# Cuntz-Nica-Pimsner algebras of product systems over groupoids

Massoud Amini and Mahdi Moosazadeh  Department of Mathematics, Faculty of
Mathematical Sciences, Tarbiat Modares University, Tehran 14115134, Iran
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $X$ be a product system over a quasi-lattice ordered groupoid $(G,P)$.
Under mild hypotheses, we associate to $X$ a $C^{*}$-algebra which is
couniversal for injective Nica covariant Toeplitz representations of $X$ which
preserve the gauge coaction. When $(G,P)$ is a quasi-lattice ordered group
this couniversal $C^{*}$-algebra coincides with the Cuntz-Nica-Pimsner algebra
introduced by Carlsen-Larsen-Sims-Vittadello, and under some mild amenability
conditions with that of Sims and Yeend. We prove related gauge invariant
uniqueness theorems in this general setup.

###### Key words and phrases:

quasi-lattice ordered groupoid, product system, Cuntz-Pimsner algebra, Hilbert
bimodule

###### 1991 Mathematics Subject Classification:

Primary 46L05

##  1\. Introduction

The Cuntz algebra $\mathcal{O}_{n}$, introduced by Joachim Cuntz in the late
70s, is the universal $C^{*}$-algebra generated by $n$ isometries on an
infinite-dimensional Hilbert space whose range projections add up to 1. These
were the first concrete examples of a separable infinite simple
$C^{*}$-algebra, and any such algebra contains a subalgebra having some
$\mathcal{O}_{n}$ as a quotient. Cuntz algebras are generalized in the early
80s by Cuntz and Krieger [11] and in the late 90s by Pimsner [40]. In the
Pimsner construction, one gets a $C^{*}$-algebra $\mathcal{O}_{X}$, now known
as the Cuntz-Pimsner $C^{*}$-algebra, for any $C^{*}$-correspondence $X$ over
a $C^{*}$-algebra $A$. Pimsner also defined a Toeplitz $C^{*}$-algebras
$\mathcal{T}_{X}$, having $\mathcal{O}_{X}$ as a quotient [40]. He showed that
his construction of $\mathcal{O}_{X}$ has crossed products by $\mathbb{Z}$ as
an special case. In this case, the Toeplitz $\mathcal{T}_{X}$ is like a
crossed product by $\mathbb{N}$.

Katsura was able to relax the assumption of isometric left action in the
Pimsner construction, and unify the construction by that of $C^{*}$-algebras
of topological graphs [28, 29, 30]. Katsura proved that a Cuntz-Pimsner
representation of a $C^{*}$-correspondence $X$ generates an isomorphic copy of
$\mathcal{O}_{X}$, precisely when the representation is injective and admits a
gauge action.

In Pimsner and Katsura construction, the representations are handled using the
Fock space, built by considering $n$-fold tensors of $X$ for $n\geq 0$, with
$A$ playing the role of the 0-fold tensor, which suggests that we are dealing
with a system of modules indexed by semigroup of non-negative integers.
Building on the notion of Fock space, Fowler introduced and studied
$C^{*}$-algebras associated to product systems of Hilbert bimodules indexed by
a semigroup, sitting in a group [21]. The idea of product systems go to
William Arveson, who first studied product systems of Hilbert spaces over the
semigroup of positive reals [3] (see also, [13]). The Fowler’s construction
employs Nica’s notion of quasi-lattice ordered groups [36]. Fowler was able to
show that $\mathcal{T}_{\rm cov}(X)$ has much of the structure of a twisted
crossed product of the coefficient algebra $A$ by the semigroup $P$, with a
twist coming from $X$.

The drawback in Fowler’s construction is that his generalized Cuntz-Pimsner
algebra $\mathcal{O}_{X}$ need not be a quotient of $\mathcal{T}_{\rm
cov}(X)$, and the canonical homomorphism from $A$ to $\mathcal{O}_{X}$ is not
necessarily injective, leaving little hope for a gauge-invariant uniqueness
theorem. Sims and Yeend [51] addressed these issues by introducing the Cuntz-
Nica-Pimsner algebra $\mathcal{NO}_{X}$ of a product system $X$ over a quasi-
lattice ordered group $(G,P)$, in such a way that it is a quotient of
$\mathcal{T}_{\rm cov}(X)$, and under relatively mild hypotheses, the
canonical representation of the product system $X$ on the Cuntz-Nica-Pimsner
algebra $\mathcal{NO}_{X}$ is isometric. Finally, Carlsen, Larsen, Sims and
Vittadello establish a gauge-invariant uniqueness theorem for
$\mathcal{NO}_{X}$, by an analysis of the fixed-point algebra (core) of the
canonical coaction of $G$ on $\mathcal{NO}_{X}$ [7].

In the above constructions, all correspondences are over a fixed
$C^{*}$-algebra $A$. This restriction could be best understood via the Fock
space. In order to tensor two bimodules over $A$ we need to have both the left
and right actions to be by the same $C^{*}$-algebra. In the algebraic
language, if the modules in a product system are indexed by elements of a
semigroup sitting in a group, it is natural to expect only one
$C^{*}$-algebra, which is then the module indexed by the identity element of
the group. Now what if we allow the possibility of some tensors to remain
undefined in the Fock space? Then, we could expect bimodules with left and
right actions of different $C^{*}$-algebras, sitting beside each other in a
tensor, exactly when the corresponding right and left module structures match,
i.e., come from actions of the same $C^{*}$-algebra. In the algebraic
language, we then have a family of bimodules indexed by a semigroupoid sitting
in a groupoid and containing its unit space, and the tensors of two bimodules
are allowed exactly when their indices are multipliable in the groupoid, i.e.,
their source and range match appropriately. In this case, we deal with
bimodules over a family of $C^{*}$-algebras indexed by the unit space of the
groupoid.

The construction of the universal $C^{*}$-algebra of a product system over a
semigroupoid is the main objective of the present paper. We work with a
compactly aligned product system $X$ of correspondences over a semigroupoid
$P$ inside a quasi-lattice ordered groupoid $G$, and construct the Cuntz-Nica-
Pimsner $C^{*}$-algebra $\operatorname{\mathcal{NO}}_{\\!X}$ and Toeplitz
$C^{*}$-algebra $\mathcal{T}_{\rm cov}{X}$. Our construction follows closely
that of Carlsen et al in [7], but there are many technicalities we have to
overcome, due to the fact that we are working with a Fell bundle over a
groupoid, whose properties are much less known compared to the fairly well
studied notion of Fell bundle over a group. This being said, we should
emphasize that our arguments here are adaptations of those already presented
by Fowler [21, 22], Yeend [58], Sims and Yeend [51], and Carlsen et al [7].
For Fell bundles over groupoids, we use the definitions and results of Kumjian
[31]. Finally, for coactions, we adapt the ideas and notions used by Exel [18]
and Quigg [42], though we extend them from coactions of groups to that of
groupoids.

Our main result gives sufficient conditions for the existence and uniqueness
of the reduced Cuntz-Nica-Pimsner $C^{*}$-algebra
$\operatorname{\mathcal{NO}}_{\\!X}^{r}$, which is couniversal for gauge-
compatible injective Nica covariant representations of $X$ (Theorem 4.1) and
the gauge-invariant uniqueness theorem (Corollary 4.18). Under a natural
condition (which is automatic in the group case), we show that the full Cuntz-
Nica-Pimsner $C^{*}$-algebra $\operatorname{\mathcal{NO}}_{\\!X}$ has gauge-
invariant uniqueness property when the canonical coaction of $G$ on the
Toeplitz algebra is normal and $G$ is exact (Corollary 4.19). We give some of
our notions and results for an étale groupoid, but in most cases, including
the above cited results, we restrict ourselves to discrete groupoids. We give
concrete examples to illustrate our construction, and in particular show, in
an appendix, that the crossed product of $C_{0}(X)$-algebras by discrete
groupoids, and $C^{*}$-algebras of multilayered topological $k$-graphs are
particular cases of our general construction. We also show how to extend the
Katsura construction of the $C^{*}$-algebra $\mathcal{O}_{X}$ of an
$A$-$A$-correspondence $X$, to construct the $C^{*}$-algebra
$\mathcal{O}_{X,Y}$ of a pair of $A$-$B$-correspondence $X$ and
$B$-$A$-correspondence $Y$, and more importantly, show that it satisfies our
gauge-invariant uniqueness property.

##  2\. Preliminaries

For an étale groupoid $G$ with Haar system $\lambda$, we write $g\mapsto
i_{G}(g)$ for the canonical inclusion of $G$ as partial isometries in the full
groupoid $C^{*}$-algebra $C^{*}(G)$. We let $\mu$ be a quasi-invariant measure
on $G^{(0)}$ and ${\rm Ind}\mu$ denote the regular representation of $G$. All
$C^{*}$-algebra tensor products are w.r.t. the minimal tensor product.

###  2.1. Hilbert bimodules and correspondences

Let $A$ and $B$ be $C^{*}$-algebras. A Hilbert $A$-$B$-bimodule is a complex
vector space $X$ endowed with a left $A$-module structure and an $A$-valued
$A$-sesquilinear form ${}_{A}\langle\cdot\,,\,\cdot\rangle$ and a right
$B$-module structure and a $B$-valued $B$-sesquilinear form
$\langle\cdot\,,\,\cdot\rangle_{B}$, with compatibility condition
${}_{A}\langle x\,,\,y\rangle\cdot z=x\cdot\langle y\,,\,z\rangle_{B}$, for
$x,y,z\in X$, such that $X$ is complete in the norm $\|x\|:=\|{}_{A}\langle
x,x\rangle\|^{1/2}=\|\langle x,x\rangle_{B}\|^{1/2}$. A map $T\colon X\to X$
is said to be $A$-adjointable if there is a map $T^{*}\colon X\to X$ such that
${}_{A}\langle Tx,y\rangle={}_{A}\langle x,T^{*}y\rangle$, for $x,y\in X$.
Every adjointable operator on $X$ is norm-bounded and linear, and the adjoint
$T^{*}$ is unique. The collection $\mathscr{L}_{A}(X)$ of $A$-adjointable
operators on $X$ endowed with the operator norm is a $C^{*}$-algebra. The
ideal of “compact” operators $\mathscr{K}_{A}(X)$ is the closed span of the
“rank” one operators $x\otimes y^{*}\colon z\mapsto x\cdot{}_{A}\langle
y,z\rangle$, with $x$ and $y$ range over $X$. The same notations are used for
$A$ replaced by $B$.

An $A$-$B$-correspondence is a right Hilbert $B$-module $X$ endowed with a
left action $\phi_{A}\colon A\to\mathscr{L}_{A}(X)$ of $A$ by adjointable
operators. Let $X$ and $Y$ be an $A$-$B$-correspondence $B$-$C$-correspondence
with left actions $\phi_{A}$ and $\phi_{B}$, respectively. The balanced tensor
product $X\otimes_{B}Y$ is the completion of the vector space spanned by
elements $x\otimes_{B}y$ with $x\in X$ and $y\in Y$, subject to $x\cdot
b\otimes_{B}y=x\otimes_{A}\phi_{B}(b)y$, in the norm coming by the inner-
product

$\langle x_{1}\otimes_{B}y_{1},x_{2}\otimes_{B}y_{2}\rangle_{B}:=\langle
y_{1},\langle x_{1},x_{2}\rangle_{B}\cdot y_{2}\rangle_{B}.$

With the right action of $C$ on $X\otimes_{B}Y$ given by $(x\otimes_{B}y)\cdot
c=x\otimes_{B}(y\cdot c)$ and left action of $A$ implemented by the
homomorphism $a\mapsto\phi_{A}(a)\otimes 1_{\mathcal{L}_{A}(Y)}$,
$X\otimes_{B}Y$ is an $A$-$C$-correspondence.

###  2.2. Semigroupoids and product systems

In this section, we define product systems over subemigroupoids of quasi-
lattice ordered groupoids. These base structures are modeled after the notion
of quasi-lattice ordered groups due to Nica [36]. An étale groupoid $G$ and a
subsemigroupoid $P$ of $G$ with $P\cap P^{-1}=G^{(0)}$, we say that $(G,P)$ is
a quasi-lattice ordered groupoid if, under the partial order $g\leq h\iff
g^{-1}h\in P$, for $g,h\in G$ with the same range, any pair $p,q$ in $G$ with
a common upper bound in $P$ have a least common upper bound $p\vee q$ in $P$
(compare with the definition in the group case [36, 22, 8, 9]). We write
$p\vee q<\infty$ when $p\vee q$ exists and $p\vee q=\infty$ when $p,q\in G$
have no common upper bound in $P$. If $p\vee q<\infty$, for all $p,q\in P$, we
say that $(G,P)$ is directed, or simply say that $P$ is directed. Finally, a
quasi-lattice ordered groupoid $(G,P)$ is called totally ordered if for each
$p,q\in P$ with the same range, either $p^{-1}q\in P$ or $q^{-1}p\in P$. In
this case, we simply say that $P$ is totally ordered.

