2.1 Étale groupoids

For a topological groupoid G, we denote the space of units by $G^{(0)}$ , the range and the source maps by r and s, respectively, and the set of composable pairs, $\{(\gamma ,\eta )\colon G\times G : s(\gamma )=r(\eta )\}$ , by $G^{(2)}$ . For $\gamma \in G$ , $\gamma ^{-1}$ denotes its inverse.

A topological groupoid G is called étale if its space of units ${G}^{(0)}$ is locally compact Hausdorff, and the range map (equivalently, the source map) $G\to {G}^{(0)}$ is a local homeomorphism when ${G}^{(0)}$ is given the subspace topology from G. The groupoid may not be Hausdorff as in [Reference Exel7]. The space of units ${G}^{(0)}$ is an open subspace of G [Reference Exel7, Proposition 3.2]. The space of units is closed in G if and only if G is Hausdorff [Reference Sims, Szabó, Williams and Perera16, Lemma 8.3.2]. Since the groupoid is locally homeomorphic to its space of units, the groupoid is locally compact and locally Hausdorff.

Fix a locally compact étale groupoid G. An open subset $U\subseteq G$ is called a bisection if the restrictions of the range and source maps to U are homeomorphisms onto open subsets of ${G}^{(0)}$ . Every bisection is a locally compact Hausdorff subspace of G. The open bisections constitute a basis, denoted by $\mathcal {B}_G$ , for the topology of G.

The vector space $\mathit {C_c}(G)$ is defined as the linear span of the complex valued functions f on G which vanish outside an open Hausdorff subspace $V\subseteq G$ and such that $f|_V \in \mathit {C_c}(V)$ . Note that an open Hausdorff subspace of G is locally compact [Reference Exel7, Proposition 3.7]. Thus, the functions in $\mathit {C_c}(G)$ need not be continuous. As $\mathcal {B}_G$ is a basis for the topology of G, $\mathit {C_c}(G)$ is spanned by the subspaces $\mathit {C_c}(U)$ , where $U\subseteq G$ is a bisection [Reference Exel7, Proposition 3.10]. Further, $\mathit {C_c}(G)$ is an involutive algebra equipped with the I-norm [Reference Exel7].

For two bisections U and V in $\mathcal {B}_G$ , define the sets

$$ \begin{align*} U^{-1} = \{u^{-1} : u\in U\} \quad \text{and} \quad UV = \{ uv : u\in U, v\in V , (u,v) \in G^{(2)} \}. \end{align*} $$

Then, $U^{-1}, UV\in \mathcal {B}_G$ ; here, $VU$ could be empty. In particular, it can be readily seen that the bisections $UU^{-1} = r(U)$ and $U^{-1} U = s(U)$ are subsets of ${G}^{(0)}$ . In fact, $\mathcal {B}_G$ is an inverse semigroup with the inverses and products of subsets [Reference Paterson13, Proposition 2.2.4].

2.2 Hilbert modules

Let A be a $\mathit {C}^*$ -algebra. By a (right) Hilbert module over A, we mean a Banach space $\mathcal H$ that is a right A-module equipped with an A-valued positive definite sesquilinear form on $\mathcal H$ that is linear in the second variable and antilinear in the first one; this form is called an A-valued inner product on $\mathcal H$ . It is assumed that the Banach space norm on $\mathcal H$ is given by $a\mapsto \lvert \lvert \langle a, a\rangle \rvert \rvert ^{1/2}$ , where $a\in \mathcal H$ . A left Hilbert A-module is defined similarly.

The Hilbert module $\mathcal H$ is called full if $\operatorname {span}\{\langle a, b\rangle : a,b \in \mathcal H\}$ is dense in A.

2.3 Bundles of $\mathit {C}^*$ -algebras

An upper semicontinuous bundle of Banach spaces is defined as in [Reference Fell and Doran9, Definition I.13.4] with the obvious modification of the continuity condition. All bundles of Banach spaces in this article are upper semicontinuous. In a bundle of Banach spaces $p\colon \mathcal {A}\to X$ , we call $\mathcal {A}$ the total space, X the base space and p the bundle map. We write p or $\mathcal {A}$ instead of $p\colon \mathcal {A}\to X$ whenever there is no ambiguity.

An upper semicontinuous Banach bundle $p\colon \mathcal {A}\to X $ is called an upper semicontinuous $\mathit {C}^*$ -bundle if each of the fibres is a $\mathit {C}^*$ -algebra; the fibrewise multiplication is a continuous mapping $\mathcal {A}^{(2)} \to \mathcal {A} $ ; and the fibrewise involution, $a \mapsto a^* $ from $\mathcal {A}\to \mathcal {A}$ , is continuous. Here,

$$ \begin{align*} \mathcal{A}^{(2)} := \{(a,a')\in \mathcal{A}\times \mathcal{A} : p(a) = p(a')\}. \end{align*} $$

Bundles of $\mathit {C}^*$ -algebras in this article are considered over locally compact Hausdorff spaces. Given a subspace $Y\subseteq X$ , $p\colon \mathcal {A}|_Y \to Y$ denotes the restricted bundle; note that the bundle map is still denoted by p.

