{-1}\gamma)\, d\lambda^{r(\gamma)}(\alpha),\quad f,g\in C_c(G,\lambda), \end{align*} $$

and the involution is defined by

$$ \begin{align*} f^*(\gamma):=\overline{f(\gamma^{-1})},\quad f\in C_c(G,\lambda). \end{align*} $$

The $C^*$ -algebra of G and a Haar system $\lambda $ is the enveloping $C^*$ -algebra of $C_c(G,\lambda )$ , and is denoted $C^*(G,\lambda )$ (see [Reference Renault26, Section 2.1] and [Reference Williams36, Section 1.4]).

Here, we provide a telegraphic account of groupoid representations that involve Hilbert spaces. Later, we discuss representations of groupoids on certain Hilbert $C^*$ -modules.

Representations of groupoids take place on Hilbert bundles that are fibred over their unit spaces. The representations of the associated convolution algebras take place on the Hilbert spaces obtained by integrating the fibres of the bundles using quasi-invariant measures on the unit spaces [Reference Williams36, Ch. 3]. How to build quasi-invariant measures for our groupoids is the focus of much of Parts II and III of this paper.

If G is a locally compact groupoid with Haar system $\lambda :=\{\lambda ^u\}_{u\in G^{(0)}}$ and if $\mu $ is a (positive) measure on $G^{(0)}$ , then one may form the measure on G that is denoted $\mu \circ \lambda $ and is defined by the formula:

$$ \begin{align*} \int_G f(\gamma)\, d\mu \circ \lambda(\gamma):= \int_{G^{(0)}}\int_G f(\gamma)\, d\lambda^{r(\gamma)}(\gamma)\,d\mu(r(\gamma)),\quad f\in C_c(G). \end{align*} $$

Composing $\mu \circ \lambda $ with the inverse map on G creates a new measure, denoted $(\mu \circ \lambda )^{-1}$ . The measure $\mu $ is called quasi-invariant in the case where $\mu \circ \lambda $ and $(\mu \circ \lambda )^{-1}$ are mutually absolutely continuous, in which case, the function $\Delta _\mu := ({d(\mu \circ \lambda )})/({d(\mu \circ \lambda )^{-1}})$ is called the modular function of $\mu $ and $\lambda $ . The modular function $\Delta _\mu $ , while defined only almost everywhere with respect to $\mu \circ \lambda $ , can be chosen so that it is a homomorphism from G to the positive real numbers [Reference Williams36, Proposition 7.6].

A (unitary) representation of a groupoid G is a triple $(\mu ,G^{(0)}\ast \mathcal {H},\hat {L})$ , where $\mu $ is a quasi-invariant measure on $G^{(0)}$ , $G^{(0)}\ast \mathcal {H}$ is a Borel Hilbert bundle on $G^{(0)}$ , and $\hat {L}:G\to {\operatorname {Iso}}(G^{(0)}\ast \mathcal {H})$ is a homomorphism of G into the unitary groupoid, ${\operatorname {Iso}}(G^{(0)}\ast \mathcal {H})$ , of the bundle $G^{(0)}\ast \mathcal {H}$ . By definition, $\text {Iso}(G^{(0)}\ast \mathcal {H}):=\{(u,U,v)\mid U\in B_{\text {iso}} (H_v,H_u)\}$ , where $B_{\text {iso}}(H_v,H_u)$ is the space of all Hilbert space isomorphisms from $H_v$ to $H_u$ . It carries a Borel structure induced by that of $G^{(0)}*\mathcal {H}$ . The product of $(u,U,v)$ and $(w,V,z)$ is defined only when $v=w$ and in that case, the product is $(u,UV,z)$ . Since G carries a Borel structure that is generated by its topology, it makes perfectly good sense to insist that $ \hat {L}$ be a Borel homomorphism from G to $\text {Iso}(G^{(0)}\ast \mathcal {H})$ as described in [Reference Williams36, Proposition 3.38]. We write

(2-1) $$ \begin{align} \hat{L}(\gamma):=(r(\gamma),L_{\gamma},s(\gamma)),\quad \gamma\in G, \end{align} $$

where $L_{\gamma }$ lies in $B_{\text {iso}}(H_{s(\gamma )},H_{r(\gamma )})$ .

Given a Borel Hilbert bundle $G^{(0)}\ast \mathcal {H}$ and (sigma-finite) Borel measure $\mu $ , one can form the Hilbert space consisting of all its Borel sections $\mathfrak {h}$ such that $\int _{G^{(0)}}\langle \mathfrak {h}(u),\mathfrak {h}(u)\rangle _u\, d\mu (u)$ is finite. This space is denoted $L^2(G^{(0)}\ast \mathcal {H},\mu )$ and the inner product of two sections $\mathfrak {h}$ and $\mathfrak {k}$ is given by the expected integral

$$ \begin{align*} ( \mathfrak{h},\mathfrak{k}): = \int_{G^{(0)}}(\mathfrak{h}(u),\mathfrak{k}(u))_{H_u} \, d\mu(u). \end{align*} $$

When the dimensions of the fibres are one, we may choose a nonvanishing cross-section $\mathfrak {k}$ of norm one in $L^2(G^{(0)}\ast \mathcal {H},\mu )$ . Then, if $\mathfrak {h}$ is another cross-section, it can be written as $\mathfrak {h}=c\cdot \mathfrak {k}$ , where c is a uniquely determined complex-valued Borel function on $G^{(0)}$ . The map $\mathfrak {h}\to c$ is a Hilbert space isomorphism from $L^2(G^{(0)}\ast \mathcal {H},\mu )$ onto the space of scalar-valued, square-integrable functions on $G^{(0)}$ , $L^2(G^{(0)},\mu )$ . It enables us to replace $L^2(G^{(0)}\ast \mathcal {H},\mu )$ with $L^2(G^{(0)},\mu )$ and to carry out computations there.

Let $(\mu ,G^{(0)}\ast \mathcal {H},\hat {L})$ be a unitary representation of G in the unitary groupoid, ${\operatorname {Iso}}(G^{(0)}\ast \mathcal {H})$ . The integrated form, $\mathfrak {L}$ , of L represents the convolution algebra of G, $C_c(G,\lambda )$ , on $L^2(G^{(0)}*\mathcal {H},\mu )$ via the formula

(2-2) $$ \begin{align} \mathfrak{L}(f)\mathfrak{h}(r(\gamma)): & = \int_{G}f(\gamma)L_{\gamma}(\mathfrak{h}(s(\gamma))\Delta(\gamma)^{-1/2}\, d\lambda^{r(\gamma)}(\gamma),\nonumber\\ & \quad\quad f\in C_c(G,\lambda), \mathfrak{h}\in L^2(G^{(0)}\ast \mathcal{H},\mu). \end{align} $$

This formula extends to all Borel functions f such that $\mathfrak {L}(f)$ makes sense as a bounded operator on $L^2(G^{(0)}\ast \mathcal {H},\mu )$ .

Turning to groupoid representations on Hilbert $C^*$ -modules and Holkar’s ideas [Reference Holkar14, Reference Holkar15], suppose A and B are $C^*$ -algebras and $\mathcal {X}$ is a (right) Hilbert module over B with a B-valued inner product $\langle \cdot , \cdot \rangle $ in the sense of Lance [Reference Lance20, Ch. 1]. We say that $\mathcal {X}$ is a $C^*$ -correspondence from A to B in the case where A is represented in the algebra of all bounded adjointable operators on $\mathcal {X}$ , $\mathcal {L}(\mathcal {X})$ . In this event, following Rieffel [Reference Rieffel33], one may induce $C^*$ -representations of A from $C^*$ -representations of B by forming balanced tensor products: for a $C^*$ -representation of B, $\pi $ , on the Hilbert space $\mathcal {H}$ , $\mathcal {X}\otimes _{\pi } \mathcal {H}$ becomes a Hilbert space in the inner product that is defined on decomposable tensors by $ \langle \xi \otimes h, \eta \otimes k\rangle : = \langle h,\pi (\langle \xi , \eta \rangle )k\rangle. $ The representation, $\operatorname {Ind}_{\pi }:A\to B(\mathcal {X} \otimes _{\pi } \mathcal {H})$ , then, is defined by $\operatorname {Ind}_{\pi }(a)(\xi \otimes h):=(a\cdot \xi )\otimes h, a\in A$ , $\xi \otimes h \in \mathcal {X}\otimes \mathcal {H}$ , that is, $\operatorname {Ind}_{\pi }(a) = L(a)\otimes I_{\mathcal {H}}$ .

Suppose that $A=C^*(H,\beta )$ and $B=C^*(G,\lambda )$ for groupoids H and G endowed with Haar systems $\beta $ and $\lambda $ . Holkar showed how to build a correspondence from A to B using what he calls a topological correspondence from $(H,\beta )$ to $(G,\lambda )$ .

We emphasize that the system of measures $\alpha $ must be proper but not necessarily full. We use the notation $\Delta _a$ for the adjoining function to distinguish it from the modular function, $\Delta $ or $\Delta _\mu $ , of a quasi-invariant measure $\mu $ . The main theorem in [Reference Holkar14, Theorem 2.10] proves that $C_c(Z)$ can be endowed with operations that give rise to a $C^*$ -correspondence from $C^*(H,\beta )$ to $C^*(G,\lambda )$ .

An equivalence $(Z,\alpha )$ , where $\alpha $ is a full $\sigma $ -system, between $(H,\beta )$ and $(G,\lambda )$ in the sense of [Reference Muhly, Renault and Williams25, Definition 2.1] is a particular example of a topological correspondence. The adjoining function $\Delta _a$ is the constant function 1 in this situation [Reference Holkar14, Example 3.9]. The $C^*$ -algebras $C^*(H,\beta )$ and $C^*(G,\lambda )$ are Morita equivalent [Reference Muhly, Renault and Williams25], and the $C^*$ -correspondence defined above becomes an imprimitivity bimodule. We specialize this setting further in Sections 5 and 9 by setting $H=G$ , $Z=G$ , and $\alpha _x=\lambda _x$ for all $x\in X$ , where $\{\lambda _{u}\}_{u\in G^{(0)}}$ is the right Haar system determined by $\lambda $ , $\int f(\gamma )\,d\lambda _{u}(\gamma ):=\int f(\gamma ^{-1})\,d\lambda ^{u}(\gamma )$ . Then, $(G,\alpha )$ is an equivalence between $(G,\beta )$ and $(G,\lambda )$ , where $\beta $ can be a different Haar system on G. When we view $(G,\alpha )$ as a topological correspondence from $(G,\lambda )$ to $(G,\lambda )$ , we denote it by $\mathcal {X}_0$ .

3 Pullbacks, imprimitivity groupoids, and Haar systems

Throughout this section, G is a groupoid with Haar system $\lambda $ ; Y is a second countable, locally compact Hausdorff space; and $\Phi :Y\to G^{(0)}$ is a continuous, open, and surjective map. We explore Haar systems on two related groupoids, $G^Z$ and $Y\ast G \ast Y$ , that are built from G, $\lambda $ , $\Phi $ , and continuous $\Phi $ -systems of measures on Y.

The space Z is defined to be the fibered product,

$$ \begin{align*} Z:=Y\,{_{\Phi}*_r}\,G:=\{(y,\gamma)\in Y\times G\,\mid\,\Phi(y)=r(\gamma)\}. \end{align*} $$

It carries the evident right G action through right translation on G:

$$ \begin{align*} (z,\alpha),\beta \to (z,\alpha\beta), \quad (z,\alpha)\in Z\ast G, \, \beta\in G. \end{align*} $$

This action of G on Z is free and proper because the action of G on G via right translation is free and proper.

To define the imprimitivity groupoid determined by G and Z, $G^Z$ , let $\sigma :Z\to G$ be the map, $\sigma (y,\gamma ):=s(\gamma )$ and set

$$ \begin{align*} Z*Z:=\{(x,y)\in Z\times Z\mid \sigma(x)=\sigma(y)\}. \end{align*} $$

Then, G acts on the right on $Z*Z$ by the diagonal action $(x,y)\cdot \gamma =(x\gamma ,y\gamma )$ . The imprimitivity groupoid determined by G and Z is $G^{Z}:=Z*_{G}Z:=(Z*Z)/G$ . The unit space of $G^{Z}$ can be identified with $Z/G$ , which in turn may be identified with Y, and the range and source maps on $G^{Z}$ are given by

$$ \begin{align*} r([x,y])=x\cdot G\quad\text{and}\quad s([x,y])=y\cdot G. \end{align*} $$

The space Z becomes a free and proper left $G^{Z}$ -space as follows. The moment map $\rho :Z\to Z/G$ is the quotient map; so, $G^{Z}*Z$ is $\{([x,y],z)\in G^{Z}\times Z\mid y\cdot G=z\cdot G\}$ . The left action of $G^{Z}$ on Z is defined via the formula

$$ \begin{align*} [x,y]\cdot z:=x\gamma, \end{align*} $$

where $\gamma $ is the unique element in G such that $z=y\cdot \gamma $ . The space Z is a free and proper left $G^{Z}$ -space and, in fact, Z is an equivalence between $G^{Z}$ and G [Reference Muhly, Renault and Williams25, Section 2].