A product system over $P$ consists of a bundle $\\{A_{u}\\}$ of
$C^{*}$-algebras over $G^{(0)}$, a semigroupoid $X$, and a semigroupoid
homomorphism $d\colon X\to P$ such that $X_{p}:=d^{-1}(p)$ is an
$A_{r(p)}$-$A_{s(p)}$-correspondence for each $p\in P$, $X_{u}=A_{u}$, for
$u\in G^{(0)}$, and the multiplication on $X$ implements isomorphisms
$X_{p}\otimes_{A_{r(q)}}X_{q}\cong X_{pq}$, for $p,q\in P\setminus\\{e\\}$
with $(p,q)\in G^{(2)}$, as well as the right $A_{s(p)}$-module structure and
the left $A_{r(p)}$-actions on each $X_{p}$. For $p\in P$, we denote the
homomorphism which implements the corresponding left action by
$\phi_{p}:A_{r(p)}\to\mathscr{L}_{A_{r(p)}}(X_{p})$. We denote the
$A_{s(p)}$-valued inner product on $X_{p}$ by $\langle\cdot,\cdot\rangle_{p}$.
For $(p,q)\in G^{(2)}$, we have $\phi_{pq}(a)(xy)=(\phi_{p}(a)x)y$, for $x\in
X_{p}$, $y\in X_{q}$ and $a\in A_{r(p)}$. Given $p,q\in P\backslash G^{(0)}$,
there is a homomorphism
$\iota^{pq}_{p}\colon\mathscr{L}_{A_{r(p)}}(X_{p})\to\mathscr{L}_{A_{r(p)}}(X_{pq})$
given by $\iota^{pq}_{p}(S)(xy)=(Sx)y,$ for $x\in X_{p}$, $y\in X_{q}$ and
$S\in\mathscr{L}_{A_{r(p)}}(X_{p})$. Identifying
$\mathscr{K}_{A_{r(p)}}(X_{r(p)})$ with $A_{r(p)}$ as above, we define
$\iota^{p}_{r(p)}\colon\mathscr{K}_{A_{r(p)}}(X_{r(p)})\to\mathscr{L}_{A_{r(p)}}(X_{p})$
by $\iota^{p}_{r(p)}=\phi_{p}$, for $p\in P$ (compare to [51, §2.2]). Finally,
under the identification $X_{p}\otimes_{A_{r(p)}}A_{r(p)}\cong X_{p}$, we
could write $\iota_{p}^{p}(S)=S\otimes{\rm id}_{A_{r(p)}}.$

A product system $X$ over $P$ is compactly aligned if $\iota^{p\vee
q}_{p}(S)\iota^{p\vee q}_{q}(T)\in\mathscr{K}_{A_{r(p\vee q)}}(X_{p\vee q})$,
for each $S\in\mathscr{K}_{A_{r(p)}}(X_{p})$ and
$T\in\mathscr{K}_{A_{r(q)}}(X_{q})$, whenever $p\vee q$ exists (cf., [22,
Definition 5.7]). Let us observe that a product system $X$ over a totally
ordered quasi-lattice ordered groupoid $(G,P)$ is automatically compactly
aligned: given $p$ and $q$ with $p\vee q<\infty$, and
$S\in\mathscr{K}_{A_{r(p)}}(X_{p}),T\in\mathscr{K}_{A_{r(q)}}(X_{q})$, we have
$p\leq p\vee q$ and $q\leq p\vee q$, thus $r(p)=r(p\vee q)=r(q)$. By total
order assumption, $p\vee q$ is either $p$ or $q$, say $p\vee q=p$. Then
$\iota_{p}^{p\vee q}(S)=\iota_{p}^{p}(S)=S\otimes{\rm
id}_{A_{r(p)}}\in\mathscr{K}_{A_{r(p)}}(X_{p})$, thus,

$\iota_{p}^{p\vee q}(S)\iota_{q}^{p\vee
q}(T)\in\mathscr{K}_{A_{r(p)}}(X_{p})=\mathscr{K}_{A_{r(p\vee q)}}(X_{p\vee
q}),$

as required.

###  2.3. Representations of product systems

Given a product system $X$ over $P$, a Toeplitz representation (or simply a
representation) $\psi$ of $X$ in a $C^{*}$-algebra $D$ is a map $\psi:X\to D$
satisfying,

  1. (1)

$\psi_{p}:=\psi|_{X_{p}}\colon X_{p}\to D$ is linear and $\psi_{u}:A_{u}\to D$
is a homomorphism;

  2. (2)

$\psi_{s(p)}(\langle x,y\rangle_{p})=\psi_{p}(x)^{*}\psi_{p}(y)$ and,
$\psi(xz)=\psi(x)\psi(z)$, if $s(p)=r(q),$

  3. (3)

$\psi(x)\psi(z)=0$, if $s(p)\neq r(q)$,

for each $p,q\in P$, $u\in G^{(0)}$, and $x,y\in X_{p}$, $z\in X_{q}$.

We would define the notion of a Nica covariant representation of a compactly
aligned product system $X$, and later show that, for such representations, we
also have,

(4) $\psi(x)\psi(y)^{*}=0$, if $s(p)\neq s(q)$, and $\psi(x)^{*}\psi(z)=0$, if
$r(p)\neq r(r)$,

for each $p,q,r\in P$, and $x\in X_{p}$, $y\in X_{q}$, $z\in X_{r}$.

A Toeplitz representation $\psi$ is injective if homomorphisms $\psi_{u}$ are
all injective, for $u\in G^{(0)}$. In this case, the linear maps $\psi_{p}$
are isometry, for $p\in P$. Given a representation $\psi$, we have
$*$-homomorphisms $\psi^{(p)}\colon\mathscr{K}_{A_{r(p)}}(X_{p})\to D$ defined
by $\psi^{(p)}(x\otimes y)=\psi_{p}(x)\psi_{p}(y)^{*}$, for $x,y\in X_{p}$
(compare to [40]). When $D=\mathscr{L}(\mathcal{H})$, for some Hilbert space
$\mathcal{H}$, we say that $\psi$ is nondegenerate if $\psi(X)\mathcal{H}$ is
a total set in $H$, and cyclic if $\psi(X)\xi$ is dense in $\mathcal{H}$, for
some $\xi\in\mathcal{H}$, called a cyclic vector. Two representations
$\psi_{1}:X\to\mathscr{L}(\mathcal{H}_{1})$ and
$\psi_{2}:X\to\mathscr{L}(\mathcal{H}_{2})$ are unitarily equivalent if there
is a unitary $U\in\mathscr{L}(\mathcal{H}_{1},\mathcal{H}_{2})$ such that
$\psi_{2}(x)={\rm ad}_{U}(\psi_{1}(x))$, for each $x\in X$.

As in [22, Proposition 2.8], we have the following result.

######  Lemma 2.1.

There is a universal $C^{*}$-algebra $\mathcal{T}_{X}$ generated by a
universal Toeplitz representation $i$ of $X$.

###### Proof.

We adapt the proof of [24, Proposition 1.3]. For a Toeplitz representation
$\psi:X\to\mathscr{L}(\mathcal{H})$, let $C^{*}(\psi)$ be the
$C^{*}$-subalgebra of $\mathscr{L}(\mathcal{H})$ generated by the range of
$\psi$, $\mathcal{H}_{\psi}$ be the closed subspace of $\mathcal{H}$ generated
by $C^{*}(\psi)\mathcal{H}$, and $P_{\psi}:\mathcal{H}\to\mathcal{H}_{\psi}$
be the corresponding orthogonal projection. Then it is easy to check that

$P\psi:X\to\mathscr{L}(\mathcal{P}_{\psi}H);\ \ x\mapsto
P_{\psi}\psi(x)P_{\psi},$

is again a Toeplitz representation of $X$, and is nondegenerate. Since
$(1-P_{\psi})\psi$ is the zero representation, it follows that $\psi$ is a
direct sum of a nondegenerate representation and a zero representation.

Next let us show that nonzero cyclic representations exist. For this we need
to construct the Fock representation. Consider the $G^{(0)}$-bundle of right
Hilbert modules, $F(X)_{u}:=\bigoplus_{r(p)=u}X_{p},$ where the right hand
side is the subset of $\prod_{p\in P}X_{p}$ consisting of all elements
$(x_{p})$ for which $\sum_{r(p)=u}\langle x_{p},x_{p}\rangle_{p}$ is summable
in $A_{r(p)}$. We write $\oplus x_{p}$ for $(x_{p})$ to indicate that the
above series is summable. Each $F(X)_{u}$ is a right Hilbert $A_{u}$-module
with the right action $(\oplus x_{p})\cdot a:=\oplus(x_{p}\cdot a)$, for $a\in
A_{u}$, and inner product $\langle\oplus x_{s},\oplus
y_{s}\rangle_{A_{u}}:=\sum_{r(p)=u}\langle x_{p},y_{p}\rangle_{p}$. Define the
Fock representation of $X$ by

$l:=\oplus l_{u}:X\to\bigoplus_{u\in G^{(0)}}\mathscr{L}_{A_{u}}(F(X)_{u}),$

where $l_{u}:X\to\mathscr{L}_{A_{u}}(F(X)_{u})$ is defined by $l_{u}(x)(\oplus
y_{p}):=\oplus z_{q},$ with $z_{q}:=xy_{r^{-1}q},$ if $r\leq q$, and
$z_{q}:=0,$ otherwise; when $x\in X_{r}$, for some $r\in P$, and the other two
indices run over $p,q\in P$ with $r(p)=r(q)=u$. Each $l(x)$ is adjointable,
and $l(x)^{*}(yz)=\langle x,y\rangle_{p}\cdot z$ for $x,y\in X_{p}$ and $z\in
X_{q}$, with $r(q)=s(p)$.

Let us observe that $l$ is a Toeplitz representation. For $x_{r}\in
X_{r},x_{s}\in X_{s}$ with $r(s)=s(r)$,

$l_{u}(x_{r})l_{u}(x_{s})(\oplus y_{p})=l_{u}(x_{r})(\oplus z_{q})=(\oplus
w_{t}),$

with $z_{q}:=x_{s}y_{s^{-1}q},$ if $s\leq q$, and $z_{q}:=0,$ otherwise, and
$w_{t}:=x_{r}z_{r^{-1}t}$, if $r\leq t$, and $w_{t}:=0,$ otherwise, for
$r(t)=r(p)=r(q)=u$. Similarly,

$l_{u}(x_{r}x_{s})(\oplus y_{p})=(\oplus w^{{}^{\prime}}_{t}),$

with $w^{{}^{\prime}}_{t}:=x_{r}x_{s}y_{s^{-1}r^{-1}t},$ if $rs\leq t$, and
$w^{{}^{\prime}}_{t}:=0,$ otherwise, for $r(t)=r(p)=r(q)=u$. Now since the
inequalities $r\leq t$ and $s\leq r^{-1}t$ both hold iff $rs\leq t$, we have
$w_{t}=x_{r}x_{s}y_{s^{-1}r^{-1}t}=w^{{}^{\prime}}_{t}$, for each $t$, that
is, $l(x_{r})l(x_{s})=l(x_{r}x_{s})$, when $r(s)=s(r)$. On the other hand, if
the inequalities $r\leq t$ and $s\leq r^{-1}t$ both hold, we have
$r(s)=r(r^{-1}t)=s(r)$, which means that if $r(s)\neq s(r)$, then $w_{t}=0$,
for each $t$, that is, $l(x_{r})l(x_{s})=0$, in this case.

Next, the restriction of $l$ to $A_{u}$ is nothing but
$\oplus_{r(p)=u}\phi_{p}$, which is known to be isometric. If we faithfully
represent the $C^{*}$-algebra $\bigoplus_{u\in
G^{(0)}}\mathscr{L}_{A_{u}}(F(X)_{u})$ in a Hilbert space $\mathcal{H}$, the
Fock representation $l$ would have a nonzero cyclic summand.

Now by Zorn lemma, every nondegenerate representation is a direct sum of
cyclic representations. Let $\mathfrak{S}$ be a set of cyclic representations
of $X$ such that every representation $\psi:X$ in a Hilbert space $H$ is
unitarily equivalent to an element of $\mathfrak{S}$. Such a set
$\mathfrak{S}$ exists an could be realized as the set of cyclic
representations on subspaces of a Hilbert space of sufficiently large
dimension. Now let $i$ be the direct sum of all the representations in
$\mathfrak{S}$, which makes sense, since for each $\psi\in\mathfrak{S}$ and
$x\in X$,

$\|\psi(x)\|=\|\psi_{s(d(x))}(\langle
x,x\rangle_{d(x)})\|^{\frac{1}{2}}\leq\|\langle
x,x\rangle_{d(x)}\|^{\frac{1}{2}}=\|x\|.$

Put $\mathcal{T}_{X}:=C^{*}(i)$, then $\mathcal{T}_{\rm cov}(X)$ is universal
among all $C^{*}$-algebras generated by range of a Toeplitz representation of
$X$, by the definition of $\mathfrak{S}$. ∎

When $X$ is a compactly aligned, a Toeplitz representation $\psi$ of $X$ is
Nica-covariant if

$\displaystyle\psi^{(p)}(S)\psi^{(q)}(T)=\begin{cases}\psi^{(p\vee
q)}\big{(}\iota^{p\vee q}_{p}(S)\iota^{p\vee q}_{q}(T)\big{)}&\text{if $p\vee
q$\ exists}\\\ 0&\text{otherwise,}\end{cases}$

for all $S\in\mathscr{K}_{A_{r(p)}}(X_{p})$ and
$T\in\mathscr{K}_{A_{r(q)}}(X_{q})$ (compare to [22, Definition 5.7]). Let
$\mathcal{T}_{\rm cov}(X)$ be the quotient of $\mathcal{T}_{X}$ by the ideal
generated by the elements

$i^{(p)}(S)i^{(q)}(T)-i^{(p\vee q)}(\iota^{p\vee q}_{p}(S)\iota^{p\vee
q}_{q}(T)),$

where $p,q\in P$, $S\in\mathscr{K}_{A_{r(p)}}(X_{p})$,
$T\in\mathscr{K}_{A_{r(q)}}(X_{q})$, with convention, $\iota^{p\vee
q}_{p}(S)\iota^{p\vee q}_{q}(T)=0$ if $p\vee q=\infty$. The composition of the
quotient map from $\mathcal{T}_{X}$ onto $\mathcal{T}_{\rm cov}(X)$ with the
universal Toeplitz representation $i$ is a Nica covariant Toeplitz
representation $i_{X}\colon X\to\mathcal{T}_{\rm cov}(X)$ with the following
universal property: if $\psi$ is a Nica covariant Toeplitz representation of
$X$ in $D$ there is a $*$-homomorphism $\psi_{*}:\mathcal{T}_{\rm cov}(X)\to
D$ such that $\psi_{*}\circ i_{X}=\psi$. When $X$ is a compactly aligned
product system of essential Hilbert bimodules, $\mathcal{T}_{\rm cov}(X)$
resembles Fowler’s algebra [22] for product systems of essential Hilbert
bimodules over quasi-lattice ordered groups.