A bundle $p\colon \mathcal {A} \to X$ is said to have enough sections if for all $x \in X$ and $a\in \mathcal {A}_x$ , there is a continuous section f with $f(x)=a$ . For a locally compact and Hausdorff base space X, the bundle $p\colon \mathcal {A}\to X$ of Banach spaces has enough sections [Reference Hofmann, Morel and Singer10].

For a bundle of $\mathit {C}^*$ -algebras $p\colon \mathcal {A}\to X$ , $\mathit {C}_0(X;\mathcal {A})$ denotes the $\mathit {C}^*$ -algebra of sections vanishing at infinity and $\mathit {C_c}(X; \mathcal {A})$ denotes the dense two-sided ideal of $\mathit {C}_0(X;\mathcal {A})$ consisting of continuous sections with compact supports. If $K\subseteq X$ is compact, $\mathit C(K; \mathcal {A}|_K)$ is the $\mathit {C}^*$ -algebra of continuous sections of $\mathcal {A}|_K \to K$ . Thus, $\mathit {C_c}(X;\mathcal {A})$ is the increasing union, and hence the inductive limit as a topological vector space, of the Banach spaces $\mathit C(K;\mathcal {A}|_K)$ , where $K\subseteq X$ is compact.

Since X is locally compact, the inductive limit topology is also generated by $\mathit {C_c}(U;\mathcal {A}|_U) \subseteq \mathit {C_c}(X;\mathcal {A})$ , where $U\subseteq X$ is open and $\mathit {C_c}(U;\mathcal {A}|_U)$ is the vector subspace consisting of sections of $\mathcal {A}\to X$ which vanish outside U.

A bundle of $\mathit {C}^*$ -algebras $p\colon \mathcal {A} \to X$ , over a locally compact Hausdorff and second countable space X, is called separable if each fibre in p is separable. In this case, the section algebra $\mathit {C}_0(X;\mathcal {A})$ is separable.

2.4 Fell bundles

We adopt the standard definition of Fell bundles in [Reference Muhly and Williams12] although our groupoids are locally compact and not necessarily Hausdorff. This can be justified by the standard construction of Fell bundles from partial dynamical systems [Reference Exel8]. The Fell bundles under consideration are not necessarily saturated.

For a Fell bundle $p\colon \mathcal {A}\to G$ as above, define

$$ \begin{align*}\mathcal{A}^{(2)} = \{(a,b)\in \mathcal{A} \times \mathcal{A} : (s(p(a)), r(p(b))) \in G^{(2)}\}. \end{align*} $$

Given $(a,b) \in \mathcal {A}^{(2)}$ , we use the shorthand $ab$ instead of $m(a, b)$ . For composable arrows $\gamma _1,\gamma _2\in G$ , define the subset

$$ \begin{align*} \mathcal{A}_{\gamma_1} \cdot \mathcal{A}_{\gamma_2} = \operatorname{span} \{ab: a\in \mathcal{A}_{\gamma_1}, b\in \mathcal{A}_{\gamma_2}\} \subseteq \mathcal{A}_{\gamma_1\gamma_2}. \end{align*} $$

The Fell bundle is called saturated if $\mathcal {A}_{\gamma _1} \cdot \mathcal {A}_{\gamma _2}$ is dense in $\mathcal {A}_{\gamma _1 \gamma _2}$ for all $(\gamma _1,\gamma _2) \in G^{(2)}$ .

For $x\in G^{(0)}$ , conditions (2)–(9) in Definition 2.4 make the fibre $\mathcal {A}_x$ a $\mathit {C}^*$ -algebra with the obvious norm, multiplication and involution. Furthermore, if $\gamma \in G$ and $a,b\in \mathcal {A}_\gamma $ , then $\mathcal {A}_\gamma $ is a right Hilbert $\mathcal {A}_{s(\gamma )}$ -module when equipped with the inner product $\langle a, b\rangle _{\mathcal {A}_{s(\gamma )}}:= a^*b$ ; and $\mathcal {A}_\gamma $ is a left Hilbert $\mathcal {A}_{r(\gamma )}$ -module if the inner product is defined by ${}_{\mathcal {A}_{r(\gamma )}}\langle a, b\rangle := ab^*$ . Thus, the Fell bundle $p\colon \mathcal {A} \to G$ is saturated if and only if $\mathcal {A}_\gamma $ is an $\mathcal {A}_{r(\gamma )}$ – $\mathcal {A}_{s(\gamma )}$ -imprimitivity bimodule for each $\gamma \in G$ .

We define $\mathit {C_c}(G; \mathcal {A})$ , the space of compactly supported sections, analogous to $\mathit {C_c}(G)$ as in [Reference Connes and Kadison4]. Thus, the elements of $\mathit {C_c}(G;\mathcal {A})$ are not necessarily continuous. With this $\mathit {C_c}(G;\mathcal {A})$ , the bundle p has enough sections. Let $\gamma \in G$ and $\xi \in \mathcal {A}_\gamma $ . For a locally compact Hausdorff neighbourhood X of $\gamma $ , consider the restricted bundle $\mathcal {A}|_X \to X$ . Then, this bundle has a section f, which is continuous on X and has compact support in X and $f(\gamma ) = \xi $ . Moreover, $f\in \mathit {C_c}(G; \mathcal {A})$ .