Proposition 3.1. The imprimitivity groupoid $G^{Z}$ is isomorphic to the pullback groupoid determined by Y and $\Phi $ , $Y*G*Y$ , where

$$ \begin{align*} Y*G*Y:= & Y{\,{_\Phi \ast_r}\,}G{\,{_s \ast_\Phi}\,}Y: = \{(x,\gamma,y)\in Y\times G\times Y\mid\Phi(x)=r(\gamma)\quad\text{and}\quad \Phi(y)=s(\gamma)\} \end{align*} $$

is endowed with the operations

$$ \begin{align*} (x,\gamma,y)\cdot(y,\delta,z):=(x,\gamma\delta,z) \end{align*} $$

and

$$ \begin{align*} (x,\gamma,y)^{-1}:=(y,\gamma^{-1},x). \end{align*} $$

Proof. It is easy to check that $\Psi :G^{Z}\to Y*G*Y$ defined via

$$ \begin{align*} \Psi([(x,\alpha),(y,\beta)])=(x,\alpha\beta^{-1},y) \end{align*} $$

is an isomorphism with the desired properties.

We next want to discuss how the Haar system $\lambda :=\{\lambda ^{u}\}_{u\in G^{(0)}}$ on G may be ‘induced’ to Haar systems on $G^Z$ and $Y\ast G \ast Y$ . We start by choosing an arbitrary continuous $\Phi $ -system of measures $\nu :=\{\nu _{u}\}_{u\in G^{(0)}}$ on Y (as per Definition 2.1). While, as the following lemma shows, we need $\nu $ to be a full $\Phi $ -system to obtain a Haar system on $G^Z$ , we proceed with arbitrary systems. We use proper nonfull systems to define groupoid correspondences in later sections. We then promote $\nu $ to a system of measures $\mathfrak {k}: = \{\mathfrak {k}_u\}_{u\in G^{(0)}}$ on Z defined by the formula:

$$ \begin{align*} \mathfrak{k}_{u}(f):=\int_{G_{u}}\int_{Y}f(y,\gamma)\,d\nu_{r(\gamma)}(y)\,d\lambda_{u}(\gamma),\quad u\in G^{(0)}, \end{align*} $$

for $f\in C_{c}(Z)$ .

The Haar system on $Y*G*Y$ defined by $\mathfrak {k}$ was studied in [Reference Williams36, Proposition 6.27], which is reproduced here as the following proposition.

Proposition 3.2. Assume that $\nu $ is a full $\Phi $ -system. The equation

(3-1) $$ \begin{align} \beta(f)(x):=\int_{G^{\Phi(x)}}\int_{Y}f(x,\gamma,y)\,d\nu_{s(\gamma)}(y)\,d\lambda^{\Phi(x)}(\gamma),\quad x\in Y, \end{align} $$

defines a Haar system, $\beta $ , on $Y*G*Y$ .

It is important for us that we can induce representations from $(G,\lambda )$ to $(G^Z,\beta )$ as in Section 4 using $\Phi $ -systems of measures that are not full by using the theory of groupoid correspondences. We provide here the main set-up in a slightly more general setting. Assume that G is a locally compact Hausdorff groupoid endowed with a Haar system $\lambda $ . Assume that Z is a right G-space with a continuous, open, and surjective anchor map $\sigma :Z\to G^{(0)}$ and let $\alpha :=\{\alpha _u\}_{u\in G^{(0)}}$ be a full $\sigma $ -system that is invariant under the action of G on Z in the sense of Definition 2.4. Since $\sigma (z\cdot \gamma ) = s(\gamma )$ , this means that $\int f(z\cdot \gamma )\,d \alpha _{r(\gamma )}(z)=\int f(z)\,d\alpha _{s(\gamma )}(z)$ . Let $G^Z:=Z{_\sigma *_\sigma }Z/G$ be the imprimitivity groupoid determined by Z. Then, $\alpha $ determines a Haar system $\beta $ on $G^Z$ via

(3-2) $$ \begin{align} \int_{G^Z}f([x,y])\,d\beta^{xG}([x,y])=\int_Z f([x,y])\,d\alpha_{s(x)}(y) \end{align} $$

for all $xG\in Z/G$ [Reference Kumjian, Muhly, Renault and Williams18, Proposition 5.2]. This Haar system on $G^Z$ is fixed for the remainder of the section.

Let $\delta $ be a nonnegative function on Z that is invariant under the action of G. This implies, of course, that the pointwise product $\delta \cdot \alpha $ is an invariant system of measures. In addition, we assume that $\delta \cdot \alpha $ is a proper invariant system, which we denote by $\alpha _{\delta }$ . This assumption places constraints on how the support of $\delta $ is distributed. We write

$$ \begin{align*} \int_Z f(x)\, d\alpha_{\delta,u}(x)=\int_Z f(x)\, d(\alpha_{\delta})_u(x) = \int_Z f(x)\delta(x) \,d\alpha_u(x). \end{align*} $$

Proof of Lemma 3.3.

Conditions (i), (ii), and (iii) of Definition 2.5 are satisfied by our assumptions. We check the fourth condition of the definition. Let $F\in C_c(G^Z*Z)$ . For $u\in G^{(0)}$ , we have

$$ \begin{align*} &\int_{Z}\int_{G^Z}F ([x,y]^{-1},z)\,d\beta^{zG}([x,y])\,d\alpha_{\delta,u}(z)\\ &\quad = \int_{Z}\int_Z F([z,y]^{-1},z)\,d\alpha_{u}(y)\,d\alpha_{\delta,u}(z)\\ &\quad = \int_Z\int_Z F([y,z],z)\delta(z)\,d\alpha_u(y)\,d\alpha_u(z)\\ &\quad = \int_Z\int_ZF([z,y],[y,z]z)\delta([y,z]z)\,d\alpha_u(y)\,d\alpha_u(z)\\ &\quad = \int_Z\int_Z F([z,y],y)\delta(y)\,d\alpha_u(y)\,d\alpha_u(z)\\ &\quad = \int_Z\int_Z F([z,y],y)(\delta(y)/\delta(z))\,d\alpha_u(y)\,\delta(z)\,d\alpha_u(z)\\ &\quad = \int_Z\int_ZF([z,y],y)\Delta_a([z,y],y)\,d\alpha_u(y)\,d\alpha_{\delta,u}(z)\\ &\quad = \int_Z\int_{G^Z}F([z,y],y)\Delta_a([z,y],y)\,d\beta^{zG}([z,y])\,d\alpha_{\delta,u}(z). \end{align*} $$

The first line follows from (3-2), the second line from the definition of $\alpha _\delta $ , and the third line follows since $(Z,\alpha )$ is an equivalence between $G^Z$ and G (see [Reference Holkar14, Examples 3.9 and 3.8]). The remaining lines follow by defining $\Delta _a([x,y],z)=\delta (y)/\delta (x)$ , from (3-2), and from the definition of $\alpha _\delta $ .

Example 3.5. Returning to the main case we study, assume that $\Phi :Y\to G^{(0)}$ is a continuous, open, surjective map. Let $\nu =\{\nu _u\}_{u\in G^{(0)}}$ be a full $\Phi $ -system of measures. Let $Z=Y*G$ and $\alpha =\{\alpha _u\}_{u\in G^{(0)}}$ be the invariant full s-system defined via

$$ \begin{align*} \int_Z f(x,\gamma)\,d\alpha_u(x,\gamma)=\int_G\int_Y f(x,\gamma)\,d\nu_{r(\gamma)}(x)\,d\lambda_u(\gamma). \end{align*} $$

That is, $\alpha _u=\mathfrak {k}_u$ of Definition 3. Let $\mathbb {d}:Y\to \mathbb {R}^+$ be a continuous function. Then, $\delta :Z\to \mathbb {R}^+$ defined via $\delta (x,\gamma )=\mathbb {d}(x)$ satisfies the hypothesis that $\delta ((x,\gamma )\eta )=\delta (x,\gamma )$ . Lemma 3.3 implies that $(Z,\alpha _\delta )$ is a topological correspondence, where

(3-3) $$ \begin{align} \int_Z f(x,\gamma)\,d\alpha_{\delta,u}(x,\gamma)=\int_G \int_Y f(x,\gamma)\mathbb{d}(x)\,d\nu_{r(\gamma)}(x)\,d\lambda_u(\gamma) \end{align} $$

and the adjoining function is defined via $\Delta _a((x,\gamma ,y),(y,\eta ))=\mathbb {d}(y)/\mathbb {d}(x)$ .

4 Inducing representations to the pullback groupoid

Continuing with the notation from Section 3, observe that the title of the current section may appear confusing because the groupoid G is not a subgroupoid of the pullback groupoid, $Y\ast G \ast Y$ . Rather, it is a quotient of $Y\ast G \ast Y$ . However, as we have shown in Proposition 3.1, $Y\ast G \ast Y$ is isomorphic to $G^Z$ , which is the imprimitivity groupoid of the right G-space $Z:=Y\ast G$ . So, it is natural to try to use the theory developed by Rieffel in [Reference Rieffel33] coupled with the equivalence theorem of [Reference Muhly, Renault and Williams25, Theorem 2.8] to induce representations of G to $G^Z$ and relate them to representations of $Y\ast G \ast Y$ using the isomorphism $\Psi $ . One of the difficulties we face when trying to do this is the necessity in this paper to use nonfull $\Phi $ -systems of measures on Y. This leads to groupoid correspondences from $Y*G*Y$ to G, and we can use [Reference Holkar14] to induce representations from $C^*(G,\lambda )$ to $C^*(Y*G*Y,\beta )$ and [Reference Renault32] to induce representations from $(G,\lambda )$ to $(Y*G*Y,\beta )$ . Our analysis leads to two different Hilbert space representations of the convolution algebra $C_c(Y\ast G \ast Y,\beta )$ that one may induce from a Hilbert space representation of the convolution algebra $C_c(G,\lambda )$ . These are denoted $\operatorname {Ind}_{\text {J}}\mathfrak {L}$ and $\operatorname {Ind}_{\text {M}}\mathfrak {L}$ , where $\mathfrak {L}$ is a prescribed Hilbert space representation of $C_c(G,\lambda )$ that comes from a representation $\hat {L}$ of the groupoid G in the isomorphism groupoid of a Hilbert bundle built over $G^{(0)}$ . Our purpose in this section is to define $\operatorname {Ind}_{\text {J}}\mathfrak {L}$ and $\operatorname {Ind}_{\text {M}}\mathfrak {L}$ , and to show that they are unitarily equivalent. (See Theorem 4.1.)

We begin with a unitary representation, $\hat {L}=(\mu ,G^{(0)}*\mathcal {H},\hat {L})$ , of $(G,\lambda )$ in the unitary groupoid ${\operatorname {Iso}}(G^{(0)} \ast \mathcal {H})$ and form its integrated form $\mathfrak {L}$ , representing the convolution algebra, $C_c(G,\lambda )$ , on $L^2(G^{(0)} \ast \mathcal {H},\mu )$ (see text following Definition 2.4). These ingredients are fixed throughout this section, as are the modular function $\Delta _{\mu }$ determined by $\mu $ and the Haar system $\lambda $ on G. We also fix a full $\Phi $ -system $\nu $ on Y that determines a full invariant s-system $\alpha $ on $Z=Y*G$ . The Haar system $\beta $ on $Y*G*Y$ is fixed and given as in (3-1).