######  Lemma 2.2.

Let $X$ be compactly-aligned product system over $P$, $\psi:X\to D$ be a Nica-
covariant Toeplitz representation of $X$, $p,q\in P$, $y\in X_{p}$, and $z\in
X_{q}$. Then $\psi(y)^{*}\psi(z)=0$, if $p\vee q=\infty$; and otherwise,

$\psi(y)^{*}\psi(z)\in\operatorname{\overline{span\\!}\,\,}\\{\psi(u)\psi(v)^{*}:u\in
X_{p^{-1}(p\vee q)},v\in X_{q^{-1}(p\vee q)}\\}.$

###### Proof.

Let us write $y=Sy^{\prime}$ with $S\in\mathscr{K}_{A_{r(p)}}(X_{p})$ and
$y^{\prime}\in X_{p}$, and $z=Tz^{\prime}$ with
$T\in\mathscr{K}_{A_{r(q)}}(X_{q})$ and $z^{\prime}\in X_{q}$. Since $\psi$ is
Nica covariant,

$\psi(y)^{*}\psi(z)=\psi(y^{\prime})^{*}\psi^{(p)}(S^{*})\psi^{(q)}(T)\psi(z^{\prime})$

which is zero if $p\vee q=\infty$, and is otherwise equal to
$\psi(y)^{*}\psi(z)=\psi(y^{\prime})^{*}\psi^{(p\vee q)}(K)\psi(z^{\prime}),$
for
$K:=(S^{*}\otimes_{A_{u}}1)(T\otimes_{A_{u}}1)\in\mathscr{K}_{A_{u}}(X_{p\vee
q})$, $u:=r(p\vee q)$. Since $K$ is compact, it can be approximated in norm by
a finite sum of operators of the form $u\otimes v^{*}$, with $u,v\in X_{p\vee
q}$, thus $\psi^{(p\vee q)}(K)$ can be approximated by finite sums of elements
in $D$ of the form $\psi(u_{1})\psi(v_{1})^{*}$. We may respectively
approximate $u_{1}$ and $v_{1}$ by finite sums of products of the form
$u_{2}u^{\prime}$ and $v_{2}v^{\prime}$, with $u_{2}\in X_{p}$, $v_{2}\in
X_{q}$, and $u^{\prime}\in X_{p^{-1}(p\vee q)}$, $v^{\prime}\in
X_{q^{-1}(p\vee q)}$. Hence $\psi(y^{\prime})^{*}\psi^{(p\vee
q)}(K)\psi(z^{\prime})$ can be approximated in norm by finite sums of
operators of the form

$\psi(y^{\prime})^{*}\psi(u_{2})\psi(u^{\prime})\psi(v^{\prime})^{*}\psi(v_{2})^{*}\psi(z^{\prime})=\psi(\langle
y^{\prime},u_{2}\rangle_{p}u^{\prime})\psi(\langle
z^{\prime},v_{2}\rangle_{q}v^{\prime})^{*}=:\psi(u)\psi(v)^{*},$

as claimed. ∎

######  Lemma 2.3.

Let $X$ be compactly-aligned product system over $P$, $\psi:X\to D$ be a Nica-
covariant Toeplitz representation of $X$, then $\psi(x)\psi(y)^{*}=0$, if
$s(p)\neq s(q)$, and $\psi(x)^{*}\psi(z)=0$, if $r(p)\neq r(r)$, for each
$p,q,r\in P$, and $x\in X_{p}$, $y\in X_{q}$, $z\in X_{r}$.

###### Proof.

Let us write $x=Sx^{\prime}$ with $S\in\mathscr{K}_{A_{r(p)}}(X_{p})$ and
$x^{\prime}\in X_{p}$, and $y=Ty^{\prime}$ with
$T\in\mathscr{K}_{A_{r(q)}}(X_{q})$ and $y^{\prime}\in X_{q}$. Then,

$\psi(x)\psi(y)^{*}=\psi^{(p)}(S)\psi(x^{\prime})\psi(y^{\prime})^{*}\psi^{(q)}(T^{*}).$

As above, let us approximate compact operators $S$ and $T^{*}$ in norm,
respectively by finite sums of the form $u\otimes v^{*}$ and
$u^{\prime}\otimes v^{\prime*}$, with $u,v\in X_{p}$ and
$u^{\prime},v^{\prime}\in X_{q}$, and observe that if $s(p)\neq s(q)$,

$\displaystyle\psi^{(p)}(u\otimes
v^{*})\psi(x^{\prime})\psi(y^{\prime})^{*}\psi^{(q)}(u^{\prime}\otimes
v^{\prime*})$
$\displaystyle=\psi_{p}(u)\psi_{p}(v)^{*}\psi_{p}(x^{\prime})\psi_{q}(y^{\prime})^{*}\psi_{q}(u^{\prime})\pi_{q}(v^{\prime})^{*}$
$\displaystyle=\psi_{p}(u)\psi_{s(p)}(\langle
v,x^{\prime}\rangle_{p})\psi_{s(q)}(\langle
y^{\prime},u^{\prime}\rangle_{q})\pi_{q}(v^{\prime})^{*}=0,$

since the product of two middle terms is zero by part (3) of the definition of
a Toeplitz representation. This proves the first statement. The second
statement, follows from Lemma 2.2, as $p\vee r=\infty$ if $r(p)\neq r(r)$. ∎

As in [22, Theorem 6.3], it follows that,

$\mathcal{T}_{\rm
cov}(X)=\operatorname{\overline{span\\!}\,\,}\\{\,i_{X}(x)i_{X}(y)^{*}\mid
x,y\in X\,\\}.$

Indeed, the right hand side is a norm closed, selfadjoint subspace of
$\mathcal{T}_{\rm cov}(X)$, which includes $i_{X}(X)$, as each $z\in X$ could
be written as $z=z^{\prime}\cdot a$, for some $z^{\prime}\in X_{p}$ and $a\in
A_{s(p)}$, that gives $i_{X}(z)=i_{X}(z^{\prime})i_{X}(a^{*})^{*}$. The
equality now follows from the fact that the right hand side is also closed
under multiplication, which in turn follows from Lemma 2.2. By a similar
argument, we have the following more general result.

######  Lemma 2.4.

If the image of a Nica covariant Toeplitz representation $\psi$ of $X$ in $D$
generates $D$ as a $C^{*}$-algebra, then
$D=\operatorname{\overline{span\\!}\,\,}\\{\,\psi(x)\psi(y)^{*}\mid x,y\in
X\,\\}$.

###  2.4. The Cuntz-Nica-Pimsner algebra $\operatorname{\mathcal{NO}}_{\\!X}$

Next, let us define Cuntz-Pimsner covariance of representations, adapted from
[51, Section 3]. A predicate statement $\mathcal{P}(s)$, with $s\in P$, is
said to be true for large $s$ if for every $q\in P$ there exists $r\geq q$
such that $\mathcal{P}(s)$ is true whenever $r\leq s$.

When $X$ is a compactly aligned over $P$, we let $I_{u}=A_{u}$, for $u\in
G^{(0)}$, and put $I_{q}:=\bigcap_{0<p\leq q}\ker(\phi_{p})$, for $q\in
P\backslash G^{(0)}$, which is a closed two-sided ideal in $A_{r(q)}$, where
the notation $0<p\leq q$ simply means that the intersections runs over those
$p\in P\backslash G^{(0)}$ satisfying $p\leq q$. We put,
$\widetilde{X}_{q}:=\textstyle{\bigoplus}_{p\leq q}X_{p}\cdot I_{p^{-1}q},$
regarded as an $A_{r(q)}$-$A_{s(q)}$-correspondence, with the homomorphism
implementing the left action denoted by $\tilde{\phi}_{q}$. When all the
homomorphisms $\tilde{\phi}_{q}$ are injective, we say that $X$ is
$\tilde{\phi}$-injective.

With the convention of $\iota^{q}_{p}(T)$ being 0 in
${\mathscr{L}_{A_{r(q)}}(X_{q})}$, for $T\in\mathscr{L}_{A_{r(p)}}(X_{p})$,
when $p\not\leq q$, we have homomorphisms
$\tilde{\iota}^{q}_{p}\colon\mathscr{L}_{A_{r(p)}}(X_{p})\to\mathscr{L}_{A_{r(q)}}(\widetilde{X}_{q})$,
for $p\in P\backslash G^{(0)}$ and $q\in P$, defined by
$\tilde{\iota}^{q}_{p}(T)=\bigoplus_{r\leq q}\iota^{r}_{p}(T)$ for
$T\in\mathscr{L}_{A_{r(p)}}(X_{p})$. Similarly, for $u\in G^{(0)}$, there is a
homomorphism
$\tilde{\iota}^{q}_{u}:\mathscr{K}_{A_{u}}(X_{u})\to\mathscr{L}_{A_{r(q)}}(\widetilde{X}_{q})$.

When $X$ is moreover $\tilde{\phi}$-injective, a Nica covariant Toeplitz
representation $\psi$ of $X$ in a $C^{*}$-algebra $D$ is called Cuntz-Nica-
Pimsner covariant (abbreviated as CNP-covariant) if for each finite subset
$F\subset P$ and any choice of $T_{p}\in\mathscr{K}_{A_{r(p)}}(X_{p})$, for
$p\in F$, the finite sum $\sum_{p\in F}\psi^{(p)}(T_{p})$ is zero in $D$,
whenever the finite sum $\sum_{p\in F}\tilde{\iota}^{q}_{p}(T_{p})$ is zero in
$\mathscr{L}_{A_{r(q)}}(\widetilde{X}_{q})$, for large $q$. We write
$\operatorname{\mathcal{NO}}_{\\!X}$ for the universal $C^{*}$-algebra
generated by a CNP-covariant representation $j_{X}$ of $X$, and call it the
Cuntz-Nica-Pimsner algebra of $X$.

Consider the closed two sided ideal $I_{\rm cov}$ of $\mathcal{T}_{\rm
cov}(X)$ generated by the set of finite sums $\sum_{p\in
F}i_{X}^{(p)}(T_{p})$, with $T_{p}\in\mathscr{K}_{A_{r(p)}}(X_{p})$, for $p\in
F\subset P$, such that $\sum_{p\in E}\tilde{\iota}\hskip
1.0pt_{p}^{s}(T_{p})=0$, for large $s$, then it follows immediately from
definition that $\operatorname{\mathcal{NO}}_{\\!X}$ is isomorphic to the
quotient $\mathcal{T}_{\rm cov}(X)/I_{\rm cov}$. Also, for each CNP-covariant
representation $\psi:X\to D$, and corresponding homomorphism
$\psi_{*}:\mathcal{T}_{\rm cov}(X)\to D$ satisfying $\psi=\psi_{*}\circ
i_{X}$, by covariance property of $\psi$, we have
$I_{cov}\subseteq\ker(\psi_{*})$, therefore, $\psi_{*}$ descends to a
homomorphism $\Pi\psi:\operatorname{\mathcal{NO}}_{\\!X}\to D$, satisfying
$\psi=\Pi\psi\circ j_{X}$.

Similar to [51, Theorem 4.1], let us show that $j_{X}$ is an injective
representation, whenever each representation $\tilde{\phi}_{p}$ is injective,
or each bounded subset of $P$ has a maximal element. This needs some
preparation.

Following [51], we define the augmented Fock representation using the
$G^{(0)}$-bundle
$\tilde{F}(X)_{u}:=\bigoplus_{r(q)=u}\tilde{X}_{q}=\bigoplus_{r(q)=u}\bigoplus_{p\leq
q}X_{p}\cdot I_{p^{-1}q}.$ Let $X$ be a compactly aligned product system over
$P$ and $l$ be its Fock representation constructed as in Lemma 2.1. For
$p,q,r\in P$ with $p\leq q$, and $r(p)=r(q)=s(r)=u$, let $x\in X_{r}$ and
$z\in X_{p}\cdot I_{p^{-1}q}$. Write $z=z^{\prime}\cdot a$ with $z^{\prime}\in
X_{p}$ and $a\in I_{p^{-1}q}\subset A_{u}$, then $xz=x(z^{\prime}\cdot
a)=(xz^{\prime})\cdot a\in X_{rp}\cdot I_{(rq)^{-1}rq}$. This observation
shows that, if we regard each $\tilde{X}_{q}$ as a submodule of $F(X)_{u}$,
and given $x\in X$, let $\tilde{l}_{u}(x)$ be the restriction of $l_{u}(x)$ to
$\tilde{F}(X)_{u}$, then
$\widetilde{l}(x)\in\mathscr{L}_{A_{u}}(\tilde{F}(X)_{u})$. Let us observe
that The map $x\mapsto\widetilde{l}(x):=\oplus\tilde{l}_{u}(x)$ is a Nica
covariant representation of $X$ in $\bigoplus_{u\in
G^{(0)}}\mathscr{L}_{A_{u}}(\tilde{F}(X)_{u})$: if $x\in X_{p}$ and $y=\oplus
y_{s}\in\tilde{F}(X)_{u}$ with $y_{s}=0$, for all $s\geq p$, we need to show
that $\widetilde{l}(x)^{*}y=0$. By linearity and continuity of
$\widetilde{l}(x)^{*}$, we may assume that $y$ has just one nonzero
coordinate, say there exists $r\in P$ such that $r\not\geq p$ and $y_{s}=0$,
for $s\not=r$. For $z\in\tilde{F}(X)_{u}$, we have,

$\langle\widetilde{l}(x)^{*}y,z\rangle_{A_{u}}=\langle
y,\widetilde{l}(x)z\rangle_{A_{u}}=\sum_{s\in P}\langle
y_{s},(\widetilde{l}(x)z)_{s}\rangle_{s}=\langle
y_{r},(\widetilde{l}(x)z)_{r}\rangle_{r}=0,$

since $r\not\geq p$.

Let $B_{X}$ (resp., $B_{\tilde{X}}$) be the $C^{*}$-algebra generated by the
range of the (resp., augmented) Fock representation $\widetilde{l}$ in
$\bigoplus_{u\in G^{(0)}}\mathscr{L}_{A_{u}}(F(X)_{u})$ (resp., in
$\bigoplus_{u\in G^{(0)}}\mathscr{L}_{A_{u}}(\tilde{F}(X)_{u})$), which is
called the (resp., augmented) Fock $C^{*}$-algebra of $X$. It follows from
Lemma 2.2 that
$B_{\tilde{X}}=\operatorname{\overline{span\\!}\,\,}\\{\widetilde{l}(x)\widetilde{l}(y)^{*}:x,y\in
X,d(x)=d(y)\\}$.

Consider the closed two sided ideal $I_{\tilde{X}}$ of $B_{\tilde{X}}$
generated by the set $K_{\tilde{X}}$ consisting of finite sums $\sum_{p\in
E}\widetilde{l}^{(p)}(T_{p})$, with $T_{p}\in\mathscr{K}_{A_{r(p)}}(X_{p})$,
for $p\in E\subset P$, such that $\sum_{p\in E}\tilde{\iota}\hskip
1.0pt_{p}^{s}(T_{p})=0$, for large $s$, then,

$I_{\tilde{X}}=\operatorname{\overline{span\\!}\,\,}\\{\widetilde{l}(x)\widetilde{l}(y)^{*}k\widetilde{l}(x^{\prime})\widetilde{l}(y^{\prime})^{*}:k\in\overline{K_{\tilde{X}}},x,y,x^{\prime},y^{\prime}\in
X,d(x)=d(y),d(x^{\prime})=d(y^{\prime})\\}.$

Let $q_{\tilde{X}}:B_{\tilde{X}}\to B_{\tilde{X}}/I_{\tilde{X}}$ be the
quotient map. Then $\psi_{\tilde{X}}:=q_{\tilde{X}}\circ\widetilde{l}$ is a
CNP-covariant representation of $X$ in $B_{\tilde{X}}/I_{\tilde{X}}$.