A Hilbert module is countably generated if and only if the $\mathit {C}^*$ -algebra of the compact operators on it is $\sigma $ -unital. Therefore, each fibre of the Fell bundle $\mathcal {A}\to G$ is separable if and only if the $\mathit {C}^*$ -algebra $\mathcal {A}_x$ is separable for $x\in {G}^{(0)}$ .

2.5 The convolution algebra of a Fell bundle

From now on, we fix a locally compact étale groupoid G and a Fell bundle $p\colon \mathcal {A}\to G$ ; the Fell bundle need not be saturated or separable. Make $\mathit {C_c}(G;\mathcal {A})$ into a $^*$ -algebra using the following convolution and involution:

$$ \begin{align*} f*g(\gamma)= \sum_{\alpha\beta=\gamma}f(\alpha)g(\beta) \quad \text{and} \quad f^*(\gamma)=f(\gamma^{-1})^*, \end{align*} $$

where $f,g\in \mathit {C_c}(G;\mathcal {A})$ . The convolution is continuous in the inductive limit topology on $\mathit {C_c}(G;\mathcal {A})$ . Since G is étale, ${G}^{(0)}$ is an open Hausdorff subset of G. Therefore, $\mathit {C_c}({G}^{(0)};\mathcal {A}|_{{G}^{(0)}})$ is a $^*$ -subalgebra of $\mathit {C_c}(G;\mathcal {A})$ .

Note that if f is a continuous section vanishing outside ${G}^{(0)}$ and $g\in \mathit {C_c}(G;\mathcal {A})$ , then

(2.1) $$ \begin{align} f*g(\gamma) = f(r(\gamma)) g(\gamma) \quad\text{and}\quad g*f(\gamma) = g(\gamma) f(s(\gamma)) \quad\text{for } \gamma\in G. \end{align} $$

Define the I-norm for $f\in \mathit {C_c}(G;\mathcal {A})$ by $\lvert \lvert f\rvert \rvert _I = \max \{\lvert \lvert f\rvert \rvert _{I,r}, \lvert \lvert f\rvert \rvert _{I,s}\}$ , where

$$ \begin{align*} \lvert\lvert f\rvert\rvert_{I,r}= \sup_{x\in G^{(0)}}\sum_{\gamma\in r^{-1}(x)} \lvert\lvert f(\gamma)\rvert\rvert \quad \text{and}\quad \lvert\lvert f\rvert\rvert_{I,s}= \sup_{x\in G^{(0)}}\sum_{\gamma\in s^{-1} (x)} \lvert\lvert f(\gamma)\rvert\rvert. \end{align*} $$

The following lemma is an analogue of [Reference Exel7, Proposition 3.11] for Fell bundles.

Proof. The proofs of claims (1)–(3) closely follow that of [Reference Exel7, Proposition 3.11], and hence we skip them. In fact, in claim (3), one can see that for $x\in r(U)$ ,

(2.2) $$ \begin{align} f*f^*(x) = \begin{cases} f(\gamma)f(\gamma)^* & \text{if } \gamma\in U\cap r^{-1}(x),\\ 0 & \text{if } x\notin U. \end{cases} \end{align} $$

A similar formula holds for $f^**f$ . Claim (4) follows from the definition of the I-norm. Claim (5) is a consequence of (2.2) and claim (4).

Proof. (1) First, consider a bisection U and let $f\in \mathit {C_c}(U; \mathcal {A}|_{U})$ . Note that ${\text {supp}}(u_{\lambda }*f)\subseteq {\text {supp}}(f)$ . Apply Lemma 2.6(4) for the bisection supported section $u_{\lambda }*f -f$ to get

$$ \begin{align*} \lvert\lvert u_{\lambda}*f-f\rvert\rvert^2_\infty & = \lvert\lvert (u_{\lambda}*f-f)*(u_{\lambda}*f-f)^*\rvert\rvert_\infty = \lvert\lvert (u_{\lambda}*f-f)*(f^**u_{\lambda}-f^*)\rvert\rvert_\infty \\ & = \lvert\lvert u_{\lambda}*(f*f^*)*u_{\lambda} - u_{\lambda}*(f*f^*) - (f*f^*)*u_{\lambda} - f* f^*\rvert\rvert_\infty. \end{align*} $$

As $f*f^*\in \mathit {C_c}({G}^{(0)};\mathcal {A}|_{{G}^{(0)}})$ and $(u_{\lambda })_{\lambda \in \Lambda }$ is an approximate unit of $\mathit {C}_0({G}^{(0)};\mathcal {A}|_{{G}^{(0)}})$ , the limit of the last term is $0$ .