We let $\mathbb {d}:Y\to \mathbb {R}^+$ be a continuous function such that $\mathbb {d}|_{\Phi ^{-1}(u)}\ne 0$ for all $u\in G^{(0)}$ . Then, $\mathbb {d}(x)$ defines a proper $\Phi $ -system of measures $\nu _{\mathbb {d}}$ via

(4-1) $$ \begin{align} \int_Y f(x)\,d\nu_{\mathbb{d},u}(x)=\int_Y f(x)\mathbb{d}(x)\,d\nu_u(x) \end{align} $$

for all $u\in G^{(0)}$ . Let $\delta (x,\gamma ):=\mathbb {d}(x)$ and let $\alpha _\delta $ be the proper invariant $\sigma $ -system defined as in (3-3). Lemma 3.3 implies that $(Z,\alpha _\delta )$ is a topological correspondence. We describe next the process of inducing the groupoid representation $\hat {L}=(\mu ,G^{(0)}*\mathcal {H},\hat {L})$ , of $(G,\lambda )$ to a groupoid representation of $(Y*G*Y,\beta )$ via $(Z,\alpha _\delta )$ following Renault’s ideas presented in [Reference Renault32]. The result is a triple, $\operatorname {Ind} \hat {L}:=(\mu _{\operatorname {Ind}},Y*\mathcal {K},\operatorname {Ind} \hat {L})$ , which we call the (unitary) induced representation of $(Y\ast G \ast Y, \beta )$ . The measure, $\mu _{\operatorname {Ind}}$ , is defined on Y, the unit space of $Y\ast G \ast Y$ , through the formula

$$ \begin{align*} \int_Y f(x)\,d\mu_{\operatorname{Ind}}(x):=\int_{G^{(0)}}\int_Y f(x)\,d\nu_{\mathbb{d},u}(x)\,d\mu(u) \end{align*} $$

for all $f\in C_c(Y)$ . The following equation, (4-2), shows that $\mu _{\operatorname {Ind}}$ is indeed quasi-invariant on Y when $Y*G*Y$ is endowed with the Haar system $\beta $ defined in (3-1). It also shows that the modular function $\Delta _{\mu _{\operatorname {Ind}}}$ is given by the equation $\Delta _{\mu _{\operatorname {Ind}}}(x,\gamma ,y)=\Delta _\mu (\gamma )\mathbb {d}(x)/\mathbb {d}(y)$ for all $(x,\gamma ,y)\in Y*G*Y$ . The computation, itself, is a conscientious application of Fubini’s theorem coupled with the quasi-invariance of the measure $\mu $ on $G^{(0)}$ :

(4-2) $$ \begin{align} \mu_{\operatorname{Ind}}\circ \beta(f) & =\int_{Y}\int_{Y*G*Y}f(x,\gamma,y)\,d\beta^x(x,\gamma,y)\,d\mu_{\operatorname{Ind}}(x) \nonumber\\ & =\int_{G^{(0)}}\int_Y\int_{G}\int_Y f(x,\gamma,y)\Delta_\mu(\gamma)(\mathbb{d}(x)/\mathbb{d}(y))\,d\nu_{r(\gamma)}(x)\,d\lambda_u(\gamma)\mathbb{d}(y)\,d\nu_u(y)\,d\mu(u)\nonumber\\ &=\int_Y\int_{Y*G*Y}f(x,\gamma,y)\Delta_\mu(\gamma)\mathbb{d}(x)/\mathbb{d}(y)\,d\beta_y(x,\gamma,y)\,d\mu_{\operatorname{Ind}}(y) \nonumber\\ &=\Delta_{\mu_{\operatorname{Ind}}} (\mu_{\operatorname{Ind}}\circ \beta)^{-1}(f) \end{align} $$

for all $f\in C_c(Y\ast G \ast Y)$ .

The Hilbert space bundle $Y*\mathcal {K}$ is the pull-back Hilbert space bundle $Y*\mathcal {H}: =\{(x,h)\mid x\in Y\quad \text {and}\quad h\in \mathcal {H}(\Phi (x))\}$ over Y. That is, $\mathcal {K}(y)=\mathcal {H}(\Phi (y))$ . The induced action of the pullback groupoid $Y*G*Y$ on $Y*\mathcal {H}$ is given by

$$ \begin{align*} (\operatorname{Ind} \hat{L})_{(x,\gamma,y)}(y,h): =(x,\hat{L}_\gamma h) \quad (x,\gamma,y)\in Y*G*Y,\, h\in \mathcal{H}(\Phi(y)). \end{align*} $$

The integrated form of $\operatorname {Ind} \hat {L}$ is denoted $\operatorname {Ind}_{\text {J}} \mathfrak {L}$ and is defined on $L^2(Y*\mathcal {H},\mu _{\operatorname {Ind}})$ via the equation

(4-3) $$ \begin{align} & ( (\operatorname{Ind}_{\text{J}} \mathfrak{L})(f) \mathfrak{h},\mathfrak{k} ) \nonumber\\ & \quad = \int_Y\int_{Y*G*Y}f(x,\gamma,y)( (\operatorname{Ind} L)_{(x,\gamma,y)}\mathfrak{h}(\Phi(y)),\mathfrak{k}(\Phi(x)) )\Delta_{\mu_{\operatorname{Ind}}}(x,\gamma,y)^{-1/2} d\beta^x(x,\gamma,y) \nonumber\\ & \quad\quad \cdots d\mu_{\operatorname{Ind}}(x) \nonumber\\ & \quad =\int_{G^{(0)}}\int_Y\int_G\int_Y f(x,\gamma,y)(L_\gamma(\mathfrak{h}(\Phi(y)))\,,\mathfrak{k}(\Phi(x)))\Delta_\mu(\gamma)^{-1/2}\mathbb{d}(x)^{1/2}\mathbb{d}(y)^{1/2}\nonumber\\ & \quad\quad \cdots d\nu_{s(\gamma)}(y)\,d\lambda^u(\gamma)\,d\nu_{u}(x)\,d\mu(u) \end{align} $$

for all $f\in C_c(Y*G*Y)$ , $\mathfrak {h},\mathfrak {k}\in L^2(Y*\mathcal {H},\mu _{\operatorname {Ind}})$ .

We abbreviate $L^2(Y*\mathcal {H},\mu _{\operatorname {Ind}})$ with $H_{\operatorname {Ind}_{\text {J}}\mathfrak {L}}$ .

We turn now to a description of Rieffel’s perspective on induced representations. Recall from Lemma 3.3 that $(Z,\alpha _\delta )$ , where $\delta (x,\gamma )=\mathbb {d}(x)$ , is topological correspondence from $(Y*G*Y,\beta )$ to $(G,\lambda )$ . Reference [Reference Holkar14, Theorem 2.10] (see Definition 2.5) implies that $C_c(Z)=C_c(Y*G)$ can be completed to a $C^*$ -correspondence from $C^*(Y*G*Y, \beta )$ to $C^*(G,\lambda )$ via the formulas listed below. The right action of $C_c(G,\lambda )$ on $C_c(Z)$ is given by the integral

(4-4) $$ \begin{align} (\xi\cdot b)(x,\gamma)=\int_G \xi(x,\gamma\eta)b(\eta^{-1})\,d\lambda^{s(\gamma)}(\eta), \quad \xi \in C_c(Z),\, b\in C_c(G,\lambda). \end{align} $$

The inner product on $C_c(Z)$ with values in $C_c(G,\lambda )$ is defined by the integral

(4-5) $$ \begin{align} \langle \xi,\eta \rangle_{C_c(G)}(\gamma) =\int_G\int_{Y} \overline{\xi(x,\zeta)}\eta(x,\zeta\gamma)\,d\nu_{\mathbb{d},r(\zeta)}(x)\,d\lambda_{r(\gamma)}(\zeta), \end{align} $$

and the left action of $C_c(Y\ast G \ast Y, \beta )$ on $C_c(Z)$ is given by the integral

(4-6) $$ \begin{align} (a\cdot \xi)(x,\gamma)=\int_G\int_{Y}a(x,\zeta,y)\xi(y,\zeta^{-1}\gamma)\mathbb{d}(y)^{1/2}/\mathbb{d}(x)^{1/2}\,d\nu_{s(\zeta)}(y)\,d\lambda^{r(\gamma)}(\zeta), \end{align} $$

where $ a \in C_c(Y\ast G \ast Y,\beta )$ and $\xi \in C_c(Z)$ .

So, we may define the representation of $C_c(Y\ast G \ast Y, \beta )$ that is induced in the sense of Rieffel from the representation $\mathfrak {L}$ which, recall, is the integrated form of the representation $\hat {L}=(\mu ,G^{(0)}\ast \mathcal {H},\hat {L})$ of the groupoid G. The Hilbert space for $\mathfrak {L}$ is $H_{\mathfrak {L}}:=L^2(G^{(0)}\ast \mathcal {H},\mu )$ .

The Hilbert space $H_{\operatorname {Ind}_{M}\mathfrak {L}}$ of Rieffel’s induced representation, $\operatorname {Ind}_{M}\mathfrak {L}$ , is the completion of the algebraic tensor product $C_c(Z)\odot H_{\mathfrak {L}}$ under the inner product

$$ \begin{align*} ( \xi\otimes \mathfrak{h},\eta\otimes \mathfrak{k} )&=( \mathfrak{L}(\langle \eta,\xi\rangle_{C^*(G)})\mathfrak{h},\mathfrak{k}) \end{align*} $$

for $\xi ,\eta \in C_c(Z)$ and $\mathfrak {h},\mathfrak {k}\in H_{\mathfrak {L}}$ . The induced representation $\operatorname {Ind}_{\text {M}}\mathfrak {L}$ acts on $H_{\operatorname {Ind}_{\text {M}}\mathfrak {L}}$ via the formula

$$ \begin{align*} (\operatorname{Ind}_{\text{M}} \mathfrak{L})(f)\xi\otimes \mathfrak{h}: =(f\cdot \xi)\otimes \mathfrak{h} \end{align*} $$

for all $f\in C_c(Y*G*Y)$ , $\xi \in C_c(Z)$ , and $\mathfrak {h}\in H_{\mathfrak {L}}$ .

The following theorem is anticipated in the discussion at the bottom of the first page of [Reference Renault32]. We need a precise statement.

Theorem 4.1. Define $\mathfrak {U}:C_c(Z)\odot H_{\mathfrak {L}}\to H_{\operatorname {Ind}_{\text {J}}\mathfrak {L}}$ via

(4-7) $$ \begin{align} \mathfrak{U}(\xi\otimes \mathfrak{h})(x)=\int_{G^{\Phi(x)}}\xi(x,\gamma)L_\gamma \mathfrak{h}(s(\gamma))\Delta_\mu(\gamma)^{-1/2}d\lambda^{\Phi(x)}(\gamma). \end{align} $$

Then, $\mathfrak {U}$ extends to a Hilbert space isomorphism from $H_{\operatorname {Ind}_{\text {M}}\mathfrak {L}}$ to $H_{\operatorname {Ind}_{\text {J}}\mathfrak {L}}$ that intertwines $\operatorname {Ind}_{\text {M}} \mathfrak {L}$ and $\operatorname {Ind}_{\text {J}} \mathfrak {L}$ .

Part II From Deaconu–Renault groupoids to proto-multiresolution analyses

5 The Deaconu–Renault groupoid

The principal player in this paper is the Deaconu–Renault groupoid of a local homeomorphism. Our goal in this section is to review some of its salient features and to show how it leads organically to the theory of transfer operators.

We begin with the following definition.

Definition 5.1. Let X be a second countable, compact Hausdorff space and let $\sigma :X\to X$ be a surjective local homeomorphism. The Deaconu–Renault Groupoid defined by X and $\sigma $ is

$$ \begin{align*}G(X,\sigma):= \{(x,k-l,y)\in X\times \mathbb{Z} \times X\mid k,l\in \mathbb{Z}_+,\,\sigma^k(x)=\sigma^l(y)\}.\end{align*} $$

The product $(x,k-l,y)(w,m-n,z)$ is defined only when $y=w$ and then

$$ \begin{align*}(x,k-l,y)(y,m-n,z):=(x,(k+m)-(l+n),z).\end{align*} $$

The inverse of $(x,k-l,y)$ is $(y,l-k,x)$ .

The unit space of $G(X,\sigma )$ is identified with X, and the maps $r:G(X,\sigma )\to X$ and $s:G(X,\sigma )\to X$ given by $r(x,k-l,y):= x$ and $s(x,k-l,y):=y$ are the range and source maps. Because $\sigma $ is assumed to be a surjective local homeomorphism, it is easy to see that $G(X,\sigma )$ viewed as a subset of $X\times \mathbb {Z} \times X$ with the relative topology is locally compact and Hausdorff, and that the maps r and s are local homeomorphisms. Consequently, $G(X,\sigma )$ is an example of what nowadays is universally known as an étale groupoid.

It is useful at times to ‘change variables’ and rewrite the defining formulas for $G(X,\sigma )$ as Valentin Deaconu did in [Reference Deaconu9]:

$$ \begin{align*} G(X,\sigma)=\{(x,n,y)\in X\times \mathbb{Z} \times X \mid \text{ there exists }\, k,l\geq 0,\, n=l-k,\sigma^k(x)=\sigma ^l(y)\}. \end{align*} $$

Two triples, $(x,n,y)$ and $(w,m,z)$ , can be multiplied if and only if $y = w$ , in which case $(x,n,y)(y,m,z)=(x,m+n,z)$ ; $(x,n,y)^{-1}=(y,-n,x)$ .