######  Lemma 2.5.

Let $X$ be a compactly aligned product system over $P$, and $\tilde{\phi}_{p}$
is injective, for $p\in P$. Then the restriction of $\psi_{\tilde{X}}$ on each
fibre $X_{u}=A_{u}$, $u\in G^{(0)}$, is faithful.

###### Proof.

We just need to show that $I_{\tilde{X}}\cap\widetilde{l}_{u}(A_{u})=\\{0\\}$,
for each $u\in G^{(0)}$. For $u\in G^{(0)}$ and $a\in A_{u}$, the restriction
of $\widetilde{l}(a)$ to the summand $\tilde{X}_{q}$ is the same as
$\tilde{\phi}_{q}(a)$. In particular,
$\|\widetilde{l}(a)|_{\widetilde{X}_{q}}\|=\|a\|$, for each $q$ with $r(q)=u$.
It therefore suffices to show that, for each $\psi\in I_{\tilde{X}}$ and
$\varepsilon>0$, there is $s\in P$ with
$\|\psi|_{\widetilde{X}_{s}}\|<\varepsilon$.

Let $K_{\tilde{X}}$ be the generating set of $I_{\tilde{X}}$, defined above.
By a continuity argument, for each $\psi\in\overline{K}$, $\varepsilon>0$, and
$q\in P$, there exists $r\geq q$ such that
$\|\psi|_{\widetilde{X}_{s}}\|<\varepsilon$, for $s\geq r$. For
$k\in\overline{K_{\tilde{X}}}$, norm one elements $x,y\in X_{p}$,
$x^{\prime},y^{\prime}\in X_{t}$, and $q\in P$ with $r(q)=u$, let us observe
that there exists $r\geq q$ such that
$\|\widetilde{l}(x)\widetilde{l}(y)^{*}k\widetilde{l}(x^{\prime})\widetilde{l}(y^{\prime})^{*}|_{\widetilde{X}_{s}}\|<\varepsilon$,
for each $s\geq r$: if $t\vee q<\infty$, choose $r^{\prime}\geq s(t)(t\vee q)$
with $\|k|_{\widetilde{X}_{s^{\prime}}}\|<\varepsilon$, for $s^{\prime}\geq
r^{\prime}$. Put $r=s(t)r^{\prime}\geq q$, then for each $s\geq r$, we have
$s(t)s\geq r^{\prime}$, and
$\|\widetilde{l}(x)\widetilde{l}(y)^{*}k\widetilde{l}(x^{\prime})\widetilde{l}(y^{\prime})^{*}|_{\widetilde{X}_{s}}\|<\varepsilon.$
Now if $t\vee q=\infty$, for $s\geq q$, $s\not\geq t$, thus
$\widetilde{l}(y^{\prime})^{*}|_{\widetilde{X}_{s}}=0$, and for $r:=q$, we get
$\widetilde{l}(x)\widetilde{l}(y)^{*}k\widetilde{l}(x^{\prime})\widetilde{l}(y^{\prime})^{*}|_{\widetilde{X}_{s}}=0$,
for each $s\geq r$, as claimed. ∎

A similar argument shows that, for an appropriate ideal $I_{X}$ of $B_{X}$ and
quotient map $q_{X}:B_{X}\to I_{X}$, $\psi_{X}:=q_{X}\circ l$ is a CNP-
covariant representation of $B_{X}/I_{X}$, which is injective on each fibre
$X_{u}=A_{u}$, $u\in G^{(0)}$, is faithful, whenever all representations
$\tilde{\phi}_{p}$ are injective.

######  Proposition 2.6.

Let $X$ be a compactly aligned product system over $P$, such that each
$\tilde{\phi}_{p}$ is injective, or each bounded subset of $P$ has a maximal
element. Then $j_{X}$ is an injective representation of $X$.

###### Proof.

Under the first assumption, by Lemma 2.5 and the universal property of
$\operatorname{\mathcal{NO}}_{\\!X}$, implies that $j_{X}$ is injective, and
so isometric, on each fibre $X_{u}=A_{u}$, for $u\in G^{(0)}$. Now for $p\in
P$ and $x\in X_{p}$, $\|x\|^{2}=\|\langle x,x\rangle_{p}\|=\|j_{X}(\langle
x,x\rangle_{r(p)})\|=\|j_{X}(x)^{*}j_{X}(x)\|=\|j_{X}(x)\|^{2},$ as required.
∎

For the rest of paper, $(G,P)$ be a quasi-lattice ordered groupoid and $X$ is
a product system over $P$ consisting of $C^{*}$-correspondences.

##  3\. Analysis of the core

In this section we discuss the fixed-point algebra of $\mathcal{T}_{\rm
cov}(X)$ under a canonical coaction $\delta$, and show that under certain
conditions, $\operatorname{\mathcal{NO}}_{\\!X}$ satisfies an analog of the
criterion (B) of Seems and Yeend [51, Section 1].

###  3.1. The core

Let us use $q_{\operatorname{CNP}}\colon\mathcal{T}_{\rm
cov}(X)\to\operatorname{\mathcal{NO}}_{\\!X}$ to denote the canonical
surjection arising from the universal property of $\mathcal{T}_{\rm cov}(X)$.

######  Lemma 3.1.

Let $\psi\colon X\to D$ be a Toeplitz representation of $X$.

$(i)$ $\psi_{t}(\iota_{p}^{t}(T)(x))=\psi^{(p)}(T)\psi_{t}(x)$, for $p\leq
t\in P$, $T\in\mathscr{K}_{A_{r(p)}}(X_{p})$, and $x\in X_{t}$, and
$\psi^{(r)}(T)\psi_{t}(x\cdot a)=0$, for $t<r\leq s\in P$,
$T\in\mathscr{K}_{A_{r(r)}}(X_{r})$, and $x\cdot a\in X_{t}\cdot I_{t^{-1}s}$,

$(ii)$ when $X$ is compactly aligned and $\psi$ is Nica covariant, also
$\psi^{(p)}(T)\psi_{t}(x\cdot a)=0$, for $p,t\leq s\in P$, $p\not\leq t$,
$T\in\mathscr{K}_{A_{r(p)}}(X_{p})$, and $x\cdot a\in X_{t}\cdot I_{t^{-1}s}$,

$(iii)$ when $X$ is compactly aligned and either the left action on each fibre
is injective $P$ is directed, and $\psi$ is injective and Nica covariant, for
any finite subset $F\subset P$, $p\in F$, and operators
$T_{p}\in\mathscr{K}_{A_{r(p)}}(X_{p})$, we have $\sum_{p\in
F}\tilde{\iota}\hskip 1.0pt^{s}_{p}(T_{p})=0$, for large $s$, whenever
$\sum_{p\in F}\psi^{(p)}(T_{p})=0$.

###### Proof.

$(i)$ For the first equality, if $u\in G^{(0)}$, the first statement follows
from $\mathscr{K}_{A_{u}}(X_{u})\cong A_{u}$ and $\iota_{u}^{t}=\phi_{t}$, and
if $p\in P\backslash G^{(0)}$, since given $t\in P$ with $r(t)=r(p)$,
$\\{\,xy\mid x\in X_{p},\,y\in X_{p^{-1}t}\,\\}$ is dense in $X_{t}$, and
$\\{\,w\otimes z^{*}\mid w,z\in X_{p}\,\\}$ is total in
$\mathscr{K}_{A_{r(p)}}(X_{p})$, we need only to observe that for $x,w,z\in
X_{p}$ and $y\in X_{p^{-1}t}$,

$\displaystyle\psi_{t}(\iota_{p}^{t}(w\otimes z^{*})(xy))$
$\displaystyle=\psi_{p}(w\cdot\langle
z,x\rangle_{A}^{p})\psi_{p^{-1}t}(y)=\psi_{p}(w)\psi_{e}(\langle
z,x\rangle_{A}^{p})\psi_{p^{-1}t}(y)$
$\displaystyle=\psi_{p}(w)\psi_{p}^{*}(z)\psi_{p}(x)\psi_{p^{-1}t}(y)=\psi^{(p)}(w\otimes
z^{*})\psi_{t}(xy).$

For the second equality, if $u\in G^{(0)}$, then $x\cdot a\in
I_{s}\subseteq\ker(\phi_{r})$. The equality
$\psi_{u}(y)\psi^{(r)}(S)=\psi^{(r)}(\phi_{r}(y)S)$ first follows for $y\in
X_{r}$ and $S:=w\otimes z^{*}$, $w,z\in X_{r}$, and then for any
$S\in\mathscr{K}_{A_{r}(r)}(X_{r})$. Taking adjoint and letting $y:=(x\cdot
a)^{*}$ and $S=T^{*}$, we get $\psi^{(r)}(T)\psi_{u}(x\cdot
a)=\psi^{(r)}(T\phi_{r}(x\cdot a))=0$, as required. When $t\in P\backslash
G^{(0)}$, for $y\in X_{t}$, $z\in X_{t^{-1}r}$, and $v\in X_{r}$, with $r,s,t$
having the same range, since $a\in I_{t^{-1}s}$, $\phi_{t^{-1}r}(a)=0$,

$\displaystyle\psi^{(r)}$ $\displaystyle(v\otimes(yz)^{*})\psi_{t}(x\cdot
a)=\psi_{r}(v)\psi_{t^{-1}r}(z)^{*}\psi_{t}(y)^{*}\psi_{t}(x\cdot
a)=\psi_{r}(v)\psi_{t^{-1}r}(z)^{*}\psi_{s(t)}(\langle y,x\cdot a\rangle_{t})$
$\displaystyle=\psi_{r}(v)\psi_{t^{-1}r}(z)^{*}\psi_{s(t)}(\langle
y,x\rangle_{t}a)=\psi_{r}(v)(\psi_{s(t)}(a^{*}\langle
x,y\rangle_{t})\psi_{t^{-1}r}(z))^{*}$
$\displaystyle=\psi_{r}(v)\psi_{t^{-1}r}(\phi_{t^{-1}r}(a^{*}\langle
x,y\rangle_{t})z)^{*}=\psi_{r}(v)\psi_{t^{-1}r}(\phi_{t^{-1}r}(a)^{*}\phi_{t^{-1}r}(\langle
x,y\rangle_{t})z)^{*}=0.$

$(ii)$ Let $(E_{k})$ be an approximate identity for
$\mathscr{K}_{A_{r(s)}}(X_{t}\cdot I_{t^{-1}s})$. Since $p,t\leq s$, $p\vee t$
exists and by the Nica covariance of $\psi$,

$\displaystyle\psi^{(p\vee t)}\big{(}\iota^{p\vee t}_{p}(T)\iota^{p\vee
t}_{t}(E_{k})\big{)}\psi_{t}(x\cdot a)$
$\displaystyle=\psi^{(p)}(T)\psi^{(t)}(E_{k})\psi_{t}(x\cdot
a)\psi^{(p)}(T)\psi_{t}(E_{k}(x\cdot a))$
$\displaystyle\to\psi^{(p)}(T)\psi_{t}(x\cdot a),$

as $k\to\infty$. Since $p\not\leq t$, we have $t<p\vee t\leq s$, and the
statement follows from the second statement in $(i)$.

$(iii)$ Given $q\in P$, list the elements of $F$ as $p_{1},\dots,p_{n}$.
Define $r_{0}:=q$, and inductively, for $1\leq i\leq n$, define
$r_{i}:=r_{i-1}\vee p_{i}$, if the right hand side exists, and
$r_{i}:=r_{i-1}$, otherwise. Then $r_{n}\geq p$ and $r_{n}\vee p$ exists, ,
for each $p\in F$, and so $r_{n}$ is an upper bound for $F$ if $P$ is
directed.

For $s\geq r_{n}$, and $x\cdot a\in X_{t}\cdot I_{t^{-1}s}$, for $r(t)=r(s)$.
By part $(i)$,

$\textstyle{\psi_{t}\big{(}\sum_{p\leq t}\iota_{p}^{t}(T_{p})(x\cdot
a)\big{)}}=\textstyle{\sum_{p\leq t}\psi_{t}(\iota_{p}^{t}(T_{p})(x\cdot
a))}=\textstyle{\sum_{p\leq t}\psi^{(p)}(T_{p})\psi_{t}(x\cdot a)},$

where the sums run over $p\in F$ with $p\leq t$. If each $\phi_{p}$ is
injective, then $I_{t^{-1}s}=0$ for $t<s$, and so $a=0$ unless $t=s$. Let
$(E_{k})$ be an approximate identity for $\mathscr{K}_{A_{r(s)}}(X_{s})$. By
assumption $p\not\leq s$ and $p\vee s=\infty$. By Nica covariance,
$\psi^{(p)}(T_{p})\psi_{s}(x)=\lim_{k}\psi^{(p)}(T_{p})\psi^{(s)}(E_{k})\psi_{s}(x)=0$,
for $T_{p}\in\mathscr{K}_{A_{r(p)}}(X_{p})$ and $x\in X_{s}$. Therefore,

$\psi_{t}(\sum_{p\leq t}\iota_{p}^{t}(T_{p})(x\cdot a))=\sum_{p\in
F}\psi^{(p)}(T_{p})\psi_{t}(x\cdot a)=0.$

The same holds when $P$ is directed, since in this case, $p\leq r\leq s$ and
part $(ii)$ applies. Since $\psi$ is injective, it follows that $\sum_{p\leq
t}\iota_{p}^{t}(T_{p})(x\cdot a)=0$, as required. ∎

By part $(i)$ in the above lemma, if $r\in tP\setminus\\{t\\}$, then the
element $\psi^{(r)}(T)\in D$ annihilates $\psi_{t}(X_{t}\cdot I_{t^{-1}s})$,
whenever $s\in rP$, and under the assumptions of part $(iii)$, we may replace
the condition $r\in tP\setminus\\{t\\}$ with the weaker condition $t\not\in
rP$, and $s=t\vee r$.

When $X$ is compactly aligned, by an argument as the one for $\mathcal{T}_{\rm
cov}(X)$ (see the last two paragraphs of subsection 2.3), it follows from the
Nica-covariance of $i_{X}$ that

$\mathcal{F}:=\operatorname{\overline{span\\!}\,\,}\\{\,i_{X}(x)i_{X}(y)^{*}\mid
x,y\in X,d(x)=d(y)\,\\}$

is a $C^{*}$-subalgebra of $\mathcal{T}_{\rm cov}(X)$, called the core of
$\mathcal{T}_{\rm cov}(X)$.