In general, consider $\sum _{i=1}^n\mathit {C_c}(U_i;\mathcal {A}|_{U_i})\subseteq \mathit {C_c}(G;\mathcal {A})$ , where $U_i$ is a bisection and $n\in \mathbb {N}$ ; let $f\in \sum _{i=1}^n\mathit {C_c}(U_i;\mathcal {A}|_{U_i})$ . Write $f =\sum _{i=1}^n f_i$ , where $f_i$ is supported in a bisection as in Lemma 2.7. Now, $\lvert \lvert u_\lambda *f-f\rvert \rvert _\infty \leq \sum _{i=1}^n \lvert \lvert u_\lambda *f_i-f_i\rvert \rvert _\infty $ and the claim follows from the last argument.

(2) The claim follows from the first claim as $u_{\lambda }*f \in \mathit {C_c}({G}^{(0)};\mathcal {A}|_{{G}^{(0)}})*\mathit {C_c}(G;\mathcal {A})$ .

3.1 Pre-representations of $\mathit {C_c}(G;\mathcal {A})$

A $^*$ -representation of $\mathit {C_c}(G;\mathcal {A})$ that is continuous in the inductive limit topology is obviously a pre-representation. Conversely, fix a pre-representation $L\colon \mathit {C_c}(G;\mathcal {A}) \to \mathrm {Lin}({\mathcal H}_{0})$ of $\mathit {C_c}(G;\mathcal {A})$ on a Hilbert space $\mathcal H$ . We shall extend L to a $^*$ -representation of $\mathit {C_c}(G;\mathcal {A})$ on $\mathcal H$ that is continuous in the inductive limit topology in Proposition 3.4. This will establish equivalence of the four notions of representations HS(1–4).

Let $1_{{G}^{(0)}} \in \mathcal M(\mathit {C}_0({G}^{(0)};\mathcal {A}|_{{G}^{(0)}}))$ denote the identity in the multiplier algebra. We fix an approximate unit $(u_{\lambda })_{\lambda \in \Lambda }$ of $\mathit {C}_0({G}^{(0)};\mathcal {A}|_{{G}^{(0)}})$ consisting of compactly supported sections.

Observation 3.2. Let $\xi ,\eta \in {\mathcal H}_{0}$ , and $f,g,k, h\in \mathit {C_c}(G;\mathcal {A})$ , then

(3.2) $$ \begin{align} \lim_{\lambda} \langle L(u_{\lambda}*f)\xi, L(u_{\lambda}*k)\eta\rangle &= \langle L(f)\xi, L(k)\eta\rangle, \end{align} $$

(3.3) $$ \begin{align} \lim_{\lambda} \langle L(f*u_{\lambda})\xi, L(k*u_{\lambda})\eta\rangle &= \langle L(f)\xi, L(k)\eta\rangle, \end{align} $$

(3.4) $$ \begin{align} \lim_{\lambda} \langle L(f*u_{\lambda}*g)\xi, L(k*u_{\lambda}*h)\eta\rangle &= \langle L(f*g)\xi, L(k*h)\eta\rangle. \end{align} $$

For the first equation, the left-hand side equals $\lim _{\lambda } \langle L((u_{\lambda }*k)^**(u_{\lambda }*f))\xi , \eta \rangle $ due to the adjointability of a pre-representation. Lemma 2.8(1) and the continuity of a pre-representation implies that $\lim _{\lambda } \langle L((u_{\lambda }*k)^**(u_{\lambda }*f))\xi , \eta \rangle = \langle L( k^* *f)\xi , \eta \rangle = \langle L(f)\xi , L(k)\eta \rangle $ . Equation (3.3) is justified similarly. Equation (3.4) follows from (3.3) since $ \langle L(f*u_{\lambda }*g)\xi , L(k*u_{\lambda }*h)\eta \rangle = \langle L(f*u_{\lambda }) (L(g)\xi ), L(k*u_{\lambda }) (L(h)\eta )\rangle $ .

Lemma 3.3. For a section $h\in \mathit {C_c}(U;\mathcal {A}|_{U})$ , where U is a bisection, define

$$ \begin{align*} q := 1_{{G}^{(0)}} \cdot \lvert\lvert h \rvert\rvert^2_\infty - (h^** h) \quad \text {and } \quad q_{\lambda} = u_{\lambda} q u_{\lambda} \end{align*} $$

so that $q\in \mathcal M(\mathit {C}_0({G}^{(0)}; \mathcal {A}|_{{G}^{(0)}}))$ and $q_{\lambda }\in \mathit {C}_0({G}^{(0)}; \mathcal {A}|_{{G}^{(0)}})$ . Then:

1. (1) q is a positive element in $\mathcal M(\mathit {C}_0({G}^{(0)}; \mathcal {A}|_{{G}^{(0)}}))$ ;

2. (2) $q_{\lambda }$ is a positive element in $\mathit {C}_0({G}^{(0)}; \mathcal {A}|_{{G}^{(0)}})$ for $\lambda \in \Lambda $ ;

3. (3) for a fixed $v\in {\mathcal H}_{00}$ ,

   $$ \begin{align*} \lim_{\lambda} (\lvert\lvert L(\lvert\lvert h\rvert\rvert_\infty u_{\lambda})v\rvert\rvert^2 - \lvert\lvert L(h*u_{\lambda})v\rvert\rvert^2) = (\lvert\lvert h\rvert\rvert_\infty \lvert\lvert v\rvert\rvert)^2 - \lvert\lvert L(h)v\rvert\rvert^2. \end{align*} $$

Proof. (1) By Lemma 2.6(3)–(5), $h^**h\in \mathit {C_c}({G}^{(0)}; \mathcal {A}|_{{G}^{(0)}})$ with $ \lvert \lvert h^* * h\rvert \rvert _\infty = \lvert \lvert h\rvert \rvert _\infty ^2 $ . Note that $q = 1_{{G}^{(0)}} \cdot \lvert \lvert h^**h\rvert \rvert _\infty -(h^**h) $ is a positive element in $\mathcal M(\mathit {C}_0({G}^{(0)}; \mathcal {A}|_{{G}^{(0)}}))$ .