The fact that $G(X,\sigma )$ is étale implies that the family of counting measures on the fibers of r, taken together, constitute a Haar system, $\lambda = \{\lambda ^x\}_{x\in X}$ , on $G(X,\sigma )$ . The formula for convolution becomes

$$ \begin{align*} f\ast g(x,k-l,y):= & \int_{G(X,\sigma)}f(\alpha)g(\alpha^{-1}\gamma)\,d\lambda^{r(\gamma)}(\alpha)\\ = & \sum f(x,m-n,z)\cdot g(z,(n+k)-(m+l),y), \end{align*} $$

where $\alpha :=(z,n-m,w)$ , $\gamma : = (x,k-l,y)$ , and where the sum ranges over all $m,n$ , and z such that $\sigma ^m (x) = \sigma ^n (z)$ . The adjoint of a function f is defined by

$$ \begin{align*} f^*(x,k-l,y):=\overline{f(y,l-k,x)}. \end{align*} $$

To get a feeling for this algebra, let U and V be open subsets of X, suppose k and l are such that the restrictions, $\sigma ^l|U$ and $\sigma ^k|V$ , are homeomorphisms with common range, i.e., $ \sigma ^k(U)=\sigma ^l(V)$ , and let

$$ \begin{align*} Z(U,V,k,l):=\{(x,k-l,y)\in G(X,\sigma)\mid x\in U,\, y\in V\}. \end{align*} $$

Then, $Z(U,V,k,l)$ is essentially the graph of $(\sigma ^l|_V)^{-1}\circ (\sigma ^k|_U)$ and is a G-set in the sense of Renault [Reference Renault26, Definition 1.10]. It is also an open subset of $G(X,\sigma )$ . Under an evident notion of composition discussed in [Reference Renault26, page 10], these open G-sets form an inverse semigroup which also is a basis for the topology of $G(X,\sigma )$ . (The germs of such G-sets are explored in [Reference Renault28].) If a G-set $Z:=Z(U,V,k,l)$ happens also to be closed, then its characteristic function $1_Z$ is an element of $C_c(G(X,\sigma ),\lambda )$ which, when properly scaled, yields a partial isometry in $C_c(G(X,\sigma ),\lambda )$ .

Another important class of subsets of $G(X,\sigma )$ is defined by the equations

(5-1) $$ \begin{align} R_{n,m}:=\{(x,n-m,y)\in G(X,\sigma)\mid \sigma^n(x)=\sigma^m(y)\}, \quad n,m\geq 0. \end{align} $$

Each $R_{n,m}$ is compact and open in $G(X,\sigma )$ , so its characteristic function $1_{R_{n,m}}$ lies in $C_c(G(X,\sigma ),\lambda )$ . Also, $1_{R_{n,m}}^* =1_{R_{m,n}}$ ; $R_{n,m}$ may be viewed as the graph of the multi-valued function $\sigma ^{-m}\circ \sigma ^n$ ; if $m=0$ , then $R_{n,0}$ is the graph of $\sigma ^n$ ; and $\{R_{n,n}\}_{n\geq 1}$ is a nested family of equivalence relations on X such that $R_{n,n}\subseteq R_{m,m}$ when $n\leq m$ .

Each $R_{n,n}$ is a subgroupoid of $G(X,\sigma )$ whose $C^*$ -algebra, $C^*(R_{n,n},\lambda )$ , is the cross-sectional $C^*$ -algebra of a matrix bundle over X and so is a $C^*$ -algebra with continuous trace. The union $\bigcup _n R_{n,n}:=R_{\infty }$ is also a subgroupoid of $G(X,\sigma )$ whose $C^{*}$ -algebra, $C^*(R_{\infty },\lambda )$ , is the inductive limit, $\underset {\longrightarrow }\lim \,C^*(R_{n,n}, \lambda )$ . These facts about the equivalence relations $R_{n,n}$ and their $C^*$ -algebras are proved and developed further in [Reference Renault30].

The set $R_{1,0} = Z(X,X,1,0) = \{(x,1,y)\mid y=\sigma (x)\}$ plays a central role in the theory because it is essentially the graph of $\sigma $ . This observation leads to the straightforward calculation:

(5-2) $$ \begin{align} 1_{R_{1,0}}\ast (1_{R_{1,0}})^*(x,k-l,y)= |\sigma^{-1}(\sigma(x))| 1_{R_{1,1}}(x,k-l,y), \end{align} $$

where $\vert \sigma ^{-1}(\sigma (x)) \vert $ is the cardinality of $\sigma ^{-1}(\sigma (x))$ . Evidently, $x \to |\sigma ^{-1}(\sigma (x))|$ is an integer-valued function in $C(X)$ , and so is finitely valued and constant on the connected components of X. Further, it is strictly positive since $\sigma $ is surjective. Its reciprocal is denoted $D_0$ . Equation (5-2) shows that S is an isometry with range projection $1_{R_{1,1}}$ .

Definition 5.2. The function S in $C_c(G(X,\sigma ))$ defined by

$$ \begin{align*} S(x,k-l,y):= D_0^{\frac{1}{2}}(x)\cdot 1_{R_{1,0}}(x,k-l,y), \quad (x,k-l,y)\in G(X,\sigma), \end{align*} $$

is called the master isometry in $C_c(G(X,\sigma ))$ and $D_0$ is called its potential.

To explain where this terminology comes from, we specialize Remark 2.2 and the discussion around it to the case where $Y=X$ and $\pi = \sigma $ . This frees us to use $\pi $ to denote the unital endomorphism of $C(X)$ given by composition with $\sigma $ . Since $\sigma $ is surjective, $\pi $ is injective. Observe that $C(X)$ may be viewed as a module over itself in two different ways: the regular module, $C(X)_{C(X)}$ , in which $C(X)$ acts on itself via pointwise multiplication—denoted simply by a —and the $\pi $ -induced module, $C(X)_{\pi (C(X))}$ , whose multiplication, $\ast $ , is defined through composition with $\pi $ : $f\ast a:= f\cdot \pi (a)$ , $f,a\in C(X)$ . We call an algebraic homomorphism from the module $C(X)_{C(X)}$ to the module $C(X)_{\pi (C(X))}$ that maps nonnegative functions to nonnegative functions a $\sigma $ -module map on $C(X)$ .

The following proposition may well be known to some, but since we do not know a precise reference and since it is an essential part of our analysis, we supply a proof.

Proposition 5.3. Every $\sigma $ -module map $\mathcal {L}$ on $C(X)$ is determined by a unique nonnegative continuous function $\psi $ on X through the formula

(5-3) $$ \begin{align} \mathcal{L}(a)(x) = \sum_{\sigma(y)=x}\psi(y)a(y),\quad a\in C(X). \end{align} $$

Proof. Every positive linear map $\mathcal {L}$ on $C(X)$ is given by a unique family of measures $\alpha =\{\alpha _x\}_{x\in X}$ via the formula $\mathcal {L}(a)(x)=\int a(y)\, d\alpha _x(y)$ . If $\mathcal {L}$ is a module map, then $\operatorname {supp} \alpha _x\subseteq \sigma ^{-1}(x)$ . Indeed, assume to the contrary that there are $x,z\in X$ such that $z\in \operatorname {supp} \alpha _x$ and $\sigma (z)\ne x$ . We can then find an open set $U_z$ such that $\sigma |_{U_z}$ is a homeomorphism onto $\sigma (U_z)$ and $x\notin \sigma (U_z)$ . Let $a\in C(X)$ be such that $a(\sigma (z))=1$ and $a(y)=0$ if $y\notin \sigma (U_z)$ . It follows that $\mathcal {L}(\pi (a))(x)=\mathcal {L}(a\circ \sigma )(x)>0$ since $z\in \operatorname {supp} \alpha _x$ . However,

$$ \begin{align*} \mathcal{L}(\pi(a))(x)=\mathcal{L}(1\cdot \pi(a))(x)=(\mathcal{L}(1)\cdot a)(x)=\mathcal{L}(1)(x)a(x)=0. \end{align*} $$

This is a contradiction. Hence, $\operatorname {supp} \alpha _x\subseteq \sigma ^{-1}(x)$ for all $x\in X$ . Observe next that since X is compact, $\sigma ^{-1}(x)$ is a finite set for each $x\in X$ . Consequently, there is a nonnegative function $\Delta $ on $X\times X$ such that

(5-4) $$ \begin{align} \mathcal{L}(f)(x)=\int_X f(y)\,d\alpha_x(y)=\sum_{\sigma(y)=x}f(y)\Delta(x,y). \end{align} $$

To prove that $\Delta $ is continuous, let U be an open subset of X on which $\sigma $ is a homeomorphism and let V be an open subset of U such that $\overline {V}\subsetneq U$ . Let $f\in C(X)$ such that $f\equiv 1$ on $\overline {V}$ and $f\equiv 0$ on the complement of U. Then, $\mathcal {L}(f)\in C(X)$ and (5-4) yields

$$ \begin{align*} \mathcal{L}(f)(x)=\Delta(x,y) \quad\text{ if }\,x\in \sigma(V) \; \text{and}\; y=\sigma^{-1}(x)\in V. \end{align*} $$

Therefore, $\Delta $ is continuous on open sets of the form $\sigma (V)\times V$ . Since the cograph of $\sigma $ is covered by such sets, $\Delta $ is supported and continuous on the cograph of $\sigma $ . Therefore, the function $\psi $ defined via the equation $\psi (y):=\Delta (\sigma (y),y)$ is continuous and nonnegative. Moreover, (5-4) implies that

$$ \begin{align*} \mathcal{L}(f)(x)=\sum_{\sigma(y)=x}f(y)\Delta(x,y)=\sum_{\sigma(y)=x}f(y)\Delta(\sigma(y),y)=\sum_{\sigma(y)=x}\psi(y)f(y). \end{align*} $$

The converse assertion and the uniqueness of $\psi $ are immediate.

If $\mathcal {L}_{\psi }$ satisfies any of these three equivalent conditions, we call $\psi $ a normalized potential.

Proof. If $\mathcal {L}_{\psi }(1)=1$ , then for all $f\in C(X)$ ,

$$ \begin{align*} (\mathcal{L}_{\psi}\circ \pi)(f)(x)= & \sum_{\sigma(y)=x} \psi(y)\pi(f)(y) = \sum_{\sigma(y)=x}\psi(y)f(\sigma(y))\\ = & \sum_{\sigma(y) = x}\psi(y)f(x) = f(x)\bigg(\sum_{\sigma(y)=x}\psi(y)\bigg) = f(x), \end{align*} $$

so $\mathcal {L}_{\psi }$ is a left inverse of $\pi $ . Write $\iota $ for the identity map on $C(X)$ . Then, the assertion that $\mathcal {L}_{\psi }$ is a left inverse of $\pi $ is equivalent to the equation $\mathcal {L}_{\psi }\circ \pi = \iota $ . In this event, $(\pi \circ \mathcal {L}_{\psi })\circ (\pi \circ \mathcal {L}_{\psi }) = \pi \circ (\mathcal {L}_{\psi }\circ \pi )\circ \mathcal {L}_{\psi } = \pi \circ \iota \circ \mathcal {L}_{\psi }= \pi \circ \mathcal {L}_{\psi }$ . So, $\pi \circ \mathcal {L}_\psi $ is a positive idempotent map on $C(X)$ such that $(\pi \circ \mathcal {L}_\psi )\circ \pi =\pi \circ (\mathcal {L}_\psi \circ \pi )=\pi $ . Thus, $\pi \circ \mathcal {L}_{\psi }$ is an idempotent map that acts like the identity on the range of $\pi $ , that is, $\pi \circ \mathcal {L}_{\psi }$ is a conditional expectation onto the range of $\pi $ . Finally, if $\pi \circ \mathcal {L}_\psi $ is a conditional expectation onto the range of $\pi $ , $(\pi \circ \mathcal {L}_{\psi }) \circ \pi = \pi $ . However, then, $\pi \circ (\mathcal {L}_{\psi }\circ \pi )=\pi $ . Since $\pi $ is injective, $\mathcal {L}_{\psi }\circ \pi = \iota $ , that is, $\mathcal {L}_{\psi }$ is a left inverse of $\pi $ . However, this manifestly implies $\mathcal {L}_{\psi }(1)=1.$

The following proposition is part of Jean Renault’s [Reference Renault30, Proposition 3.1].

Proposition 5.7. Every full normalized potential D satisfies an equation of the form

$$ \begin{align*} D=D_0\cdot b, \end{align*} $$

where b is a nonvanishing function in $\pi (C(X))$ and $D_0$ is the potential of the master isometry from Definition 5.2.