For $u\in G^{(0)}$, let us observe that

$\mathcal{F}_{u}:=\operatorname{\overline{span\\!}\,\,}\\{\,i_{X}(x)i_{X}(y)^{*}\mid
x,y\in X,d(x)=d(y)\in G^{u}\,\\}$

is a $C^{*}$-subalgebra of $\mathcal{F}$. Given $x_{1},y_{1}\in X_{p}$ and
$x_{2},y_{2}\in X_{q}$ with $r(p)=r(q)=u$, the element
$i_{X}(x_{1})i_{X}(y_{1})^{*}i_{X}(x_{2})i_{X}(y_{2})^{*}$ is zero if $p\vee
q=\infty$, and otherwise is a limit of linear combinations of elements of the
form

$i_{X}(x_{1})i_{X}(u)i_{X}(v)^{*}i_{X}(y_{2})^{*}=i_{X}(x_{1}u)i_{X}(y_{2}v)^{*},$

with $u\in X_{p^{-1}(p\vee q)}$ and $v\in X_{q^{-1}(p\vee q)}$, which are
again in $\mathcal{F}_{u}$, as $d(x_{1}u)=pp^{-1}(p\vee q)=p\vee q\in G^{u}$
and $d(y_{2}v)=qq^{-1}(p\vee q)=p\vee q\in G^{u}$, showing that
$\mathcal{F}_{u}$ is closed under multiplication, and so a $C^{*}$-subalgebra
of $\mathcal{F}$.

###  3.2. Gauge coaction

When $G$ is discrete groupoid with Haar system consisting of counting
measures, there is a $*$-homomorphism $\delta_{G}\colon C^{*}(G)\to
C^{*}(G)\otimes C^{*}(G)$, given by $\delta_{G}(i_{G}(g))=i_{G}(g)\otimes
i_{G}(g)$, for $g\in G$. This is a coaction in the following sense (compare to
[42]). We should warn the reader that our notion of groupoid coaction on
$C^{*}$-algebras deviates from the one defined on $W^{*}$-algebras by
Yamanouchi [57] based on Connes’ notion of fibred products (cf., [38], [53]).

Recall that a $*$-homomorphism $\phi:A\to D$ between $C^{*}$-algebras is
called nondegenerate if $\phi(A)D$ is dense in $D$. This is automatic when $A$
and $D$ are unital and $\phi$ is a unital map. In general, if $\phi$ maps an
approximate identity of $A$ to one for $D$, then $\phi$ is nondegenerate. A
nondegenerate $*$-homomorphism $\phi:A\to D$ has a unique extension to an
$*$-homomorphism $\tilde{\phi}:M(A)\to M(D)$ between the multiplier algebras.
Indeed, we have the following useful characterization, which would be needed
later. This is stated and proved in [33, Proposition 2.5] for the case where
$D=\mathscr{L}_{B}(E)$, for some $C^{*}$-algebra $B$ and right Hilbert
$B$-module $E$, but the same proof essentially works in the general case as
well. Recall that the strict topology on $M(A)$ is the locally convex topology
given by the family of seminorms $x\mapsto\|ax\|$ and $x\mapsto\|xa\|$ on
$M(A)$, for $a$ running over $A$.

######  Lemma 3.2.

For a $*$-homomorphism $\phi:A\to D$ between $C^{*}$-algebras, the following
are equivalent:

$(i)$ $\phi$ is nondegenerate,

$(ii)$ $\phi$ has an extension to a unital $*$-homomorphism
$\tilde{\phi}:M(A)\to M(D)$, which is strictly continuous on the closed unit
ball of $M(A)$,

$(iii)$ $\phi$ maps some approximate identity of $A$ to an approximate
identity of $D$,

$(iv)$ the net $(\phi(e_{i}))$ converges strictly to $1_{M(D)}$, for some
approximate identity of $A$.

It turns out that such a notion of nondegeneracy is too strong for coactions
of groupoids on $C^{*}$-algebras. Here we propose a weaker notion. A subset
$D_{0}$ of a $C^{*}$-algebra $D$ is called selfadjoint if $x\in D_{0}$ iff
$x^{*}\in D_{0}$, for each $x\in D$.

######  Definition 3.3.

Let $\phi:A\to D$ be a $*$-homomorphism between $C^{*}$-algebras. Let
$\mathfrak{Z}$ be a $C^{*}$-subalgebra of $M(D)$ whose set of nonzero
projections is nonempty. We say that $\phi$ is nondegenerate modulo
$\mathfrak{Z}$, or simply nondegenerate (mod $\mathfrak{Z}$), if there is a
selfadjoint total subset $D_{0}\subseteq D$ such that for each $x\in D_{0}$
there is a nonzero projection $p\in\mathfrak{Z}$ such that $px$ is in the
closed linear span of elements $\phi(a)b$, with $a\in A,b\in D$.

The above definition looks nonsymmetric, but in fact if $p$ is the nonzero
projection such that $px^{*}$ is within $\varepsilon$ of the span of
$\phi(A)D$, then so is $xp$, and vice versa, so the above definition has an
equivalent right version. Also, note that when
$\mathfrak{Z}=\mathbb{C}1_{M(D)}$, the notion of nondegeneracy (mod
$\mathfrak{Z}$) is the same as the classical notion of nondegeneracy.

When $D=A\otimes C^{*}(G)$, for a discrete groupoid $G$, there is a natural
candidate for $\mathfrak{Z}$, namely $\mathfrak{Z}=C_{0}(G^{(0)})$, identified
with its copy $1_{M(A)}\otimes C_{0}(G^{(0)})$ inside $M(A\otimes C^{*}(G))$,
and if we choose $D_{0}$ to be the set of all basic tensors $a\otimes f$ with
$a\in A$ and $f\in C_{c}(G)$, then we say that a $*$-homomorphism $\phi:A\to
A\otimes C^{*}(G)$ is nondegenerate (mod $G^{(0)}$) if for each
$\varepsilon>0$, $a\in A$ and $g\in G$, there is a unit element $u\in G^{(0)}$
such that $a\otimes(1_{u}*1_{g})$ is within $\varepsilon$ of the linear span
of $\phi(A)(A\otimes C^{*}(G))$. It is easy to see that being nondegenerate
(mod $G^{(0)}$) is the same as being nondegenerate (mod $C_{0}(G^{(0)})$), in
the above sense.

######  Definition 3.4.

A (full) coaction of a discrete groupoid $G$ on a $C^{*}$-algebra $A$ is a
$*$-homomorphism $\delta\colon A\to A\otimes C^{*}(G)$, satisfying the
identity
$(\delta\otimes\operatorname{id})\circ\delta=(\operatorname{id}_{A}\otimes\delta_{G})\circ\delta$,
which is nondegenerate (mod $G^{(0)}$).

When $G$ is a discrete group, $G^{(0)}$ is a singleton consisting of the
identity of $G$, and being nondegenerate (mod $G^{(0)}$) is the same as being
nondegenerate in the usual sense.

For a coaction $\delta$ of a discrete groupoid $G$ on a $C^{*}$-algebra $A$,
the fixed-point algebra is by definition, the cross-sectional $C^{*}$-algebra
of the $G^{(0)}$-bundle $A_{u}^{\delta}:=\\{a\in A\mid\delta(a)=a\otimes
i_{G}(u)\\}$, $u\in G^{(0)}$. There is a coaction of $G$ on $\mathcal{T}_{\rm
cov}(X)$ whose fixed-point algebra is the core $\mathcal{F}$.

######  Lemma 3.5.

Let $G$ be discrete and $X$ be compactly aligned. The map $\psi$ defined by
$\psi:X\to\mathcal{T}_{\rm cov}(X)\otimes C^{*}(G);\ x\mapsto i_{X}(x)\otimes
i_{G}(d(x))$, is a Nica covariant representation of $X$.

###### Proof.

The fact that $\psi$ is a representation follows from Lemma 2.1 and the fact
that $\delta_{p}*\delta_{q}=\delta_{pq}$ when $s(p)=r(q)$, and zero otherwise,
in $C^{*}(G)$. To see that $\psi$ is Nica covariant, first observe that,

$\displaystyle\psi^{(p)}(x\otimes y^{*})$
$\displaystyle=\psi_{p}(x)\psi_{p}(y)^{*}=\big{(}i_{X}(x)\otimes
i_{G}(p)\big{)}\big{(}i_{X}(y)^{*}\otimes i_{G}(p^{-1})\big{)}$
$\displaystyle=i_{X}(x)i_{X}(y)^{*}\otimes i_{G}(r(p))=i_{X}^{(p)}(x\otimes
y^{*})\otimes i_{G}(r(p)),$

for each $p\in P,x,y\in X_{p}$. Thus,

$\psi^{(p)}(S)=i_{X}^{(p)}(S)\otimes i_{G}(r(p)),\ \ (p\in
P,S\in\mathscr{K}_{A_{r(p)}}(X_{p})),$

Now, $\psi$ is Nica covariant, as $i_{X}$ is so, by the paragraph after Lemma
2.1. ∎

######  Lemma 3.6.

Let $G$ be discrete and $X$ be compactly aligned. There is a coaction $\delta$
of $G$ on $\mathcal{T}_{\rm cov}(X)$, satisfying
$\delta(i_{X}(x))=i_{X}(x)\otimes i_{G}(d(x))$, for $x\in X$.

###### Proof.

Since the representation $\psi:X\to\mathcal{T}_{\rm cov}(X)\otimes C^{*}(G);\
x\mapsto i_{X}(x)\otimes i_{G}(d(x))$ is Nica covariant by Lemma 3.5, by the
universal property of $\mathcal{T}_{\rm cov}(X)$, one gets a $*$-homomorphism
$\delta:\mathcal{T}_{\rm cov}(X)\to\mathcal{T}_{\rm cov}(X)\otimes C^{*}(G)$,
satisfying $\delta(i_{X}(x))=\psi(x)$, for $x\in X$. The coaction identity is
immediate on generators, and so holds everywhere.

To see that $\delta$ is nondegenerate (mod $G^{(0)}$), let
$(\theta_{\lambda})$ be an approximate identity for $\mathcal{F}$, which is
then also an approximate identity for $\mathcal{T}_{\rm cov}(X)$, and observe
that for $x,y\in X$ with $d(x)=d(y)$, $\theta_{\lambda}i_{X}(x)i_{X}(y)^{*}\to
i_{X}(x)i_{X}(y)^{*},$ and $i_{X}(x)i_{X}(y)^{*}\theta_{\lambda}\to
i_{X}(x)i_{X}(y)^{*},$ as $\lambda\to\infty$. For $p:=d(x)\in P$,
$x=z\cdot\langle z,z\rangle_{p}=(z\otimes z^{*})(z)$, for some $z\in X$, and
$i_{X}(x)i_{X}(y)^{*}=i_{X}^{(p)}(z\otimes z^{*})i_{X}(z)i_{X}(y)^{*}$. Since
$i_{X}^{(p)}(z\otimes z^{*})$ is in $\mathcal{F}$,
$\theta_{\lambda}i_{X}^{(p)}(z\otimes z^{*})\to i_{X}^{(p)}(z\otimes z^{*})$,
that is, $\theta_{\lambda}i_{X}(x)i_{X}(y)^{*}\to i_{X}(x)i_{X}(y)^{*}$, as
claimed. The same holds, by a similar argument, when $\theta_{\lambda}$ is
multiplied from right. Next, observe that for basic tensors of the form
$i_{X}(x)i_{X}(y)^{*}\otimes i_{G}(g)$, for $x\in X_{p},y\in X_{q}$ and $g\in
G$, let $u:=r(p)=r(q)$, then, $1_{u}*1_{g}=1_{g}$, when $r(g)=u$, and zero
otherwise, and in the former case,

$\displaystyle\delta\big{(}i_{X}(x)i_{X}(y)^{*}\big{)}(\theta_{\lambda}\otimes
1_{qp^{-1}g})$ $\displaystyle=(i_{X}(x)i_{X}(y)^{*}\otimes
1_{pq^{-1}})(\theta_{\lambda}\otimes 1_{qp^{-1}g})$
$\displaystyle=i_{X}(x)i_{X}(y)^{*}\theta_{\lambda}\otimes 1_{ug}$
$\displaystyle=i_{X}(x)i_{X}(y)^{*}\theta_{\lambda}\otimes 1_{g}$
$\displaystyle\to i_{X}(x)i_{X}(y)^{*}\otimes 1_{g}$
$\displaystyle=(1_{M(\mathcal{T}_{\rm cov}(X))}\otimes
1_{u})(i_{X}(x)i_{X}(y)^{*}\otimes 1_{g}),$

when $\lambda\to\infty$, as required. ∎

The above coaction $\delta$ of $G$ on $\mathcal{T}_{\rm cov}(X)$ is called the
gauge coaction. Note that unlike the group case [7, Proposition 3.5], $\delta$
is not injective in general. We would comment on the kernel of $\delta$ soon
(see Lemmas 3.9, 3.12, 3.14). Also, note that $\delta$ is not nondegenerate in
general. However, we have some sort of nondegeneracy for the restriction of
$\delta$ to the core (see Lemma 3.15). This restricted version of
nondegeneracy later appears in the definition of induced ideals (see
Definition 4.4)).

######  Lemma 3.7.

Let $G$ be discrete and let $X$ be compactly aligned.

$(i)$ The core $\mathcal{F}$ is the same as the cross-sectional
$C^{*}$-algebra $\mathcal{T}_{\rm cov}(X)_{0}^{\delta}$ of the fixed-point
$G^{(0)}$-bundle with fibres $\mathcal{T}_{\rm
cov}(X)_{u}^{\delta}=\\{a\in\mathcal{T}_{\rm cov}(X)\mid\delta(a)=a\otimes
i_{G}(u)\\}$, $u\in G^{(0)}$.

$(ii)$ $\mathcal{F}=\operatorname{\overline{span\\!}\,\,}\\{i^{(p)}_{X}(T)\mid
p\in P\text{ and }T\in\mathscr{K}_{A_{r(p)}}(X_{p})\\}.$

###### Proof.

$(i)$ Since $G^{(0)}$ is discrete, $\mathcal{T}_{\rm cov}(X)_{0}^{\delta}$ is
nothing but the direct sum of the fibres [31, Example 3.5(i)]. By the above
lemma, for $x,y\in X_{p}$,

$\delta(i_{X}(x)i_{X}(y)^{*})=\delta(i_{X}(x))\delta(i_{X}(y))^{*}=i_{X}(x)i_{X}(y)^{*}\otimes
i_{G}(s(p)),$

thus, $\mathcal{F}\subseteq\mathcal{T}_{\rm cov}(X)_{0}^{\delta}$. The reverse
inclusion follows similarly.