(2) Let k be the positive square root of q in the multiplier algebra. Then, $u_{\lambda }qu_{\lambda } = (k u_{\lambda })^*(ku_{\lambda })$ .

(3) For $v\in {\mathcal H}_{00}$ , compute the difference

$$ \begin{align*} \lvert\lvert L(\lvert\lvert h\rvert\rvert_\infty u_{\lambda})v\rvert\rvert^2 - \lvert\lvert L(h*u_{\lambda})v\rvert\rvert^2 = \langle L(\lvert\lvert h\rvert\rvert_\infty u_{\lambda})v \mathbin{,} L(\lvert\lvert h\rvert\rvert_\infty u_{\lambda})v \rangle - \langle L(h*u_{\lambda})v, L(h*u_{\lambda})v\rangle. \end{align*} $$

As $v\in {\mathcal H}_{00}$ , we can write $v=\sum _{i=1}^k L(f_i)\xi _i$ for $f_i\in \mathit {C_c}(G;\mathcal {A})$ , $\xi _i\in {\mathcal H}_0$ and $k\in \mathbb {N}$ . Substitute this value of v in the last difference and compute

$$ \begin{align*} &\biggl\langle L(\lvert\lvert h\rvert\rvert_\infty u_{\lambda}) \sum_{i=1}^k L(f_i)\xi_i, L(\lvert\lvert h\rvert\rvert_\infty u_{\lambda}) \sum_{i=1}^k L(f_i)\xi_i\biggr\rangle \\ &\qquad - \bigg\langle L(h*u_{\lambda}) \sum_{i=1}^k L(f_i)\xi_i , L(h*u_{\lambda})\sum_{i=1}^k L(f_i)\xi_i\bigg\rangle\\ & \quad = \sum_{i,j} (\lvert\lvert h\rvert\rvert_\infty^2\langle L(u_{\lambda} * f_i)\xi_i, L(u_{\lambda}*f_j)\xi_j\rangle - \langle L(h*u_{\lambda}* f_i)\xi_i, L(h*u_{\lambda} * f_j)\xi_j\rangle). \end{align*} $$

From (3.2) and (3.4), the limit of the preceding difference over $\Lambda $ is

$$ \begin{align*} &\sum_{i,j} (\lvert\lvert h\rvert\rvert_\infty^2 \langle L(f_i)\xi_i, {L(f_j)\xi_j}\rangle - \langle L(h*f_i)\xi_i, L(h*f_j)\xi_j\rangle)\\& \quad= \lvert\lvert h\rvert\rvert_\infty^2 \bigg\langle\sum_i^kL(f_i)\xi_i, {\sum_i^kL(f_i)\xi_i}\bigg\rangle - \bigg\langle L(h) \sum_i^kL(f_i)\xi_i, L(h) \sum_i^kL(f_i)\xi_i\bigg\rangle, \end{align*} $$

which proves the claim.

Proof. (1) We prove this in two steps.

Step I. Assume that ${\text {supp}}(h)$ lies inside an open bisection. We intend to prove that

(3.5) $$ \begin{align} \lvert\lvert M'(h)v\rvert\rvert\leq \lvert\lvert h\rvert\rvert_\infty \lvert\lvert v\rvert\rvert \quad \text{for } v \in {\mathcal H}_{00}. \end{align} $$

If the last inequality is proved, then, due to the density of ${\mathcal H}_{00}\subseteq \mathcal H$ , $M'(h)$ can be extended to a bounded operator $M(h)\colon \mathcal H \to \mathcal H$ .

To prove (3.5), consider an approximate unit $(u_{\lambda })_{\lambda \in \Lambda }$ of $\mathit {C}_0({G}^{(0)};\mathcal {A}|_{{G}^{(0)}})$ , where $u_{\lambda }\in \mathit {C_c}({G}^{(0)};\mathcal {A}|_{{G}^{(0)}})$ (see Lemma 2.2). Let $v\in {\mathcal H}_{00}$ , so that $v = \sum _{i=1}^k L(f_i)\xi _i$ for some $k\in \mathbb {N}$ , $f_i\in \mathit {C_c}(G;\mathcal {A})$ and $\xi _i\in {\mathcal H}_0$ . By Lemma 3.3(3),

$$ \begin{align*} (\lvert\lvert h\rvert\rvert_\infty \lvert\lvert v\rvert\rvert)^2 - \lvert\lvert M'(h)v\rvert\rvert^2 = \lim_{\lambda} (\lvert\lvert h\rvert\rvert_\infty \lvert\lvert L(u_{\lambda})v\rvert\rvert)^2 - \lvert\lvert L(h*u_{\lambda})v\rvert\rvert^2. \end{align*} $$