Remark 5.8. For our purposes, every (proper) potential may be taken to be normalized. For if $\psi $ is a potential and if

$$ \begin{align*}D:=\psi/\mathcal{L}_{\psi}(1)\circ\sigma,\end{align*} $$

then D is a normalized potential that vanishes precisely at the same points as $\psi $ . Full normalized potentials may always be taken to be $D_0$ . Sections 10 and 11 are devoted to constructing nonfull normalized potentials from full nonnormalized potentials. This is how we build scaling functions and associated mother wavelets, that is, quadrature mirror filters.

We want to draw attention, also, to some additional relations that hold between $\pi $ and one of its left inverses $\mathcal {L}$ .

1. (1) First, recall that by Proposition 5.5, $\pi \circ \mathcal {L}$ is a conditional expectation of $C(X)$ onto the range of $\pi $ and every conditional expectation of $C(X)$ onto the range of $\pi $ is of this form for a suitable $\mathcal {L}$ . So, $\pi ^n\circ \mathcal {L}^n:= \boldsymbol {E}_n$ is a conditional expectation onto the range of $\pi ^n$ for every nonnegative integer n.

2. (2) Further still, we have the relations

   $$ \begin{align*} \boldsymbol{E}_n\boldsymbol{E}_m = \boldsymbol{E}_m\boldsymbol{E}_n = \boldsymbol{E}_n,\quad m\leq n. \end{align*} $$
   That is, the sequence $\{\boldsymbol {E}_n\}_{n=0}^{\infty }$ is a decreasing sequence of projections on the $C^*$ -algebra $C(X)$ , sometimes called a reverse martingale. We want to understand their relationship with the representation theory of $C_c(G(X,\sigma ),\lambda )$ that we develop later in this paper.

3. (3) Observe that computations like those in (5-3) show every normalized potential, full or not, leads to an isometry in the convolution algebra of $G(X,\sigma )$ . In fact, one may do a bit better. If $\mathfrak {u}$ is a continuous function on X such that $|\mathfrak {u}|^2$ is a normalized potential D, and if

   (5-5) $$ \begin{align} S_{\mathfrak{u}}(x,k-l,y):= \mathfrak{u}(x)1_{R_{1,0}}(x,k-l,y), \end{align} $$
   then (5-3) shows that $S_{\mathfrak {u}}$ satisfies
   $$ \begin{align*} f = S_{\mathfrak{u}}^* \ast (f\circ \sigma)\ast S_{\mathfrak{u}}= S_{\mathfrak{u}}^* \ast \pi(f) \ast S_{\mathfrak{u}} = \mathcal{L}_D\circ \pi (f) \end{align*} $$
   for $f\in C(X), |\mathfrak {u}|^2=D$ . Additionally, in general, $\mathcal {L}^n_D(f) = (S^*_{\mathfrak {u}})^n\ast f \ast (S_{\mathfrak {u}})^n$ .

4. (4) Finally, we have the easily proved fact that if the $\mathcal {L}$ leaves a probability measure $\mu $ invariant, the conditional expectations $\boldsymbol {E}_n$ satisfy the equations

   (5-6) $$ \begin{align} \int_X \boldsymbol{E}_n(f)g\,d\mu=\int_X f\boldsymbol{E}_n(g)\,d\mu, \quad f,g\in C(X). \end{align} $$

It is convenient to regard the isometry $S_{\mathfrak {u}}$ as an element of $\mathcal {L}(\mathcal {X}_0)$ , where $\mathcal {X}_0$ is the $C^*$ -correspondence defined by the equivalence $(G,\alpha )$ between $(G,\lambda )$ and $(G,\lambda )$ as described in the paragraph preceding Section 3. There, the operator $S_{\mathfrak {u}}\in \mathcal {L}(\mathcal {X}_0)$ is defined via

(5-7) $$ \begin{align} S_{\mathfrak{u}}\xi(x,t,y)=\mathfrak{u}(x)\xi(\sigma(x),t-1,y). \end{align} $$

It may seem that we have introduced confusion by using ‘ $S_{\mathfrak {u}}$ ’ to denote both an isometry in $C^*(G,\lambda )$ and the isometry from (5-7) on $\mathcal {L}(\mathcal {X}_0)$ . However, this ambiguity should cause no significant problem. The two isometries are the same, up to a unitary equivalence between the two Hilbert modules involved.

6 Pullbacks by projective limits

In nearly every operator-theoretic approach to wavelets that we have encountered, there are isometries like our $S_{\mathfrak {u}}$ and efforts are made to understand unitary extensions of them. We want to do that here as well. However, we also want to emphasize that the extensions we produce do not live in $C^*(G(X,\sigma ))$ , but in larger $C^*$ -algebras that we reveal as certain pullbacks. Their description requires several ingredients.

The first ingredient is a pullback of $G(X,\sigma )$ in the sense of Proposition 3.1 that is determined by a pair consisting of a space $X_{\infty }$ and map $p:X_{\infty }\to X$ . The space $X_{\infty }$ is the projective limit space induced by $\sigma $ :

$$ \begin{align*}X_{\infty}:=\{(x_0,x_1,x_2,\ldots)\in X^{\infty}\mid x_i=\sigma(x_{i+1})\}.\end{align*} $$

Here, of course, $X^{\infty }$ is the infinite product of copies of X indexed by $\mathbb {N}$ . Since X is compact, and $\sigma $ is continuous and surjective, $X_{\infty }$ is a nonempty closed, and therefore compact, subset of $X^{\infty }$ . We write its elements as $\underline {x}:=(x_0,x_1,\ldots )$ . The map p is simply the projection onto the first coordinate: $p(\underline {x}):=x_0$ . Since $\sigma $ is open, p is also open, as well as continuous, and it is surjective because $\sigma $ is surjective.

So, using the notation from Proposition 3.1, we write the pullback of $G(X,\sigma )$ by $X_{\infty }$ as $X_{\infty }\ast G(X,\sigma )\ast X_{\infty }$ . To shorten the notation, we usually write $G_{\infty }(X,\sigma )$ for this groupoid. We also use some self-evident variations of this notation.

The main purpose of $X_{\infty }$ is to carry an invertible ‘extension’ of $\sigma $ . Define $\sigma _{\infty }$ on $X_{\infty }$ by the formula

$$ \begin{align*} \sigma_{\infty}(\underline{x}):= (\sigma(x_0),x_0,x_1\cdots),\quad \underline{x}= (x_0,x_1\cdots)\in X_{\infty}. \end{align*} $$

The inverse of $\sigma _{\infty }$ is given by the equation

$$ \begin{align*} \sigma_{\infty}^{-1}(x_0,x_1,x_2\cdots):= (x_1,x_2\cdots). \end{align*} $$

The map $\sigma _{\infty }$ is not really an extension of $\sigma $ , but it does satisfy the important ‘intertwining’ equation

$$ \begin{align*} \sigma\circ p= p\circ \sigma_{\infty}. \end{align*} $$

By Proposition 3.1, $G_{\infty }(X,\sigma )$ is a locally compact Hausdorff groupoid with unit space $X_{\infty }$ . The range and source maps are the obvious projections that are, however, not local homeomorphisms. Consequently, $G_{\infty }(X,\sigma )$ is not étale. So, we must find Haar systems that are not given by counting measures. These are built from potentials for transfer operators associated to $\sigma $ (see Remark 5.4).

Using a normalized potential D on X, one can construct a system of probability measures on $G_{\infty }(X,\sigma )$ , indexed by $X_{\infty }$ , that is denoted $\nu ^D*\lambda := \{(\nu ^D*\lambda )^{\underline {x}}\}_{\underline {x}\in X_{\infty }}$ . The symbol $\nu ^D$ by itself refers to a system of measures $\{\nu ^D_{x_0}\}_{x_0\in X}$ on $X_{\infty }$ that is parameterized by X and is defined as follows.

Recall that the product topology on $X^{\infty }$ is such that the collection of complex-valued functions of the form $f=f_0\cdot f_1\cdot f_2 \cdots f_n$ , where the $f_i$ lie in $C(X)$ and

$$ \begin{align*} f(\underline{x}): = f_0(x_0)\cdot f_1(x_1)\cdot f_2(x_2)\cdots f_n(x_n),\quad \underline{x}=(x_0,x_1,x_2, \cdots)\in X^{\infty}, \end{align*} $$

generate $C(X^{\infty })$ . Consequently, their restrictions to $X_{\infty }$ generate $C(X_{\infty })$ . Recall, too, that Kolmogorov’s extension theorem [Reference Rosenblatt34, Appendix A.3] guarantees that the formula

(6-1) $$ \begin{align} & \int_{X_\infty} f_{0}(x_{0})\cdot f_{1}(x_{1})\cdots f_{n}(x_{n})\,d\nu^D_{x_{0}}(\underline{x}) \nonumber\\ & \quad := f_{0}(x_{0})\bigg(\!\sum_{\sigma(x_{1})= x_{0}}D(x_{1})f_{1}(x_{1}) \bigg(\!\sum_{\sigma(x_{2})=x_1} D(x_{2})f_{2}(x_{2}) \cdots \bigg(\!\sum_{\sigma(x_{n})=x_{n-1}}D(x_{n})f_{n}(x_{n})\bigg)\cdots\bigg)\bigg), \end{align} $$

which is valid for all $f=f_{0}f_{1}\cdots f_{n}$ with $f_{i}\in C(X)$ , yields a well-defined probability measure supported on $p^{-1}({x_{0}})$ that we have denoted $\nu ^D_{{x_{0}}}$ .

Equation (6-1) becomes more manageable when one uses the transfer operator $\mathcal {L}_D$ to abbreviate the sums. The result is

(6-2) $$ \begin{align} &\int_{X_\infty} f_{0}(x_{0})\cdot f_{1}(x_{1})\cdots f_{n}(x_{n})\,d\nu^D_{x_{0}}(\underline{x}) \nonumber\\ & \quad :=(f_{0}\mathcal{L}_D(f_1\mathcal{L}_D(f_2 \cdots \mathcal{L}_D(f_{n-1}\mathcal{L}_D(f_n))\cdots )))(x_0), \end{align} $$

which not only makes it clear that $\nu ^D_{x_0}$ is a probability measure on $X_{\infty }$ , it shows also that the family $\{\nu ^D_{x_0}\}_{x_0\in X}$ may be viewed as an element of $C(X,\mathfrak {M}_1(X_{\infty }))$ , where $\mathfrak {M}_1(X_{\infty })$ is the space of all probability measures on $X_{\infty }$ with the weak- $\ast $ topology. Thus, $\{\nu ^D_{x_0}\}_{x_0\in X}$ is a continuous family of probability measures on $X_{\infty }$ indexed by X. Because the family $\nu ^D = \{\nu ^D_{x_0}\}_{x_0\in X}$ is a system of measures on $X_{\infty }$ that is fibered over X by p (that is, each $\nu ^D_{x_0}$ is supported on $p^{-1}(x_0), x_0\in X$ ), we refer to $\nu ^D$ as the p-system of measures on $X_{\infty }$ determined by D.

Finally, letting $\lambda := \{\lambda ^x\}_{x\in X}$ denote the counting measure Haar system on $G(X,\sigma )$ (paragraphs following Definition 5.1), the formula

(6-3) $$ \begin{align} & \int_{G_{\infty}(X,\sigma)}f(\underline{x},(p(\underline{x}),t, p(\underline{y})),\underline{y})\, d(\nu^D\ast \lambda)^{\underline{x}}\nonumber\\ & \quad := \int_{G(X,\sigma)}\int_{X_{\infty}}f(\underline{x},(p(\underline{x}),t,p(\underline{y})), \underline{y})\, d\nu^D_{p(\underline{y})}(\underline{y})\, d\lambda^{p(\underline{x})}(p(\underline{x}),t,p(\underline{y})), \end{align} $$

where $f\in C_c(G_{\infty }(X,\sigma ))$ , defines the system of measures $\nu ^D*\lambda : = \{(\nu ^D*\lambda )^{\underline {x}}\}_{\underline {x}\in X_{\infty }}$ on $G_\infty (X,\sigma )$ . Close inspection reveals that (6-3) is a special case of (3-1), that is, $\nu ^D\ast \lambda = \beta $ specialized to $G_{\infty }(X,\sigma )$ . Thus, $\nu ^D\ast \lambda $ is a Haar system on $G_{\infty }(X,\sigma )$ if $\nu ^D$ is full. The following proposition describes when this happens.

From now on, we usually drop the subscript, $0$ , on $\nu ^D_{x_0}$ .

7 p-systems of measures and conditional expectations

We begin with a continuation of the paragraph following Definition 5.2.