$(ii)$ This is immediate, since
$i^{(p)}_{X}\colon\mathscr{K}_{A_{r(p)}}(X_{p})\to\mathcal{T}_{X}$ satisfies
$i^{(p)}_{X}(x\otimes y)=i_{X}(x)i_{X}(y)^{*}$, for $x,y\in X_{p}$, and
$\mathscr{K}_{A_{r(p)}}(X_{p})$ is the closed linear span of the set of
elements $x\otimes y$, with $x,y\in X_{p}$. ∎

A subset $F$ of $P$ is said to be closed if for $p,q\in F$, $p\vee q\in F$,
whenever it exists. Let $\mathcal{P}_{\rm fin}^{\vee}(P)$ denote the set of
finite closed subsets of $P$, which is directed under inclusion, by an
argument as in [22, p. 367]. Also, each bounded $F\in\mathcal{P}_{\rm
fin}^{\vee}(P)$ has a maximal element.

For $p\in P$, let $B_{p}$ be the $C^{*}$-subalgebra
$i_{X}^{(p)}(\mathscr{K}_{A_{r(p)}}(X_{p}))$ of $\mathcal{T}_{\rm cov}(X)$.
For finite closed subset $F$ of $P$, let $B_{F}:=\textstyle{\sum}_{p\in
F}B_{p}$, which is a linear subspace of $\mathcal{T}_{\rm cov}(X)$, then
$\bigcup_{F\in\mathcal{P}_{\rm fin}^{\vee}(P)}B_{F}$ is dense in
$\mathcal{F}$. Let us observe that when $X$ is compactly aligned, $B_{F}$ is
indeed a $C^{*}$-subalgebra of $\mathcal{F}$: $B_{F}$ is closed under adjoint
and multiplication, by the Nica covariance of the universal representation
$i_{X}$ of $X$ in $\mathcal{T}_{\rm cov}(X)$. To show that it is norm closed,
we proceed by induction on $n:=|F|$. If $n=1$, say $F=\\{p\\}$, then
$B_{F}=B_{p}=i_{X}^{(p)}(\mathscr{K}_{A_{r(p)}}(X_{p}))$ is the range of a
$C^{*}$-homomorphism and hence closed. Assuming this for $|F|\leq k$, take
$F\subset P$ with $n=|F|=k+1$, and take a minimal element $m$, in the sense
that, for $p\in F\setminus\\{m\\}$, $p\not\leq m$. By the induction
hypothesis, $B_{m}$ and $B_{F\setminus\\{m\\}}$ are $C^{*}$-subalgebras of
$\mathcal{F}$. Choose $p\in F$, $p\neq m$, then $p\not\leq m$ by minimality of
$m$, and $p\vee m\neq m$. Thus, for $S\in\mathscr{K}_{A_{r(p)}}(X_{p})$ and
$T\in\mathscr{K}_{A_{r(m)}}(X_{m})$,
$i_{X}^{(p)}(S)i_{X}^{(m)}(T)=i_{X}^{(p\vee m)}(\iota^{p\vee
m}_{p}(S)\iota^{p\vee m}_{m}(T))$ is in $i_{X}^{(p\vee
m)}(\mathscr{K}_{A_{r(p\vee m)}}(X_{p\vee m}))\subseteq
B_{F\setminus\\{m\\}}$. Similarly, $i_{X}^{(m)}(T)i_{X}^{(p)}(S)\in
B_{F\setminus\\{m\\}}$, therefore $B_{F}=B_{m}+B_{F\setminus\\{m\\}}$ is norm
closed [14, Corollary 1.8.4].

######  Lemma 3.8.

Let $X$ be compactly aligned. Suppose that either the left action on each
fibre is injective, or $P$ is directed. Let $\psi\colon X\to D$ be an
injective Nica covariant representation of $X$, and let
$\psi_{*}:\mathcal{T}_{\rm cov}(X)\to D$ be homomorphism satisfying
$\psi=\psi_{*}\circ i_{X}$. Then
$\ker(\psi_{*})\cap\mathcal{F}\subset\ker(q_{\operatorname{CNP}})$.

###### Proof.

By an argument similar to that of the proof of [1, Lemma 1.3], and above
observations, it is enough to show that $\ker(\psi_{*})\cap
B_{F}\subset\ker(q_{\operatorname{CNP}})$, for each $F\in\mathcal{P}_{\rm
fin}^{\vee}(P)$. For a typical element $c:=\sum_{p\in F}i_{X}^{(p)}(T_{p})$ of
$B_{F}$, if $c\in\ker(\psi_{*})$, since $\psi$ is injective, by part $(iii)$
of Lemma 3.1, $\sum_{p\in F}\tilde{\iota}\hskip 1.0pt^{s}_{p}(T_{p})=0$, for
large $s$. Since $j_{X}$ is CNP-covariant,
$\textstyle{q_{\operatorname{CNP}}(c)=\sum_{p\in F}j_{X}^{(p)}(T_{p})=0}$,
that is, $c\in\ker(q_{\operatorname{CNP}})$. ∎

The above lemma gives a way to control the kernel of canonical coaction
$\delta$ of $G$ on $\mathcal{T}_{\rm cov}(X)$ defined in Lemma 3.6.

######  Lemma 3.9.

Let $X$ be compactly aligned. Suppose that either the left action on each
fibre is injective, or $P$ is directed. Then for the canonical coaction

$\delta:\mathcal{T}_{\rm cov}(X)\to\mathcal{T}_{\rm cov}(X)\otimes C^{*}(G);\
i_{X}(x)\mapsto i_{X}(x)\otimes i_{G}(d(x)),\ \ (x\in X),$

we have $\ker(\delta)\cap\mathcal{F}\subset\ker(q_{\operatorname{CNP}})$,
where $q_{\operatorname{CNP}}:\mathcal{T}_{\rm
cov}(x)\to\operatorname{\mathcal{NO}}_{\\!X}$ is the quotient map.

###### Proof.

Let $\psi:X\to\mathcal{T}_{\rm cov}(X)$ be the Nica covariant representation
defined in Lemma 3.5. Then $\psi_{*}=\delta$, and the result follows from
Lemma 3.6. ∎

We are now ready to show that $\operatorname{\mathcal{NO}}_{\\!X}$ satisfies
analog of criterion (B) of Sims and Yeend [51, Section 1], whenever the left
actions on fibres of $X$ are all injective, or $P$ is directed. This in
particular gives a gauge-invariant uniqueness theorem for
$\operatorname{\mathcal{NO}}_{\\!X}$, when $G$ is a discrete amenable
groupoid.

######  Lemma 3.10.

Let $X$ be compactly aligned. Assume that either the left actions on the
fibres of $X$ are all injective, or $P$ is directed and $X$ is
$\tilde{\phi}$-injective. Let $\psi\colon X\to D$ be a CNP-covariant
representation of $X$ in a $C^{*}$-algebra $D$. Then the induced homomorphism
$\Pi{\psi}:\operatorname{\mathcal{NO}}_{\\!X}\to D$ is injective on
$q_{\operatorname{CNP}}(\mathcal{F})$ if and only if $\psi$ is injective as a
Toeplitz representation.

###### Proof.

Suppose that $\Pi{\psi}$ is injective on
$q_{\operatorname{CNP}}(\mathcal{F})$. By Proposition 2.6, $j_{X}$ is
injective, and $\psi=\Pi{\psi}\circ(j_{X})$. Conversely, if $\psi$ is
injective, by Lemma 3.8,

$\ker(\psi_{*})=\ker(\Pi\psi\circ
q_{\operatorname{CNP}})\cap\mathcal{F}\subset\ker(q_{\operatorname{CNP}}),$

that is, $\Pi{\psi}$ is injective on $q_{\operatorname{CNP}}(\mathcal{F})$. ∎

######  Lemma 3.11.

Let $G$ be discrete and $X$ be compactly aligned. For the canonical Nica
covariant representation

$\psi:X\to\mathcal{T}_{\rm cov}(X)\otimes C^{*}(G);\ x\mapsto i_{X}(x)\otimes
i_{G}(d(x)),$

the induced representation

$\tilde{\psi}:=(q_{CNP}\otimes{\rm
id})\circ\psi:X\to\operatorname{\mathcal{NO}}_{\\!X}\otimes C^{*}(G);\
x\mapsto j_{X}(x)\otimes i_{G}(d(x)),$

is a CNP-covariant representation of $X$. When each $\tilde{\phi}_{p}$ is
injective, or each bounded subset of $P$ has a maximal element, $\tilde{\psi}$
is also injective as a representation.

###### Proof.

First observe that,

$\displaystyle\tilde{\psi}^{(p)}(x\otimes y^{*})$
$\displaystyle=\tilde{\psi}_{p}(x)\tilde{\psi}_{p}(y)^{*}=\big{(}j_{X}(x)\otimes
i_{G}(p)\big{)}\big{(}j_{X}(y)^{*}\otimes i_{G}(p^{-1})\big{)}$
$\displaystyle=j_{X}(x)j_{X}(y)^{*}\otimes i_{G}(r(p))=j_{X}^{(p)}(x\otimes
y^{*})\otimes i_{G}(r(p)),$

for each $p\in P,x,y\in X_{p}$, thus,

$\psi^{(p)}(S)=j_{X}^{(p)}(S)\otimes i_{G}(r(p)),\ \ (p\in
P,S\in\mathscr{K}_{A_{r(p)}}(X_{p})),$

Now, $\psi$ is CNP-covariant, as $j_{X}$ is so by the definition.

Next, assume that each $\tilde{\phi}_{p}$ is injective, or each bounded subset
of $P$ has a maximal element. Given $u\in G^{(0)}$, let $1_{u}$ be the
characteristic function of singleton $\\{u\\}$, then for each $n\geq 1$,
scalars $c_{1},\cdots c_{n}$, and elements $g_{1},\cdots,g_{n}\in G$ with the
same range,

$\sum_{i,j=1}^{n}c_{i}\bar{c}_{j}1_{u}(g^{-1}_{i}g_{j})=\sum_{(i,j)\in
I}c_{i}\bar{c}_{j}=\sum_{i=1}^{n}|c_{i}|^{2}+2{\rm Re}\big{(}\sum_{(i,j)\in
J}c_{i}\bar{c}_{j}\big{)}\geq 0,$

where $I:=\\{(i,j):g_{i}=g_{j},s(g_{i})=u\\}$ and $J:=\\{(i,j)\in I:i<j\\}$;
that is, $1_{u}$ is positive definite on $G$ [39, Proposition 3], and so
$1_{u}:C_{c}(G)\to\mathbb{C}$ extends to a bounded linear functional on
$C^{*}(G)$ [39, Proposition 4], mapping $i_{G}(u)$ to 1. Now for $a\in A_{u}$,
if $\tilde{\psi}(a)=j_{X}(a)\otimes i_{G}(u)=0$, then, $0=({\rm id}\otimes
1_{u})\tilde{\psi}(a)=j_{X}(a),$ which in turn implies that $a=0$, by
Proposition 2.6. ∎

######  Lemma 3.12.

Let $X$ be compactly aligned. Suppose that either the left action on each
fibre is injective, or $P$ is directed and $X$ is $\tilde{\phi}$-injective.
Then for the canonical coaction

$\delta:\mathcal{T}_{\rm cov}(X)\to\mathcal{T}_{\rm cov}(X)\otimes C^{*}(G);\
i_{X}(x)\mapsto i_{X}(x)\otimes i_{G}(d(x)),\ \ (x\in X),$

induces a coaction of $G$ on $\operatorname{\mathcal{NO}}_{\\!X}$, again
denoted by $\delta$, given by,

$\delta:\operatorname{\mathcal{NO}}_{\\!X}\to\operatorname{\mathcal{NO}}_{\\!X}\otimes\
C^{*}(G);\ j_{X}(x)\mapsto j_{X}(x)\otimes i_{G}(d(x)),\ \ (x\in X),$

which is injective on $q_{\operatorname{CNP}}(\mathcal{F})$.

###### Proof.

Let $\tilde{\psi}:X\to\operatorname{\mathcal{NO}}_{\\!X}$ be the CNP-covariant
representation defined in Lemma 3.11. Then $\Pi\tilde{\psi}=\delta$ on
$\operatorname{\mathcal{NO}}_{\\!X}$, and the result follows from Lemmas 3.10
and 3.11. ∎

Karla Oty introduced the notion of groupoids with uniformly bounded fibres in
her study of the Fourier-Stieltjes algebras of groupoids [37], which are those
with a common upper bound on the cardinality of the sets $G^{u}$ of elements
with a given range $u$. We give examples of such groupoids in Section 5. In
the proof of the next lemma, we use notations of [39].

######  Lemma 3.13.

If $G$ has uniformly bounded fibres with uniform bound $N$, then for each
finite subset $F\subseteq G$, there is a linear functional $1_{F}$ on
$C^{*}(G)$, of norm at most $N$, such that $1_{F}(i_{G}(g))=1$, for each $g\in
F$.

###### Proof.

For $u\in G^{(0)}$, $1_{u}$ is a positive definite function on $G$ by the
proof of Lemma 3.11. For a finite subset $F\subseteq G$, by [39, Theorem 1],
the positive definite function $1_{s(F)}=\sum_{u\in s(F)}1_{u}$, the
coefficient function $(\xi,\xi)$ of some representation $L$ of $G$ on a
$G$-Hilbert bundle $E$. Moreover, $\|\xi\|^{2}=\|1_{s(F)}\|_{\infty}=1$, by
[39, Proposition 4]. Next, it follows from [39, Proposition 10] that the left
convolution by $1_{F}$ extends to a bounded linear operator on $E^{2}$ of norm
at most $\|1_{F}\|_{I,r}=\sup_{u\in G^{(0)}}\sum_{g\in
G^{u}}|1_{F}(g)|=\sup_{u}|G^{u}|\leq N$. For $g\in F$, we have
$1_{g}*1_{u}=1_{g}$, if $u=s(g)$, and zero otherwise, thus
$1_{F}=1_{F}*1_{s(F)}$, which is then the coefficient function
$(\xi,1_{F}*\xi)$ of $L$. Now $L$ integrates to a representation of
$C^{*}(G)$, again denoted by $L$, and we have,

$\langle 1_{F},a\rangle=\langle L(a)\xi,1_{F}*\xi\rangle,\ \ (a\in C^{*}(G)),$

that is, $1_{F}$ could be regarded as a bounded linear functional on
$C^{*}(G)$ of norm at most $\|\xi\|\|1_{F}*\xi\|\leq N$. Moreover,

$\langle 1_{F},f\rangle=\sum_{g\in G}f(g)1_{F}(g)=\sum_{g\in F}f(g),$

for $f\in C_{c}(G)$. In particular, $1_{F}(i_{G}(g))=\langle
1_{F},1_{g}\rangle=1$, for each $g\in F$, as required. ∎

######  Lemma 3.14.

If $G$ has uniformly bounded fibres, the canonical coactions of $G$ on
$\mathcal{T}_{\rm cov}(X)$ and $\operatorname{\mathcal{NO}}_{\\!X}$ are
injective.