Therefore, to prove (3.5), it suffices to show that $(\lvert \lvert h\rvert \rvert _\infty \lvert \lvert L(u_{\lambda })v\rvert \rvert )^2 - \lvert \lvert L(h*u_{\lambda })v\rvert \rvert ^2 \geq 0$ for all $\lambda \in \Lambda $ . For $\lambda \in \Lambda $ , this difference is

$$ \begin{align*} (\lvert\lvert h\rvert\rvert_\infty \lvert\lvert L(u_{\lambda})v\rvert\rvert)^2 - \lvert\lvert L(h*u_{\lambda})v\rvert\rvert^2 &= \lvert\lvert h\rvert\rvert^2_\infty \bigg\lvert\bigg\lvert \sum_{i=1}^k L(u_{\lambda}*f_i)\xi_i\bigg\rvert\bigg\rvert^2 - \bigg\lvert\bigg\lvert \sum_{i=1}^k L(h*u_{\lambda} * f_i)\xi_i\bigg\rvert\bigg\rvert^2\\ & = \bigg\langle\lvert\lvert h\rvert\rvert^2_\infty \bigg(\sum_{i=1}^k L(u_{\lambda}*f_i)\xi_i\bigg), \sum_{i=1}^k L(u_{\lambda} *f_i)\xi_i\bigg\rangle\\ &\quad - \bigg\langle \sum_{i=1}^k L(h * u_{\lambda}* f_i)\xi_i, \sum_{i=1}^k L(h * u_{\lambda}* f_i)\xi_i\bigg\rangle. \end{align*} $$

By the adjointability of the pre-representation, the last term becomes

(3.6) $$ \begin{align} \sum_{i, j=1}^k \langle ( L(\lvert\lvert h\rvert\rvert^2_\infty u_{\lambda}*u_{\lambda}* f_i))\xi_i, {L(f_j)\xi_j} \rangle - \sum_{i, j=1}^k \langle L(u_{\lambda}*h^* * h * u_{\lambda} * f_i)\xi_i, L(f_j)\xi_j \rangle. \end{align} $$

Since $h^**h \in \mathit {C_c}({G}^{(0)};\mathcal {A}|_{{G}^{(0)}})$ , we have $u_{\lambda }*(h^**h) *u_{\lambda } = u_{\lambda }\cdot (h^**h) \cdot u_{\lambda }\in \mathit {C_c}({G}^{(0)}; \mathcal {A}|_{{G}^{(0)}})$ . Moreover, (2.1) gives the convolutions

$$ \begin{align*} ( u_{\lambda}*(h^**h)*u_{\lambda}) * f_i &= (u_{\lambda}\cdot (h^**h) \cdot u_{\lambda})\circ r \cdot f_i = (u_{\lambda}\circ r \cdot (h^**h)\circ r \cdot u_{\lambda}\circ r) \cdot f_i, \\ u_{\lambda}^2 * f_i &= (u_{\lambda}^2 \circ r) \cdot f_i = (u_{\lambda} \circ r)^2 \cdot f_i. \end{align*} $$

Therefore, the difference (3.6) can be written as

$$ \begin{align*} & \sum_{i, j=1}^k \langle L( \lvert\lvert h\rvert\rvert^2_\infty \cdot (u_{\lambda}\circ r)^2\cdot f_i)\xi_i, L(f_j)\xi_j\rangle \\&\qquad - \sum_{i, j=1}^k \langle L( (u_{\lambda}\circ r)\cdot (h^**h)\circ r \cdot(u_{\lambda}\circ r) \cdot f_i)\xi_i, L(f_j)\xi_j\rangle \\&\quad = \sum_{i, j=1}^k \langle L((\lvert\lvert h\rvert\rvert^2_\infty \cdot (u_{\lambda}\circ r)^2 - (u_{\lambda}\circ r)\cdot (h^**h)\circ r \cdot(u_{\lambda}\circ r) )\cdot f_i)\xi_i, L(f_j)\xi_j\rangle \\&\quad = \sum_{i, j=1}^k \langle L((u_{\lambda}\circ r (\lvert\lvert h\rvert\rvert^2_\infty\cdot 1_{{G}^{(0)}} - (h^**h) \circ r ) u_{\lambda}\circ r) \cdot f_i)\xi_i, L(f_j)\xi_j\rangle, \end{align*} $$

where $1_{{G}^{(0)}}\in \mathcal M(\mathit {C}_0({G}^{(0)}); \mathcal {A}|_{{G}^{(0)}})$ is the unit in the multiplier algebra. Here, we substitute $q =\lvert \lvert h\rvert \rvert ^2_\infty \cdot 1_{{G}^{(0)}} - (h^**h) \in \mathcal M(\mathit {C}_0({G}^{(0)}); \mathcal {A}|_{{G}^{(0)}})$ and $q_{\lambda } = u_{\lambda } q u_{\lambda }\in \mathit {C_c}({G}^{(0)};\mathcal {A}|_{{G}^{(0)}})$ . Then, the last term in the above computation becomes

$$ \begin{align*} \sum_{i, j=1}^k \langle L((q_{\lambda} \circ r) \cdot f_i)\xi_i, L(f_j)\xi_j\rangle. \end{align*} $$