First, note that the defining formula for $\nu ^D$ reveals it to be a map from $C(X_{\infty })$ to $C(X)$ that is linear, positive, and unital:

(7-1) $$ \begin{align} \nu^D(f)(x): = \int_{X_{\infty}}f(\underline{y})\, d\nu_x^D(\underline{y}),\quad f\in C(X_{\infty}). \end{align} $$

Most important for our purposes, (7-1) makes it clear that $\nu ^D$ is a left inverse of the map $p_*:C(X)\to C(X_{\infty })$ that is dual to p:

$$ \begin{align*} p_*(f)(\underline{x}):=f(p(\underline{x})),\quad f\in C(X). \end{align*} $$

Indeed,

(7-2) $$ \begin{align} [(\nu^D\circ p_*)(f)](x) & = [\nu^D(p_*(f))](x) = \int_{X_\infty}p_*(f)(\underline{y})\,d\nu^D_{x}(\underline{y})\nonumber\\ & = \int_{X_{\infty}}f(p(\underline{y}))\, d\nu^D_{x}(\underline{y}) = \int_{X_{\infty}} f(x) \, d\nu^D_x(\underline{y}) = f(x). \end{align} $$

The penultimate equation reflects the fact that each $\nu _x^D$ is a probability measure on $X_{\infty }$ supported on $p^{-1}(x)$ . Consequently, also, (7-2) shows that $\nu ^D$ is a surjective map from $C(X_{\infty })$ onto $C(X)$ .

In analogy with the notation following Remark 5.8, we find that $p_*\circ \nu ^D$ is a conditional expectation, $\mathcal {E}_0$ , on $C(X_{\infty })$ , whose definition expressed in terms of integrals is

(7-3) $$ \begin{align} \mathcal{E}_0(f)(\underline{x}) & = [(p_*\circ \nu^D)(f)](\underline{x}) = [p_*(\nu^D(f))](\underline{x})\nonumber\\ & = (\nu^D(f))(p(\underline{x})) = \int_{X_{\infty}} f(\underline{y})\, d\nu^D_{p(\underline{x})}(\underline{y}), \quad f\in C(X_{\infty}). \end{align} $$

This equation shows that $\mathcal {E}_0$ is a unital, positive map from $C(X_{\infty })$ to $C(X_{\infty })$ , while the fact that $\mathcal {E}_0$ is idempotent is immediate from (7-2).

Finally, the fact that the range of $\mathcal {E}_0$ consists of all functions in $C(X_{\infty })$ that depend only on the first variable is clear from (7-3). Evidently, such functions form an algebra isomorphic to $C(X)$ , and $\mathcal {E}_0$ is a bimodular map over it. Consequently, we informally, but unambiguously, write $\mathcal {E}_0(a\cdot f \cdot b)=a\cdot \mathcal {E}_0(f)\cdot b$ for all $f\in C(X_{\infty })$ and $a,b\in C(X)$ .

The following equation arises later in our analysis (see (9-2)). We believe it is helpful to parse it now. For $f\in C(X_\infty )$ ,

(7-4) $$ \begin{align} \sum_{\sigma(z)=x}D(z)\int_{X_\infty}f(\sigma_\infty(\underline{z}))\,d\nu_{z}^D(\underline{z})=\int_{X_\infty}f(\underline{x})\,d\nu_{x}^D(\underline{x}) \end{align} $$

for all $x\in X$ . This equation may be easier to understand if it is rewritten in terms of the maps that are implicit in it. First, let $\pi _{\infty }$ be defined by the formula

$$ \begin{align*} \pi_{\infty}(f):= f\circ \sigma_{\infty}, \quad f\in C(X_\infty). \end{align*} $$

Thus, $\pi _{\infty }$ is an analog of the endomorphism $\pi $ of $C(X)$ defined in Proposition 5.7. This time, however, $\pi _{\infty }$ is an automorphism of $C(X_{\infty })$ , and its inverse is given by composing with $\sigma _{\infty }^{-1}$ , that is, $\pi _{\infty }^{-1}(f)= f\circ \sigma _{\infty }^{-1}$ for all $f\in C(X_{\infty })$ . So, now we can parse (7-4) in terms of maps:

(7-5) $$ \begin{align} \mathcal{L}_D\circ \nu^D\circ \pi_{\infty} = \nu^D, \end{align} $$

or a bit more symmetrically, we may write

(7-6) $$ \begin{align} \mathcal{L}_D\circ \nu^D = \nu^D\circ \pi_{\infty}^{-1}. \end{align} $$

Keep in mind that both sides of each of these equations are maps from $C(X_{\infty })$ to $C(X)$ .

For our purposes here, the most important role of the p-system of measures $\nu ^D$ is revealed in Equation (7-10), below, that is based on the following lemma.

Lemma 7.1. The map $\Phi :G_{\infty }(X,\sigma ) \to G(X,\sigma )$ defined by the equation

$$ \begin{align*} \Phi((\underline{x},k-l,\underline{y})):=(p(\underline{x}),k-l,p(\underline{y})),\quad (\underline{x},k-l,\underline{y})\in G_{\infty}(X,\sigma), \end{align*} $$

is a continuous, open, proper and surjective groupoid homomorphism mapping the pullback groupoid $G_{\infty }(X,\sigma )$ onto $G(X,\sigma )$ .

As a consequence of this lemma, $\Phi _*$ , defined by the equation

(7-7) $$ \begin{align} \Phi_*(f):= f\circ \Phi, \quad f\in C_c(G,\sigma), \end{align} $$

is a linear map from $C_c(G,\sigma )$ into $C_c(G_{\infty },\sigma )$ that is continuous with respect to the inductive limit topologies on each space. It is also clear from (6-3) that for all $f\in C_c(G(X,\sigma ))$ ,

(7-8) $$ \begin{align} \int_{G_{\infty}(X,\sigma)}\Phi_*(f)(\underline{x},t, \underline{y})\, d(\nu^D\ast \lambda)^{\underline{x}}:= \int_{G(X,\sigma)}f(x_0,t,y_0)\, d\lambda^{x_0}(x_0,t,y_0). \end{align} $$

A succinct way to express (7-8) is to write everything in terms of the maps determined by the systems of measures involved:

(7-9) $$ \begin{align} p_*\circ \lambda = (\nu^D*\lambda)\circ \Phi_*, \end{align} $$

where $\lambda :=\{\lambda ^x\}_{x\in X}$ is viewed as a map from $C_c(G(X,\sigma ))$ to $C(X)$ , $p_*:C(X)\to C(X_{\infty })$ is the map dual to p, $\nu ^D * \lambda $ is the map determined by the system $\{\nu ^D_{p(\underline {x})}*\lambda ^{p(\underline {x})}\}_{\underline {x}\in X_{\infty }}$ , and $\Phi _*$ is the map dual to $\Phi $ acting on functions.

The image of $\Phi _*$ thus consists of those functions in $C_c(G_{\infty }(X,\sigma ),\nu ^D*\lambda )$ that depend only on the zeroth-coordinate. The equations in the paragraphs following the proof of Lemma 7.1, in particular, (7-9), make it clear that $\Phi _*$ defines an algebra isomorphism from $C_c(G(X,\sigma ),\lambda )$ onto its image in $C_c(G_{\infty }(X,\sigma ),\nu ^D\ast \lambda )$ that is a homeomorphism in the inductive limit topologies involved.

Recall that $1_{R_{0,0}}$ is the identity of $C_c(G(X,\sigma ),\lambda )$ (see the paragraphs following (5-1)). Consequently, $\Phi _{*}(1_{R_{0,0}})$ is a projection in $C_c(G_{\infty }(X,\sigma ),\nu ^D\ast \lambda )$ . However, a projection in a $\ast $ -algebra determines a projection on the algebra via ‘cornering’. So, in our setting, $f\to \Phi _{*}(1_{R_{0,0}})\ast f \ast \Phi _{*}(1_{R_{0,0}}) $ defines an idempotent, positivity-preserving, linear map, which we denote by $\mathbb {E}$ , whose range is the image of $\Phi _*$ . In fact, it is a straightforward calculation to see that

(7-10) $$ \begin{align} \mathbb{E}(f)(\underline{x},k-l,\underline{y}):= \ & \Phi_{*}(1_{R_{0,0}})\ast f \ast \Phi_{*}(1_{R_{0,0}})(\underline{x},k-l,\underline{y})\nonumber\\ = & \int_{X_{\infty}} \int_{X_{\infty}} f(\underline{u},k-l,\underline{v})\,d\nu^D_{p(\underline{x})}(\underline{u})\,d\nu^D_{p(\underline{y})}(\underline{v}) \end{align} $$

for all $f\in C_c(G_{\infty }(X,\sigma ),\nu ^D\ast \lambda )$ . This equation, in turn, coupled with (7-3), justifies our writing

$$ \begin{align*} \mathbb{E} = \mathcal{E}_0\otimes \iota \otimes \mathcal{E}_0, \end{align*} $$

restricted to $C_c(G_{\infty }(X,\sigma ))$ , where we are regarding $G_{\infty }(X,\sigma )$ as a locally closed subset of $X_{\infty } \times \mathbb {Z} \times X_{\infty }$ and where $\iota $ denotes the identity map on $C_c(\mathbb {Z})$ .

It is also easy to see that $\mathbb {E}$ is bimodular in the sense that $\mathbb {E}(\Phi _{*}(b_1)\ast f \ast \Phi _{*}(b_2))=\Phi _{*}(b_1)\ast \mathbb {E}(f)\ast \Phi _{*}(b_2)$ for all $f\in C_c(G_{\infty }(X,\sigma ),\nu ^D\ast \lambda )$ and $b_i\in C_c(G(X,\sigma ),\lambda )$ , $i=1,2$ .

Thus, $\mathbb {E}$ has all the trappings of a conditional expectation, except one: $\mathbb {E}$ is not unital. That is not important here. The defining equation (7-10) means that $\mathbb {E}$ is a ‘cornering’ projection and so extends to a completely positive, contractive, and idempotent map on any $C^*$ -algebra that is generated by an $\ast $ -representation of $C_c(G_{\infty }(X,\sigma ),\nu ^D\ast \lambda )$ .

8 Representations of $G_{\infty }(X,\sigma )$ induced from $G(X,\sigma )$

We fix a full normalized potential D and we consider the Haar system $\nu ^D*\lambda $ on $G_\infty (X,\sigma )$ determined by D. This Haar system on $G_\infty (X,\sigma )$ is fixed throughout the section. We induce representations of $C^*(G(X,\sigma ),\lambda )$ to representations of $C^*(G_\infty (X,\sigma ),\nu ^D*\lambda )$ using nonfull normalized potentials derived from D via the techniques developed in Section 4. Since X and $\sigma $ are fixed, we write G for $G(X,\lambda )$ and $G_{\infty }$ for $G_{\infty }(X,\sigma )$ when the notation becomes cluttered.

We consider a nonnegative continuous function $b:X\to \mathbb {R}_+$ such that the function $D_b(x):=b(x)D(x)$ is another normalized potential for $\sigma $ , not necessarily full: $ \sum _{\sigma (y)=x}D_b(y)=1$ for all $x\in X$ . The potential $D_b$ defines a p-system of measure $\nu ^{D_b}$ on $X_\infty $ via (6-1) or, equivalently, (6-2). Moreover, $\nu _u^{D_b}$ is absolutely continuous with respect to $\nu _u^D$ since

$$ \begin{align*} \int_{X_\infty} f_0(x_0)f_1(x_1)\cdots f_n(x_n)\,d\nu_{x_0}^{D_b}(\underline{x}) = \int_{X_\infty} f_0(x_0)f_1(x_1)\cdots f_n(x_n)\prod_{i=1}^n b(x_i)\,d\nu_{x_0}^{D}(\underline{x}) \end{align*} $$

for all $f=f_0f_1\cdots f_n$ with $f_i\in C(X)$ and $n\ge 0$ . We let $\mathbb {d}$ be the Radon–Nykodim derivative such that $\nu _u^{D_b}=\mathbb {d} \nu _u^D$ for all $u\in X$ . In the notation of (4-1), one would write $\nu _{\mathbb {d}}^{D_b}$ for $\nu ^{D_b}$ . We prefer the new notation since $\nu _{\mathbb {d}}^{D_b}$ is hard to read and $\mathbb {d}$ is determined by b. We define a groupoid correspondence $(Z,\alpha _b)$ from $(G_{\infty }(X,\sigma ),\nu ^D*\lambda )$ to $(G,\lambda )$ using the framework described in the paragraphs preceding Lemma 3.3, where

$$ \begin{align*} Z: = X_{\infty}\,{_p\ast_r}\, G(X,\sigma) = \{(\underline{x},(p(\underline{x}),t,y))\mid \underline{x}\in X_{\infty},(p(\underline{x}),t,y)\in G(X,\sigma)\}, \end{align*} $$

and $\alpha _b=\nu ^{D_b}*\lambda $ .