###### Proof.

Let $\delta$ be the canonical coaction of $G$ on $\mathcal{T}_{\rm cov}(X)$,
defined by

$\delta(i_{X}(x)i_{X}(y)^{*})=i_{X}(x)i_{X}(y)^{*}\otimes i_{G}(pq^{-1}),\ \
\big{(}x,y\in X,r(d(x))=r(d(y))\big{)}.$

Let $a\in\ker(\delta)$, and given $\varepsilon>0$, choose a finite sum
$b=\sum_{i=1}^{n}i_{X}(x_{i})i_{X}(y_{i})^{*}\in\mathcal{T}_{\rm cov}(X)$,
within $\varepsilon$ of $a$. Let $x_{i}\in X_{p_{i}},y_{i}\in X_{q_{i}}$, for
$p_{i},q_{i}\in P$, and put $g_{i}:=p_{i}q_{i}^{-1}$, for $1\leq i\leq n$.
Since $\delta$ is norm decreasing, $\delta(b)$ is of norm at most
$\varepsilon$. Put $F:=\\{g_{i};1\leq i\leq n\\}$ and let $1_{F}$ be the
bounded functional on $C^{*}(G)$ constructed in Lemma 3.13, then

$\displaystyle(\operatorname{id}\otimes 1_{F})\delta(b)$
$\displaystyle=(\operatorname{id}\otimes
1_{F})\sum_{i}i_{X}(x_{i})i_{X}(y_{i})^{*}\otimes i_{G}(g_{i})$
$\displaystyle=\sum_{i}i_{X}(x_{i})i_{X}(y_{i})^{*}=b,$

is of norm at most $N\varepsilon$. Therefore, $a$ is of norm at most
$(N+1)\varepsilon$. Since $\varepsilon$ was arbitrary, it follows that $a=0$,
as required. The argument for the coaction of $G$ on
$\operatorname{\mathcal{NO}}_{\\!X}$ is similar, with $i_{X}$ replaced by
$j_{X}$. ∎

Finally let us prove a restricted version of nondegeneracy for the canonical
coactions of $G$ on $\mathcal{T}_{\rm cov}(X)$ and
$\operatorname{\mathcal{NO}}_{\\!X}$, as promised.

######  Lemma 3.15.

If $G$ has uniformly bounded fibres, the restriction of the gauge coaction
$\delta$ to the core $\mathcal{F}$ is a coaction of $G^{(0)}$ on
$\mathcal{F}$. Moreover, the restricted $*$-homomorphism
$\delta:\mathcal{F}\to(\mathcal{F}\otimes C^{*}(G))\cap\delta(\mathcal{T}_{\rm
cov}(X))$ is surjective. Similarly, the restriction of the canonical coaction
of $G$ on $\operatorname{\mathcal{NO}}_{\\!X}$ (again denoted by $\delta$) to
$q_{\rm CNP}(\mathcal{F})$ is a coaction of $G^{(0)}$ on $q_{\rm
CNP}(\mathcal{F})$ and $\delta:q_{\rm CNP}(\mathcal{F})\to\big{(}q_{\rm
CNP}(\mathcal{F})\otimes C^{*}(G)\big{)}\cap\delta(\mathcal{T}_{\rm cov}(X))$
is a $*$-epimomorphism.

###### Proof.

It is clear that $\delta(\mathcal{F})\subseteq\mathcal{F}\otimes C^{*}(G)$, so
$\delta$ restricts to a coaction of $G$ on $\mathcal{F}$. Put
$D:=(\mathcal{F}\otimes C^{*}(G))\cap\delta(\mathcal{T}_{\rm cov}(X))$. To
prove surjectivity of the restricted map $\delta:\mathcal{F}\to D$, it is
enough to show that the set of basic tensors
$\delta(i_{X}(x)i_{X}(y)^{*})=i_{X}(x)i_{X}(y)^{*}\otimes i_{G}(r(p))$, with
$p\in P$ and $x,y\in X_{p}$, is a total set in $D$. Given $\varepsilon>0$ and
$b\in D$, choose a finite sum $\sum_{i}i_{X}(x_{i})i_{X}(y_{i})^{*}\otimes
i_{G}(g_{i})\in\mathcal{F}\odot C^{*}(G)$, with $p_{i}\in P$ and
$x_{i},y_{i}\in X_{p_{i}}$, within $\varepsilon$ of $b$. Choose
$a\in\mathcal{T}_{\rm cov}(X)$ with $b=\delta(a)$. Finally, choose a finite
sum $c:=\sum_{j}i_{X}(z_{j})i_{X}(w_{j})^{*}$, with $r_{j},q_{j}\in P$,
$r(r_{j})=r(q_{j})$, and $z_{j}\in X_{r_{j}},w_{j}\in X_{q_{j}}$, within
$\varepsilon$ of $a$. Since $\delta$ is norm decreasing, $b=\delta(a)$ is
within $\varepsilon$ of $\delta(c)=\sum_{j}i_{X}(z_{j})i_{X}(w_{j})^{*}\otimes
i_{G}(r_{j}q^{-1}_{j})$. Thus the following finite sum is of norm at most
$2\varepsilon$,

$d:=\sum_{i}i_{X}(x_{i})i_{X}(y_{i})^{*}\otimes
i_{G}(g_{i})-\sum_{j}i_{X}(z_{j})i_{X}(w_{j})^{*}\otimes
i_{G}(r_{j}q^{-1}_{j}).$

For the finite set $F\subseteq G$ including elements $g_{i}$ and
$r_{j}q^{-1}_{j}$, for each $i$ and $j$, let $1_{F}$ be the bounded linear
functional of norm at most $N$ constructed in the proof of Lemma 3.13. Then,

$\langle\operatorname{id}\otimes
1_{F},b\rangle=\sum_{i}i_{X}(x_{i})i_{X}(y_{i})-\sum_{j}i_{X}(z_{j})i_{X}(w_{j})^{*}$

is of norm at most $2N\varepsilon$. Applying $\delta$ to the right hand side,
the following finite sum is of norm at most $2N\varepsilon$,

$\sum_{i}i_{X}(x_{i})i_{X}(y_{i})^{*}\otimes
i_{G}(r(p_{i}))-\sum_{j}i_{X}(z_{j})i_{X}(w_{j})^{*}\otimes
i_{G}(r_{j}q^{-1}_{j}).$

Therefore, $b$ is within $(2N+1)\varepsilon$ of
$\sum_{i}i_{X}(x_{i})i_{X}(y_{i})^{*}\otimes i_{G}(r(p_{i}))$, proving the
claim. The proof for $\operatorname{\mathcal{NO}}_{\\!X}$ is similar, with
$i_{X}$ replaced by $j_{X}$ in the above argument. ∎

##  4\. gauge-invariant uniqueness theorem

In this section we give a gauge-invariant uniqueness theorem for product
systems over quasi-lattice ordered groupoids. A Toeplitz representation
$\psi\colon X\to D$ is gauge-compatible if there is a coaction $\beta$ of $G$
on $D$ such that $\beta(\psi(x))=\psi(x)\otimes i_{G}(d(x))$, for $x\in X$.
For gauge-compatible Toeplitz representations $\psi_{i}:X\to D_{i}$, with
coaction $\beta_{i}$ of $G$ on $D_{i}$ satisfying
$\beta_{i}(\psi_{i}(x))=\psi_{i}(x)\otimes i_{G}(d(x))$, for $x\in X$ and
$i=1,2$, an $*$-homomorphism $\phi:D_{1}\to D_{2}$ is equivariant if
$\phi\circ\psi_{1}=\psi_{2}$ and
$(\phi\otimes\operatorname{id})\circ\beta_{1}=\beta_{2}\circ\phi$.

The main result of this section is an extension of [7, Theorem 4.1].

######  Theorem 4.1.

Let $G$ be discrete and $X$ be compactly aligned. Suppose that either the left
action on each fibre is injective, or $P$ is directed and $X$ is
$\tilde{\phi}$-injective. Then there exists a triple
$(\operatorname{\mathcal{NO}}_{\\!X}^{r},j_{X}^{r},\delta^{n})$ which is
couniversal for gauge-compatible injective Nica covariant representations of
$X$ in the following sense:

  1. (1)

$\operatorname{\mathcal{NO}}_{\\!X}^{r}$ is a $C^{*}$-algebra, $j^{r}_{X}$ is
an injective Nica covariant representation of $X$ whose image generates
$\operatorname{\mathcal{NO}}_{\\!X}^{r}$, and $\delta^{n}$ is a normal
coaction of $G$ on $\operatorname{\mathcal{NO}}_{\\!X}^{r}$ with
$\delta^{n}(j^{r}_{X}(x))=j^{r}_{X}(x)\otimes i_{G}(d(x))$, for $x\in X$.

  2. (2)

If $\psi\colon X\to D$ is an injective Nica covariant gauge-compatible
representation whose image generates $D$ then there is a surjective
$*$-homomorphism $\phi\colon D\to\operatorname{\mathcal{NO}}_{\\!X}^{r}$ such
that $\phi(\psi(x))=j^{r}_{X}(x)$, for $x\in X$.

Moreover, the representation $j^{r}_{X}$ is CNP-covariant, the coaction
$\delta^{n}$ is normal, and
$(\operatorname{\mathcal{NO}}_{\\!X}^{r},j^{r}_{X},\nu^{n})$ is the unique
triple satisfying (1) and (2), in the sense that, if $(C,\rho,\gamma)$
satisfies the same two conditions, then there is an isomorphism $\phi\colon
C\to\operatorname{\mathcal{NO}}_{\\!X}^{r}$ such that
$j^{r}_{X}=\phi\circ\rho$ and $\phi$ is equivariant for $\gamma$ and
$\nu^{n}$.

To prove Theorem 4.1, we essentially adapt argument of the proof of [7,
Theorem 4.1] to the groupoid case. We first need to recall some facts about
Fell bundles over groupoids and their cross-sectional $C^{*}$-algebras [31].
Then we need to slightly extend the known results on the relationship between
topologically graded $C^{*}$-algebras and $C^{*}$-algebras associated to Fell
bundles as in [18], to the case where both gradings and bundles are over
groupoids. Finally we need to extend the connection between discrete coactions
and Fell bundles, as explored in [42], to this more general setting.

In [42, Definition 3.5], Quigg introduced the notion of the reduced
$C^{*}$-algebra of a Fell bundle over a discrete group and its coaction. Note
that the constructions in [18] and [42] are compatible [17, page 749]. We need
to extend some of these results to our setting.

Let us recall the notion of a Fell bundle over an étale groupoid $G$ [31],
[56]. Let $p:E\to G$ be a Banach bundle, and set

$E^{2}=\\{(e_{1},e_{2})\in E\times E\mid(p(e_{1}),p(e_{2}))\in G^{(2)}\\}.$

A multiplication on $E$ is a continuous map: $E^{2}\to E;(e_{1},e_{2})\mapsto
e_{1}e_{2}$, satisfying:

$(i)$ $p(e_{1}e_{2})=p(e_{1})p(e_{2})$,

$(ii)$ the induced map: $E_{g}\times E_{h}\to E_{gh}$ is bilinear,

$(iii)$ $(e_{1}e_{2})e_{3}=e_{1}(e_{2}e_{3})$,

$(iv)$ $\|e_{1}e_{2}\|\leq\|e_{1}\|\|e_{2}\|,$

for $e_{1},e_{2},e_{3}\in E$ and $g,h\in G$, whenever the multiplications in
$E$ or $G$ are defined.

An involution on $E$ is a continuous map: $E\to E;e\mapsto e^{*}$, satisfying:

$(v)$ $p(e^{*})=p(e)^{-1}$,

$(vi)$ the induced map: $E_{g}\to E_{g^{-1}}$ is conjugate linear,

$(vii)$ $e^{**}=e,$

for $e\in E,g\in G$. The bundle $E$ together with these structure maps is
called a Fell bundle over $G$ if in addition the following conditions hold:

$(viii)$ $(e_{1}e_{2})*=e^{*}_{1}e^{*}_{2}$,

$(ix)$ $\|e^{*}e\|=\|e\|^{2}$,

$(x)$ $e^{*}e\geq 0$,

for $e,e_{1},e_{2}\in E$, whenever the multiplications in the first equality
are defined. We say that $E$ is nondegenerate if all fibres $E_{g}$ are
nonzero, and saturated if $E_{g}E_{h}$ is total in $E_{gh}$, for $(g,h)\in
G^{(2)}$. Let $E^{0}$ be the restriction of the bundle $E$ to $G^{(0)}$.

Next let us define the full and reduced cross-sectional $C^{*}$-algebra of $E$
[56], [31]. Regarding $E$ as a Banach bundle over $G$, let $C_{c}(E)$ and
$C_{0}(E)$ be respectively the normed and Banach spaces consisting of
continuous sections which are respectively compactly supported and vanishing
at infinity. Given $h,h_{1},h_{2}\in C_{c}(E)$, define the convolution and
involution by,

$h_{1}*h_{2}(g)=\sum_{g=g_{1}g_{2}}h_{1}(g_{1})h_{2}(g_{2}),\ \
h^{*}(g)=h(g^{-1})^{*},\ \ (g\in G),$

where the sum runs over pairs for which the multiplication is defined. Then
$C_{c}(E)$ is a $*$-algebra. Note that each fibre $E_{g}$ is a Hilbert
$E_{r(g)}$-$E_{s(g)}$-bimodule with inner products $\langle
e_{1},e_{2}\rangle_{E_{s(g)}}:=e^{*}_{1}e_{2}$ and ${}_{E_{r(g)}}\langle
e_{1},e_{2}\rangle:=e_{1}e^{*}_{2}$. Let $P:C_{c}(E)\to C_{c}(E^{0})$ be the
restriction map, and consider the $C_{c}(E^{0})$-valued inner product on
$C_{c}(E)$ defined by

$\langle h_{1},h_{2}\rangle:=P(h_{1}*h_{2}),\ \ (h_{1},h_{2}\in
C_{c}(E^{0})),$

turning it to a pre-Hilbert $C_{0}(E^{0})$-module [31, Proposition 3.2].