By Lemma 3.3, $q_{\lambda }$ is a positive element in $\mathit {C_c}({G}^{(0)};\mathcal {A}|_{{G}^{(0)}}) \subseteq \mathit {C}_0({G}^{(0)};\mathcal {A}|_{{G}^{(0)}})$ . Let $s_{\lambda }\in \mathit {C}_0({G}^{(0)};\mathcal {A}|_{{G}^{(0)}})$ be its unique positive square root; then $s_{\lambda }\in \mathit {C_c}({G}^{(0)};\mathcal {A}|_{{G}^{(0)}})$ . By substituting $q_{\lambda }=s_{\lambda }^2$ , the last term of the main computation becomes

$$ \begin{align*} \sum_{i, j=1}^k \langle L((s_{\lambda}\circ r) \cdot f_i)\xi_i, L((s_{\lambda}\circ r) \cdot f_j)\xi_j\rangle & = \sum_{i, j=1}^k \langle L(s_{\lambda}* f_i)\xi_i, L(s_{\lambda} * f_j)\xi_j\rangle\\ & = \bigg\langle\sum_{i=1}^nL(s_{\lambda}* f_i)\xi_i, \sum_{i=1}^nL(s_{\lambda} * f_i)\xi_i\bigg\rangle \geq 0. \end{align*} $$

Thus, (3.5) is proved.

Step II. For general $h\in \mathit {C_c}(G, \mathcal {A})$ , write $h= \sum _{i=1}^p h_i$ , where each $h_i$ is supported in a bisection. Now, (3.5) implies that $\lvert \lvert M'(h)v\rvert \rvert \leq (\sum _{i=1}^p \lvert \lvert h_i\rvert \rvert _\infty ) \lvert \lvert v\rvert \rvert \leq p \lvert \lvert h\rvert \rvert _\infty \lvert \lvert v\rvert \rvert $ for $v\in {\mathcal H}_{00}$ ; here, the number p depends on a representation of h as a sum of the $h_i$ . However, for the fixed representation, p is fixed. Now we can extend $M'(h)$ to a bounded operator $M(h)\colon \mathcal H \to \mathcal H$ using the density of ${\mathcal H}_0\subseteq \mathcal H$ . Finally, note that $\lvert \lvert M(h)\rvert \rvert \leq p\lvert \lvert h\rvert \rvert _\infty $ . Therefore, $M\colon \mathit {C_c}(G;\mathcal {A}) \to \mathbb B(\mathcal H)$ is continuous in the inductive limit topology.

(3) By the construction of $M(h)$ in part (1), it is clear that $M(h)|_{{\mathcal H}_{00}} = L(h)|_{{\mathcal H}_{00}}$ . Since ${\mathcal H}_{00}\subseteq {\mathcal H}_0$ is dense, $M(h)|_{{\mathcal H}_0} = L(h)$ .

(2) Let $h\in \mathit {C_c}(G;\mathcal {A})$ . From part (3) and the adjointability of a pre-representation, $M(h^*) = M(h)^*$ . Finally, the nondegeneracy of M follows from the nondegeneracy of a pre-representation and the fact that the linear subspace of $\mathit {C_c}(G;\mathcal {A})$ spanned by $\mathit {C_c}(G;\mathcal {A})*\mathit {C_c}(G;\mathcal {A})$ is uniformly dense (see Lemma 2.8(2)).

For the Fell bundle $p\colon \mathcal {A} \to G$ , define the collections:

1. (1) $\mathcal {R}_R$ , the collection of representations;

2. (2) $\mathcal {R}_{H}$ , the collection of I-norm bounded representations;

3. (3) $\mathcal {R}_{\mathrm {In}}$ the collection of representations continuous in the inductive limit topology;

4. (4) $\mathcal {R}_P$ , the collection of all representations induced from pre-representations as in Proposition 3.4.

With this notation, we can state the following theorem.

Theorem 3.5. Let $p\colon \mathcal {A} \to G$ be a (not necessarily saturated) Fell bundle over an étale groupoid G (not necessarily Hausdorff). For $f\in \mathit {C_c}(G;\mathcal {A})$ , define

$$ \begin{align*} \lvert\lvert f\rvert\rvert_R = \sup_{\varrho\in \mathcal{R}_R}\{\lvert\lvert \varrho(f)\rvert\rvert \}, \ \lvert\lvert f\rvert\rvert_{H} = \sup_{\varrho\in \mathcal{R}_{H}}\{\lvert\lvert \varrho(f)\rvert\rvert \}, \ \lvert\lvert f\rvert\rvert_{\mathrm{In}} = \sup_{\varrho\in \mathcal{R}_{\mathrm{In}}}\{\lvert\lvert \varrho(f)\rvert\rvert \}, \ \lvert\lvert f\rvert\rvert_P = \sup_{\varrho\in \mathcal{R}_P}\{\lvert\lvert \varrho(f)\rvert\rvert \}. \end{align*} $$