Our goal now is to relate the representation theory of $G(X,\sigma )$ to the representation theory of $G_{\infty }(X,\sigma )$ using the potential $D_b$ . It is fixed for the remainder of this section.

Recall that $C_c(Z)$ can be completed to a $C^*$ -correspondence that we denote $\mathcal {X}_\infty $ from $C^*(G_{\infty }(X,\sigma ),\nu ^D*\lambda )$ to $C^*(G(X,\sigma ),\lambda )$ via (4-4), (4-5), and (4-6). In particular, $C^*(G_{\infty }(X,\sigma ),\nu ^D*\lambda )$ acts as multipliers on $\mathcal {X}_\infty $ via

(8-1) $$ \begin{align} & \mathbb{L}(a)\xi(\underline{x},(p(\underline{x}),t,y)):=a\cdot \xi(\underline{x},(p(\underline{x}),t,y))\nonumber\\ & \quad= \int_{G_{\infty}} a(\underline{x}, s, \underline{w})\xi(\underline{w},(p(\underline{w}),t-s,y))\mathbb{d}(\underline{w})^{1/2}/\mathbb{d}(\underline{x})^{1/2}\, d(\nu^D\ast \lambda)^{\underline{x}}(\underline{x},s,\underline{w}). \end{align} $$

Fix a representation $\hat {L}= (\mu ,X\ast \mathcal {H}, \hat {L})$ of the groupoid G and let $\mathfrak {L}$ be its integrated form representing $C^*(G,\lambda )$ on $L^2(X\ast \mathcal {H},\mu )$ as described in the paragraph containing (2-1). We want to explain how to induce $\hat {L}$ , $\mathfrak {L}$ , and all related structures to the pullback groupoid, $G_{\infty }(X,\sigma )$ . The relevance of the normalized potential, $D_b$ , is tracked in the process.

The induced Hilbert bundle determined by $G^{(0)}\ast \mathcal {H} = X\ast \mathcal {H}$ is the bundle $X_{\infty }\ast \mathcal {H} = G_{\infty }(X,\sigma )^{(0)}\ast \mathcal {H}$ that is the pull back of $X\ast \mathcal {H}$ by p, i.e., the fiber $H_{\underline {x}}:= H_{p(\underline {x})}$ . This bundle is the same for all the other induced structures we consider. The isomorphism groupoid of $X_\infty \ast \mathcal {H}$ , ${\operatorname {Iso}}(X_{\infty }\ast \mathcal {H})$ , is similarly identified:

$$ \begin{align*} {\operatorname{Iso}}(X_{\infty}\ast \mathcal{H}):=\{(\underline{x},U,\underline{y})\mid U\in B_{{\operatorname{Iso}}}(H_{\underline{y}}, H_{\underline{x}})\}= \{(\underline{x},U,\underline{y})\mid U\in B_{{\operatorname{Iso}}}(H_{p(\underline{y})}, H_{p(\underline{x})})\}. \end{align*} $$

The representation $\hat {L}_{\infty }$ of $G_{\infty }(X,\sigma )$ on $X_{\infty }\ast \mathcal {H}$ induced by $\hat {L}$ is given by the formula

$$ \begin{align*} \hat{L}_\infty(\underline{x},t,\underline{y})=(\underline{x},L_{\infty,(\underline{x},t,\underline{y})},\underline{y}), \end{align*} $$

where

$$ \begin{align*} L_{\infty,(\underline{x},t,\underline{y})}(\mathfrak{h}(\underline{y})): = L_{(p(\underline{x}),t,p(\underline{y}))}\mathfrak{h}(p(\underline{y})), \quad (\underline{x},t,\underline{y})\in G_{\infty}, \mathfrak{h}\in X_{\infty}\ast \mathcal{H}. \end{align*} $$

Its action on $X_{\infty }\ast \mathcal {H}$ , too, is independent of the potential $D_b$ .

What depends upon $D_b$ is the quasi-invariant measure on $G^{(0)}_{\infty }=X_{\infty }$ . Viewed as a map acting from $C(G^{(0)}_{\infty })$ to $\mathbb {C}$ , it is defined by the equation

$$ \begin{align*} \mu_{\infty}^{D_b}:=\mu\circ \nu^{D_b}. \end{align*} $$

Note that $\mu _{\infty }^{D_b}$ is well defined: $\nu ^{D_b}$ is a map from $C(G^{(0)}_{\infty })$ to $C(G^{(0)}(X,\sigma ))=C(X)$ and $\mu $ is a map from $C(X)$ to $\mathbb {C}$ . Recall from (4-2) that the modular function is

$$ \begin{align*} \Delta_{\infty}(\underline{x},k-l,\underline{y}) =\Delta_\mu(p(\underline{x}),k-l,p(\underline{y}))\mathbb{d}(\underline{x})/\mathbb{d}(\underline{y}). \end{align*} $$

We write $\mathfrak {L}_\infty $ for the integrated form of $L_\infty$ , which is a $C^\ast$ -representation of $C^*(G_\infty (X,\sigma ),\nu ^D*\lambda )$ on the space $L^2(X_\infty *\mathcal {H}_\infty ,\mu _\infty ^{D_b})$ . For $f\in C_c(G_\infty )$ , $\mathfrak {L}_{\infty }(f)$ acts on $L^2(X_\infty *\mathcal {H}_\infty ,\mu _\infty ^{D_b})$ via

(8-2) $$ \begin{align} &\mathfrak{L}_\infty(f)\mathfrak{f}(\underline{x})\nonumber\\ & \quad= \int_{G_{\infty}}f(\underline{x},k-l,\underline{y})L_{(p(\underline{x}),k-l,p(\underline{y}))}\mathfrak{f}(\underline{y})\Delta_\infty(\underline{x},k-l,\underline{y})^{-1/2}\,d(\nu^D*\lambda)^{\underline{x}}(\underline{x},k-l,\underline{y}) \end{align} $$

for all $\mathfrak {f}\in L^2(X_\infty *\mathcal {H}_\infty ,\mu _\infty )$ .

9 Proto-multiresolution analyses and their representations

In this section, we return to the elements of Rieffel’s theory that are developed preceding Theorem 4.1. We continue with the same notation and we let $\mathfrak {u}\in C(X)$ be a function such that $|\mathfrak {u}|^2=D_b$ . Our first goal is to define a unitary operator $U_{\mathfrak {u}}$ on $\mathcal {X}_{\infty }$ that conjugates the image of $C^*(G_\infty (X,\sigma ),\nu ^D\ast \lambda )$ in $\mathcal {L}(\mathcal {X}_{\infty })$ into itself and is a unitary extension of the isometry $S_{\mathfrak {u}}$ defined in (5-7). This unitary is used to generate what we call a proto-multiresolution analysis. In a sense we make clear, constructing a proto-multiresolution analysis constitutes the first step when building an MRA. Proto-multiresolution analyses put into evidence precisely the variables that can be ‘tuned’ to produce MRAs.

Again, we abbreviate $G(X,\sigma )$ to G and $G_{\infty }(X,\sigma )$ to $G_{\infty }$ .

The putative unitary extension of $S_{\mathfrak {u}}$ , $U_{\mathfrak {u}}$ , is defined initially on $C_c(X_\infty *G)$ via

(9-1) $$ \begin{align} (U_{\mathfrak{u}} \xi)(\underline{x},(p(\underline{x}),n,y)):=\mathfrak{u}(p(\underline{x}))\xi(\sigma_{\infty}(\underline{x}),(p(\sigma_\infty(\underline{x})),n-1,y)) \end{align} $$

for all and $(\underline {x},(p(\underline {x}),n,y)))\in X_\infty *G$ .

Before proving that $U_{\mathfrak {u}}$ is a unitary extension of $S_{\mathfrak {u}}$ , we first show that $U_{\mathfrak {u}}$ is an isometry and compute its adjoint. Let $\xi \in C_c(X_\infty *G)$ and compute, using (4-5), (7-4), and its compact form (7-5):

(9-2) $$ \begin{align} \langle U_{\mathfrak{u}}& \xi, U_{\mathfrak{u}}\xi\rangle_{C_c(G)}(x,n,y)\nonumber\\ & := \int_G \int_{X_\infty}\overline{U_{\mathfrak{u}} \xi(\underline{z},(p(\underline{z}),m,x))}U_{\mathfrak{u}}\xi(\underline{z},(p(\underline{z}),m+n,y))\,d\nu^{D_b}_z(\underline{z})\,d\lambda_x(z,m,x) \nonumber\\ & =\int_G \int_{X_\infty}\overline{\mathfrak{u}(z)}\overline{\xi(\sigma_\infty(\underline{z}),(p(\sigma_\infty(\underline{z})),m-1,x))} \nonumber\\ & \quad \cdots\mathfrak{u}(z)\xi(\sigma_\infty(\underline{z}),(p(\sigma_\infty(\underline{z})),m+n-1,y))\,d\nu^{D_b}_z(\underline{z})\,d\lambda_x(z,m,x) \nonumber\\ & = \int_G \sum_{\sigma(z)=t}D_b(z)\int_{X_\infty} \overline{\xi(\sigma_\infty(\underline{z}),(p(\sigma_\infty(\underline{z})),m-1,x))}\nonumber\\ & \quad \cdots\xi(\sigma_\infty(\underline{z}),(p(\sigma_\infty(\underline{z})),m+n-1,y))\,d\nu^{D_b}_z(\underline{z})\,d\lambda_x(t,m-1,x) \nonumber\\ & = \int_G \mathcal{L}_{D_b}\circ \nu_t^{D_b}\circ \pi_\infty(\overline{\xi(\cdot,(\cdot,m-1,x))}\xi(\cdot,(\cdot,m+n-1,y)))\,d\lambda_x(t,m-1,x) \nonumber\\ & =\int_G \nu_t^{D_b}(\overline{\xi(\cdot,(\cdot,m-1,x))}\xi(\cdot,(\cdot,m+n-1,y)))\,d\lambda_x(t,m-1,x) \nonumber\\ & =\int_G\int_{X_\infty} \overline{\xi(\underline{t},(p(\underline{t}),m,x))} \xi(\underline{t},(p(\underline{t}),m+n,y))\,d\nu^{D_b}_z(\underline{t})\,d\lambda_x(t,m,x) \nonumber\\ & =\langle \xi,\xi\rangle_{C_c(G)}(x,n,y), \end{align} $$

where the penultimate equality follows by relabeling the integers, that is, by the invariance of the Haar system $\lambda $ .

To complete the proof that $U_{\mathfrak {u}}$ is a unitary, we must examine first how the zero set of $D_b$ , $Z_{D_b}$ , affects the calculation of the $C_c(G(X,\sigma ))$ -norm of elements in $C_c(X_\infty *G)$ . Let $z_1\in Z_{D_b}$ and let $z_0:=\sigma (z_1)$ . Consider an open neighborhood A of $z_1$ such that $\sigma |_A:A\to \sigma (A)$ is a homeomorphism onto the open set $\sigma (A)$ . Let $f\in C(X_\infty )$ be supported in $\sigma (A)*A*X_\infty $ . Then,

$$ \begin{align*} \nu_{z_0}^{D_b}(f)=D_b(z_1)\nu_{z_1}^{D_b}(f(z_0,z_1,\cdot))=0. \end{align*} $$

Assume that $\xi \in C_c(X_\infty *G)$ is of the form $\xi (\underline {x},(p(\underline {x}),n,y))=f_1(\underline {x})f_2(p(\underline {x}),n,y)$ such that $f_1$ is supported in $\sigma (A)*A*X_\infty $ . Let

$$ \begin{align*} Z_{2,D_b}=\{\underline{x}\in X_\infty\,\mid\,x_1\in Z_{D_b}\}. \end{align*} $$

If $\tilde {f}_1=f_1 1_{Z_{2,D}^c}$ and $\tilde {\xi }:=\tilde {f}_1f_2$ , then $\langle \xi -\tilde {\xi },\xi -\tilde {\xi }\rangle _{C_c(G)}=0$ . That is, in the inner product, an element $\xi \in C_c(X_\infty *G)$ depends only on the values outside the set $Z_{2,D}$ . Hence, the operator $U_{\mathfrak {u}}^*$ defined on $C_c(X_\infty *G)$ via

$$ \begin{align*} U_{\mathfrak{u}}^*\xi(\underline{x},(p(\underline{x}),n,y))=\frac{1}{\mathfrak{u}(p(\sigma_\infty^{-1}(\underline{x})))}\xi(\sigma_\infty^{-1}(\underline{x}),(p(\sigma_\infty^{-1}(\underline{x})),n+1,y)) \end{align*} $$

for all $(\underline {x},(p(\underline {x}),n,y))\in X_\infty *G$ such that $\mathfrak {u}(x_1)\ne 0$ is well defined in the $C_c(G)$ -inner product (4-5) since $p(\sigma _\infty ^{-1}(\underline {x}))=x_1$ . One can check that $U_{\mathfrak {u}}^*$ is an isometry using version (7-6) of (7-4). We leave the details to the reader. The fact that $U_{\mathfrak {u}}^*$ is the inverse of $U_{\mathfrak {u}}$ is a straightforward computation.