Then, $\lvert \lvert f\rvert \rvert _R = \lvert \lvert f\rvert \rvert _{H} = \lvert \lvert f\rvert \rvert _{\mathrm {In}}= \lvert \lvert f\rvert \rvert _{P}$ and $\lvert \lvert f\rvert \rvert := \lvert \lvert f\rvert \rvert _R$ is a $\mathit {C}^*$ -norm on $\mathit {C_c}(G;\mathcal {A})$ . Moreover, the $\mathit {C}^*$ -algebra $\overline {\mathit {C_c}(G;\mathcal {A})}^{\lvert \lvert \cdot \rvert \rvert _R}$ is the same as the $\mathit {C}^*$ -algebra defined by Muhly and Williams [Reference Muhly and Williams12] for a saturated Fell bundle over a locally compact Hausdorff second countable groupoid.

The $\mathit {C}^*$ -algebra $\overline {\mathit {C_c}(G;\mathcal {A})}^{\lvert \lvert \cdot \rvert \rvert }$ is the full $\mathit {C}^*$ -algebra of $p\colon \mathcal {A}\to G$ . This $\mathit {C}^*$ -algebra is denoted by $\mathit {C}^*(G;\mathcal {A})$ .

Assume in Theorem 3.5 that G is locally compact Hausdorff and second countable étale, and $\mathcal {A}$ is saturated and separable. Then, Muhly and Williams define the full $\mathit {C}^*$ -algebra of $\mathcal {A}$ using $\mathcal {R}_H, \mathcal {R}_{\mathrm {In}}$ and $\mathcal {R}_P$ (and unitary representations of p). They show that these three norms are the same. We have added one more norm $\mathcal {R}_R$ . The $\mathit {C}^*$ -algebra given by the last theorem is the same as that of Muhly and Williams.

For the following remarks, a Fell bundle $p\colon \mathcal {A} \to G$ over an étale groupoid is fixed.

Remark 3.8. The $\mathit {C}^*$ -norm of $f\in \mathit {C_c}(G;\mathcal {A})$ which is supported on a bisection is

$$ \begin{align*} \lvert\lvert f\rvert\rvert^2 = \lvert\lvert f*f^*\rvert\rvert = \lvert\lvert f*f^*\rvert\rvert_\infty = \lvert\lvert f\rvert\rvert_\infty^2, \end{align*} $$

where the first equality is the $\mathit {C}^*$ -identity, the second equality comes about because $f*f^* \in \mathit {C}_0({G}^{(0)}; \mathcal {A}|_{{G}^{(0)}})$ and the last is Lemma 2.6(4). Moreover, Lemma 2.6(4) says that $\lvert \lvert f\rvert \rvert _I = \lvert \lvert f\rvert \rvert _\infty $ .

Example 3.11. Let G be the groupoid of the trivial equivalence relation on the set $\{0, 1\}$ , that is, $G = \{(0, 0), (0,1), (1,0),(1,1)\}$ . Then, G is an étale groupoid. Let A and B be two $\mathit {C}^*$ -algebras and X a (not necessarily full) Hilbert A–B-bimodule. Let $X^*$ be the dual module of X.

Consider the Fell bundle $p\colon \mathcal {A} {\kern-1pt}\to{\kern-1pt} G$ as follows: the fibres are $\mathcal {A}_{(0,0)} {\kern-1pt}={\kern-1pt} A$ , $\mathcal {A}_{(0,1)} {\kern-1pt}={\kern-1pt} X$ , $\mathcal {A}_{(1,0)} = X^*$ and $\mathcal {A}_{(1,1)} = B$ . The multiplication of the Fell bundle $\mathcal {A}_{(0,1)} \times \mathcal {A}_{(1,0)} \to \mathcal {A}_{(0,0)}$ is defined by $\xi \cdot \eta := {}_A \langle \xi , \eta ^*\rangle $ and for $\mathcal {A}_{(1,0)} \times \mathcal {A}_{(0,1)} \to \mathcal {A}_{(1,1)}$ by $\eta \cdot \xi := \langle \eta ^*, \xi \rangle _B$ , where $\xi \in X$ and $\eta \in X^* $ . This Fell bundle is unsaturated unless X is an imprimitivity bimodule. Let L be the linking algebra associated with the Hilbert bimodule X defined by Brown et al. [Reference Brown, Mingo and Shen2]. Define $\varphi \colon \mathrm {C}(G;\mathcal {A}) \to L$ by

$$ \begin{align*} \varphi(f) = \begin{pmatrix} f(0,0) & f(0,1) \\ f(1,0) & f(1,1) \end{pmatrix} \end{align*} $$

for $f\in \mathrm {C}(G;\mathcal {A})$ . Then, $\varphi $ is a bijective $^*$ -homomorphism. Since L is a $\mathit {C}^*$ -algebra, by the uniqueness of the $\mathit {C}^*$ -norm, $\mathit {C}^*(G;\mathcal {A}) \simeq L$ . This example is an unsaturated version of Kumjian’s example [Reference Kumjian11, Example 3.5(ii)].