Recall that $\mathcal {X}_0$ is the $C^*$ -correspondence defined by the $(G,\lambda )-(G,\lambda )$ equivalence $(G,\alpha )$ , where $\alpha _x=\lambda _x$ , as described in the paragraph following (5-6). The isometry $S_{\mathfrak {u}}\in \mathcal {K}(\mathcal {X}_0)$ is defined via (5-7). To prove that $U_{\mathfrak {u}}$ is a unitary extension of $S_{\mathfrak {u}}$ , we define $V_{\infty ,0}:\mathcal {X}_0\to \mathcal {X}_\infty $ via

$$ \begin{align*} V_{\infty,0}\xi(\underline{x},(p(\underline{x}),t,y))=\xi(p(\underline{x}),t,y) \quad\text{for all}\,\xi\in C_c(Z). \end{align*} $$

Proposition 9.1. The map $V_{\infty ,0}$ extends to an adjointable isometry that intertwines $U_{\mathfrak {u}}$ and $S_{\mathfrak {u}}$ , $U_{\mathfrak {u}} V_{\infty ,0}=V_{\infty ,0}S_{\mathfrak {u}}$ , such that

$$ \begin{align*} V_{\infty,0}^*\eta(x,t,y)=\int_{X_\infty}\eta(\underline{x},(x,t,y))\,d\nu_x^{D_b}(\underline{x}) \end{align*} $$

for all $\eta \in C_c(X_\infty *G)$ . Moreover, $V_{\infty ,0}(\xi \cdot a)=V_{\infty ,0}\xi \cdot a$ for all $\xi \in \mathcal {X}_0$ and $a\in C^*(G)$ .

Proof. The fact that $V_{\infty ,0}$ is an isometry follows from the following computation that uses the fact that $D_b$ is unital and, thus, $\nu _x^{D_b}(1)=1$ for all $x\in X$ :

$$ \begin{align*} \langle V_{\infty,0}\xi,V_{\infty,0}\eta\rangle_{C_c(G)}(x,t,y)& =\sum_{(z,s,x)\in G}\int_{X_\infty} \overline{V_{\infty,0}\xi(\underline{z},(z,s,x))}V_{\infty,0}\eta(\underline{z},(z,s+t,y))\,d\nu_z^{D_b}(\underline{z})\\ & =\sum_{(z,s,x)\in G}\int_{X_\infty}\overline{\xi(z,s,x)}\eta(z,s+t,y)\,d\nu_z^{D_b}(\underline{z})\\ & =\sum_{(z,s,x)\in G}\overline{\xi(z,s,x)}\eta(z,s+t,y) =\langle \xi,\eta\rangle_{C_c(G)}(x,t,y). \end{align*} $$

The fact that $V_{\infty ,0}$ intertwines $S_{\mathfrak {u}}$ and $U_{\mathfrak {u}}$ is also an easy verification:

$$ \begin{align*} U_{\mathfrak{u}} V_{\infty,0}\xi (\underline{x},(p(\underline{x}),t,y)) & = \mathfrak{u}(p(\underline{x}))V_{\infty,0}\xi(\sigma_\infty(\underline{x}),(p(\sigma_\infty(\underline{x})),t-1,y))\\ & =\mathfrak{u}(p(\underline{x}))\xi(\sigma(p(\underline{x})),t-1,y)=V_{\infty,0}S_{\mathfrak{u}}(\underline{x},(p(\underline{x}),t,y)). \end{align*} $$

The last statement of the proposition follows from the following computation:

$$ \begin{align*} & \langle V_{\infty,0}\xi,\eta\rangle_{C_c(G)}(x,t,y) = \sum_{(z,s,x)\in G}\int_{X_\infty}\overline{V_{\infty,0}\xi(\underline{z},(z,s,x))}\eta(\underline{z},(z,s+t,y))\,d\eta_z^{D_b}(\underline{z})\\ & \quad =\sum_{(z,s,x)\in G}\overline{\xi(z,s,x)}\int_{X_\infty}\eta(\underline{z},(z,s+t,y))\,d\nu_z^{D_b}(\underline{z}) =\langle \xi,V_{\infty,0}^*\eta\rangle_{C_c(G)}(x,t,y).\\[-46pt] \end{align*} $$

Recall that $\Phi _{*}(f)$ is the image of $f\in C_c(G(X,\sigma ),\lambda )$ in $C_c(G_\infty (X,\sigma ),\nu ^D*\lambda )$ under the map $\Phi _*$ defined in (7-7). Define $\mathbb {E}_0:=\mathbb {L}(\Phi _{*}(1_{R_{0,0}}))$ , where $\mathbb {L}$ is the left action of $C^*(G_\infty (X,\sigma ),\nu ^D*\lambda )$ on $\mathcal {X}_\infty $ defined in (8-1). Then, $\mathbb {E}_0$ is a projection since $\Phi _{*}(1_{R_{0,0}})$ is a projection (see paragraph above (7-10)).

Definition 9.2. The sequence of projections $\{\mathbb {E}_j\}^{\infty }_{j= -\infty }$ on $\mathcal {L}(\mathcal {X}_{\infty })$ defined by the formula

$$ \begin{align*} \mathbb{E}_j:=U_{\mathfrak{u}}^j \mathbb{E} U_{\mathfrak{u}}^{-j}, \quad j\in \mathbb{Z}, \end{align*} $$

together with $U_{\mathfrak {u}}$ , is called the proto-multiresolution analysis determined by $(X,\sigma ,D,b)$ .

Proof. The proof is a consequence of the definition of the projections $\mathbb {E}_j$ , the fact that $1_{R_{0,0}}$ is the unit of $C^*(G(X,\sigma ),\lambda )$ , and the fact that $\mathbb {E}_j=\mathbb {L}(\Phi _{*}(1_{R_{j,j}}))$ for all $j\ge 0$ . We check this property for $j=1$ for simplicity:

$$ \begin{align*} & U_{\mathfrak{u}}\mathbb{E}_0U_{\mathfrak{u}}^{-1}\xi(\underline{x},(p(\underline{x},t,y))) =\mathfrak{u}(p(\underline{x}))\mathbb{E}_0U^{-1}\xi (\sigma_\infty(\underline{x}),(p(\sigma_\infty(\underline{x})),t-1,y))\\ & \quad=\mathfrak{u}(p(\underline{x}))\int_G\int_{X_\infty}\Phi_*(1_{R_{0,0}})(\sigma_\infty(\underline{x}),s,\underline{w})(U_{\mathfrak{u}}^{-1}\xi)(\underline{w},(w,t-s-1,y))\frac{\mathbb{d}(\underline{w})^{1/2}}{\mathbb{d}(\sigma_\infty(\underline{x}))^{1/2}}\\ & \quad\quad\cdots d\nu_w^D(\underline{w})\,d\lambda^{\sigma(x)}(\sigma(x),s,w)\\ & \quad=\mathfrak{u}(p(\underline{x}))\int_{X_\infty}\frac1{\mathfrak{u}(p(\sigma_\infty^{-1}(\underline{x})))}\xi(\sigma_{\infty}^{-1}(\underline{x}),(p(\sigma_\infty^{-1}(\underline{x})),t,y))\frac{\mathbb{d}(\underline{w})^{1/2}}{\mathbb{d}(\sigma_\infty(\underline{x}))^{1/2}}\,d\nu_{\sigma(x)}^D(\underline{w})\\ & \quad=\sum_{\sigma(u)=\sigma(x)}\int_{X_\infty}\xi(\underline{u},(p(\underline{u}),t,y))\frac{\mathbb{d}(\underline{u})^{1/2}}{\mathbb{d}(\underline{x})^{1/2}}\,d\nu_u^D(\underline{u})=\mathbb{L}(1_{R_{1,1}})\xi(\underline{x},(p(\underline{x}),t,y)). \end{align*} $$

For example, the last property follows from the fact that if $a\in C^*(G(X,\sigma ),\lambda )$ , then

$$ \begin{align*} \mathbb{L} (\Phi_{*}(a))=\mathbb{L} (\Phi_{*}(1_{R_{0,0}}*a*1_{R_{0,0}}))=\mathbb{E}_0\mathbb{L}(\Phi_{*}(a))\mathbb{E}_0.\\[-34pt] \end{align*} $$

Let $\hat {L}=(\mu ,X*\mathcal {H},\hat {L})$ be a representation of $G(X,\sigma )$ . As before, we write $\mathfrak {L}$ for the integrated representation of $C^*(G(X,\sigma ),\lambda )$ on $L^2(X*\mathcal {H},\mu )$ defined by $\hat {L}$ via (4-3). The normalized potential $D_b$ defines a topological correspondence from $G_{\infty }(X,\sigma )$ to $G(X,\sigma )$ (see Section 8). Theorem 4.1 implies that the induced $C^*(G_{\infty }(X,\sigma ),\nu ^{D}*\lambda )$ -representation of $\mathfrak {L}$ is unitarily equivalent to the integrated form $\mathfrak {L}_\infty $ of the induced $G_{\infty }(X,\sigma )$ representation $\hat {L}_\infty =(\mu _\infty ^{D_b},X_\infty *\mathcal {H}_\infty ,\hat {L}_\infty )$ defined in Section 8. Set $\tilde {S}_{\mathfrak {u}}:=\mathfrak {L}(S_{\mathfrak {u}})\in \mathcal {B}(L^2(X*\mathcal {H},\mu ))$ , $\tilde {U}:=\mathfrak {L}_\infty (U_{\mathfrak {u}})\in \mathcal {B}(L^2(X_\infty *\mathcal {H}_\infty ,\mu _\infty ^{D_b}))$ , and $V_j:=\mathfrak {L}_\infty (\mathbb {E}_j)L^2(X_\infty *\mathcal {H}_\infty ,\mu _\infty ^{D_b})$ for all $j\in \mathbb {Z}$ . Theorem 9.3 implies the following result.

Part III Mallat’s theorem redux

10 Potentials, filters, and scaling functions

We fix a transfer operator $\mathcal {L}_{\psi }$ for $\sigma $ as in the paragraph following Definition 5.2 and we assume the potential $\psi $ is full. In this section, we define and develop the concept of a filter associated to the potential $\psi $ . A filter is a function that determines a new potential that is normalized but not necessarily full. In fact, to produce an analog of the scaling function that appears in (1-1), the new potential is definitely not full (see the paragraph before Section 11). A filter defines a p-system of measures on the projective limit $X_\infty $ , where, recall, $p:X_\infty \to X$ is the projection onto the first component. The main result of this section is Proposition 10.1, which shows that a scaling function for $(X,\sigma )$ exists if and only if the system of measures is atomic.

The full potential $\psi $ is used to define the Haar system on the pullback groupoid $X_\infty *G*X_\infty $ via (3-1). To keep the notation shorter, we assume in the remainder of the paper that $\psi $ is an unital potential in addition to being a full potential. If $\psi $ is not unital, then one needs to replace $\mathcal {L}_\psi $ with $\mathcal {L}_{D_0}$ , where $D_0(x)=\psi (x)/\mathcal {L}_\psi (1)(\sigma (x))$ in the formula below and in the construction of the p-system of measures on $X_\infty $ (see Remark 5.8).

A function $\boldsymbol {m}\in C(X)$ is called a $\psi $ -filter if $\mathcal {L}_\psi (\vert \boldsymbol {m}\vert ^2)=1$ . That is,

(10-1) $$ \begin{align} \sum_{\sigma(y)=x}\psi(y)\vert \boldsymbol{m}(y)\vert^2=1 \end{align} $$

for all $x\in X$ .

If $\boldsymbol {m}$ is a $\psi $ -filter, we write $\mathfrak {u}$ for the continuous function $\mathfrak {u}:X\to \mathbb {C}$ defined via $\mathfrak {u}(x)=\sqrt {\psi (x)}\boldsymbol {m}(x)$ and we define $D:X\to \mathbb {R}_+$ via

$$ \begin{align*} D(x)=\vert \mathfrak{u}(x)\vert^2=\psi(x)\vert \boldsymbol{m}(x)\vert^2 \quad \text{for all }x\in X. \end{align*} $$