Let $L^{2}(E)$ be the completion of $C_{c}(E)$ in the norm coming from the
above $C_{c}(E^{0})$-valued inner product. Then $L^{2}(E)$ is the canonical
Hilbert module associated to the $G^{(0)}$-bundle of Hilbert modules [31, 1.7]
with fibres $V_{u}:=\bigoplus_{s(g)=u}E_{g}$. When moreover $E$ is saturated,
$V_{r(g)}\otimes_{E_{r(g)}}E_{g}\simeq V_{s(g)}$, for $g\in G$ [31, 3.3]. The
left multiplication by elements of $C_{c}(E)$ is adjointable and induces a
$*$-monomorphism
$\lambda_{E}:C_{c}(E)\hookrightarrow\mathscr{L}_{C_{0}(E^{0})}(L^{2}(E))$. The
reduced cross-sectional $C^{*}$-algebra $C^{*}_{r}(E)$ of $E$ is the closure
of the copy of $C_{c}(E)$ inside $\mathscr{L}_{C_{0}(E^{0})}(L^{2}(E))$ under
the above monomorphism. In particular, $C^{*}_{r}(E^{0})=C_{0}(E^{0})$. For
$u\in G^{(0)}$, we have a representation
$\pi_{u}:C^{*}_{r}(E)\to\mathscr{L}_{E_{u}}(V_{u})=:\mathscr{L}_{u}$, and the
reduced norm is given by $\|a\|_{r}=\sup_{u\in
G^{(0)}}\|\pi_{u}(a)\|_{\mathscr{L}_{u}}$. Each $h\in C_{b}(G^{(0)})$ is
identified with an element $M_{g}$ of the multiplier algebra
$M(C^{*}_{r}(E))$, satisfying $M_{h}f(g)=h(r(g))f(g)$, for $g\in G,f\in
C_{c}(E)$. The restriction map $P:C_{c}(E)\to C_{c}(E^{0})$, extends to a
faithful conditional expectation $\Phi_{r}:C^{*}_{r}(E)\to C_{0}(E^{0})$ [31,
Propositions 3.6, 3.10]. For the full cross-sectional $C^{*}$-algebra
$C^{*}(E)$, we simply complete $C_{c}(E)$ in the maximal $C^{*}$-norm coming
from bounded representations of this $*$-algebra [56]. The above monomorphism
extends to a $*$-epimorphism $\lambda_{E}:C^{*}(E)\twoheadrightarrow
C^{*}_{r}(E)$, called the regular representation.

Following [42], let us next define the notion of a representation for Fell
bundles over groupoids. Let $E$ and $F$ be Fell bundles over étale groupoids
$G$ and $H$. Let $M(F)$ be the multiplier bundle of $F$ [20, VIII.2.14] with
fibres $M_{h}(F)$, $h\in H$. We identify $M_{v}(F)$ with $M(F_{v})$, for $v\in
F^{(0)}$. A map $\phi:E\to M(F)$ is called a morphism if there is a (not
necessarily unique) continuous groupoid homomorphism $\eta:G\to H$ such that,

$(i)$ $\phi|_{E_{g}}:E_{g}\to M_{\eta(g)}(F)$ is linear,

$(ii)$ $\phi(ab)=\phi(a)\phi(b),\phi(a^{*})=\phi(a)^{*}$,

$(iii)$ $\phi|_{E_{u}}:E_{u}\to M(F_{\eta(u)})$ is nondegenerate,

for $g\in G$, $u\in G^{(0)}$, and $a,b\in E,$ with $(a,b)\in E^{2}$. When
$E=F$ and $\eta={\operatorname{id}}_{G}$, this is just a $G$-bundle morphism.
For a $C^{*}$-algebra $D$, a representation of $E$ in $M(D)$ is a morphism
$\phi:E\to M(D)$, with $D$ regarded as a trivial $G$-bundle. A representation
$\phi:E\to M(D)$ lifts uniquely to a nondegenerate $*$-homomorphism
$\phi:C^{*}(E)\to M(D)$. When $\phi(E)\subseteq D$, we call $\phi$ a
representation of $E$ in $D$.

######  Lemma 4.2.

Let $G$ be a discrete groupoid.

$(i)$ If $G$ coact by $\delta$ on a $C^{*}$-algebra $A$, and
$A^{\delta}_{g}:=\\{a\in A\mid\delta(a)=a\otimes i_{G}(g)\\}$ is the spectral
subspace of $A$ at $g\in G$. The Banach bundle with fibre
$A^{\delta}_{g}\times\\{g\\}$ at $g$ is a Fell bundle over $G$.

$(ii)$ If $E$ is a Fell bundle over $G$, there is a canonical coaction
$\delta_{E}$ on the full cross-sectional algebra $C^{*}(E)$ such that
$\delta_{E}(a)=a\otimes i_{G}(g)$, for $g\in G$, $a\in E_{g}$.

###### Proof.

$(i)$ Let $E_{A}$ be the disjoint union of the fibres
$A^{\delta}_{g}\times\\{g\\}$ (identified with $A^{\delta}_{g}$) and
$p:E_{A}\to G$ be the projection on the second leg. Then $E^{2}_{A}$ consists
of all pairs $(a,b)$ with $a\in A^{\delta}_{g}$ and $b\in A^{\delta}_{h}$, for
some $g,h\in G$ with $r(h)=s(g)$. All the required conditions listed above now
follow from the fact that $A$ is a $C^{*}$-algebra.

$(ii)$ The morphism $\delta_{E}:E\to C^{*}(E)\otimes C^{*}(G)$ lifts to a
representation $\delta_{E}:C^{*}(E)\to C^{*}(E)\otimes C^{*}(G)$. The coaction
identity follows as it is trivial on elements of $E$. Let us observe that
$\delta_{E}$ is nondegenerate (mod $G^{(0)}$). For $g,h\in G$ and a basic
tensor $a\otimes i_{G}(h)$ with $a\in E_{g}$, consider the projection
$p:=1_{r(g)}$ for finite subset $F\subseteq G^{(0)}$, then
$(1_{M(C^{*}(E))}\otimes p)(a\otimes i_{G}(h))$ is nonzero when $r(h)=r(g)$,
in which case, for an approximate identity $(e_{i})$ of $C^{*}(G)$,

$\displaystyle\delta_{E}(a)(e_{i}\otimes i_{G}(g^{-1}h))$
$\displaystyle=(a\otimes i_{G}(g))(e_{i}\otimes i_{G}(g^{-1}h))=ae_{i}\otimes
i_{G}(r(g)h)$ $\displaystyle=ae_{i}\otimes i_{G}(r(h)h)=ae_{i}\otimes
i_{G}(h)\to a\otimes i_{G}(h),$

as $i\to\infty$, as required. ∎

If $E$ is a Fell bundle over $G$, a cross-sectional algebra of $E$ is the
$C^{*}$-completion $C^{*}_{\gamma}(E)$ of the algebra $C_{c}(E)$ with respect
to a $C^{*}$-norm $\gamma$. The bundle $E$ is topologically graded if there
exists a contractive conditional expectation from $E$ to $E_{u}$ vanishing on
each $E_{g}$, for $g\in G\backslash G^{(0)}$ (compare to [18, Definition
3.4]). For topologically graded bundles, the reduced cross-sectional algebra
$C^{*}_{r}(E)$ is minimal among all cross-sectional algebras
$C^{*}_{\gamma}(E)$. Indeed, by an argument as in [18, Theorem 3.3], there
exists a surjective homomorphism $\lambda_{\gamma}:C^{*}_{\gamma}(E)\to
C^{*}_{r}(E)$ such that $\lambda_{\gamma}\circ\iota_{\gamma}=\iota_{\rm min}$,
where $\iota_{\gamma}$ and $\iota_{r}$ are the embeddings of $C_{c}(E)$ into
$C^{*}_{\gamma}(E)$ and $C^{*}_{r}(E)$, respectively. On the other hand, since
each representation of $C_{c}(E)$ extends to a representation of $C^{*}(E)$,
the inclusion $\iota_{\gamma}$ extends to a $*$-epimorphism
$\rho_{\gamma}\colon C^{*}(E)\to C^{*}_{\gamma}(E)$ such that
$\rho_{\gamma}\circ\iota_{\rm max}=\iota_{\gamma}$ where
$\gamma_{\mathcal{A}}$ is the embedding of the algebra of finitely supported
sections on $\mathcal{A}$ into $C^{*}(\mathcal{A})$. In particular,
$\lambda_{\rm max}=\rho_{\rm min}=\lambda_{E}$ is the regular representation.

Let $\delta$ be a coaction of an étale groupoid $G$ on a $C^{*}$-algebra $A$,
and $B$ be a unital $C^{*}$-algebra. Each nondegenerate $*$-homomorphism
$\sigma:C_{0}(G)\to B$, with $\sigma|_{C_{0}(G^{(0)})}\equiv 1_{B}$, induces a
coaction $\delta^{\sigma}$ of $G$ on $B$, defined by

$\delta^{\sigma}:B\to B\otimes C^{*}(G);\ \ b\mapsto{\rm
Ad}_{(\sigma\otimes{\rm id})(i_{G})}(b\otimes 1),$

where $i_{G}$ is the element in $M(C_{0}(G)\otimes C^{*}(G))$ associated to
the function $g\mapsto i_{G}(g)$ in $C_{b}(G,C^{*}(G))$. The element
$u:=(\sigma\otimes{\rm id})(i_{G})$ is a unitary element in $M(B\otimes
C^{*}(G))$, since $i_{G}(g)i^{*}_{G}(g)=i_{G}(s(g))$ and
$i^{*}_{G}(g)i_{G}(g)=i_{G}(r(g))$, that is, $i_{G}i^{*}_{G},i^{*}_{G}i_{G}\in
C_{b}(G,C_{0}(G^{(0)}))$, and so $(\sigma\otimes{\rm
id})(i_{G}i^{*}_{G})=(\sigma\otimes{\rm id})(i^{*}_{G}i_{G})=1$ in $M(B\otimes
C^{*}(G))$. In particular, $\delta^{\sigma}$ is a homomorphism. The coaction
identity is immediate, and nondegeneracy follows by an argument verbatim to
that of [43, Lemma 1.11].

For a $C^{*}$-algebra $D$, a covariant representation of $(A,G,\delta)$ in
$M(B)$ is a pair $(\pi,\sigma)$ of nondegenerate representations $\pi:A\to
M(D)$ and $\sigma:C_{0}(G)\to M(D)$, with $\sigma|_{C_{0}(G^{(0)})}\equiv
1_{M(D)}$, such that $(\pi\otimes{\rm
id})\circ\delta=\delta^{\sigma}\circ\pi$, that is $\pi:A\to M(D)$ is
equivariant for the coactions $\delta$ and $\delta^{\sigma}$.

######  Lemma 4.3.

Let $\delta$ be a coaction of a discrete groupoid $G$ on $A$ and $E_{A}$ be
the corresponding Fell bundle, then,

$(i)$ $E_{A}$ is topologically graded,

$(ii)$ the full cross-sectional $C^{*}$-algebra of $E_{A}$ is isomorphic to
$A$.

###### Proof.

$(i)$ Given $u\in G^{(0)}$, we need to build a contractive conditional
expectation onto $(E_{A})_{u}$ vanishing on fibres over $G\backslash G^{(0)}$.
Let $1_{u}$ be the characteristic function of singleton $\\{u\\}$, and extend
$1_{u}:C_{c}(G)\to\mathbb{C}$ to a bounded linear functional on $C^{*}(G)$ as
in Lemma 3.11. Let $B(G)$ consist of coefficient functions of representations
of $G$ on Hilbert bundles, and for $f\in B(G)$, put $\delta_{f}:=({\rm
id}\otimes f)\circ\delta$, then we claim that $\delta_{u}:=\delta_{1_{u}}:A\to
A$ is the desired conditional expectation. Indeed, we only need to check that
$\delta_{u}(A)=A^{\delta}_{u}$. Let $a\in A$ and put $b:=\delta_{u}(a)$, then,

$\displaystyle({\rm id}\otimes f)[\delta(b)-b\otimes i_{G}(u)]$
$\displaystyle=({\rm id}\otimes f)\delta\big{(}({\rm id}\otimes
1_{u})\delta(a)\big{)}-f(u)b$ $\displaystyle=({\rm id}\otimes{\rm id}\otimes
f)({\rm id}\otimes{\rm id}\otimes 1_{u})(\delta\otimes{\rm
id})\delta(a)-f(u)b$ $\displaystyle=({\rm id}\otimes{\rm id}\otimes
f1_{u})({\rm id}\otimes\delta_{G})\delta(a)-f(u)b$ $\displaystyle=f(u)({\rm
id}\otimes 1_{u})\delta(a)-f(u)b=0,$

and since the coefficient functions of the representations separate the points
of $G$, we conclude that $\delta(b)=b\otimes i_{G}(u)$, i.e.,
$\delta_{u}(A)\subseteq A^{\delta}_{u}$. The reverse inclusion is immediate.

$(ii)$ Let $g\in G$, then $1_{g}*1_{s(g)}=1_{g}$, thus $1_{g}\in B(G)$ by [39,
Proposition 10]. An argument as in $(i)$, shows that
$\delta_{g}(A):=\delta_{1_{g}}(A)=A^{\delta}_{g}$. Now the result follows from
universality of cross sectional $C^{*}$-algebras of the corresponding Fell
bundles $(A^{\delta}_{g}\times\\{g\\})_{g\in G}$ and
$(\delta_{g}(A)\times\\{g\\})_{g\in G}$. ∎

######  Definition 4.4.

Let $\delta$ be a coaction of a discrete groupoid $G$ on a $C^{*}$-algebra
$A$. Let $I$ be a closed ideal of $A$. We say that $I$ is an induced ideal
(with respect to $\delta$) if $\bigcup_{u\in G^{(0)}}A(I\cap A^{\delta}_{u})A$
is a total set in $I$. If moreover, the restricted map

$\delta:I\to(I\otimes C^{*}(G))\cap\delta(A)$

is nondegenerate as a $*$-homomorphism between $C^{*}$-algebras, we say that
$I$ is a strongly induced ideal (with respect to $\delta$).

Two comments on the above definition is in order: First, if $I$ is an induced
ideal, then $\delta(I)\subseteq(I\otimes C^{*}(G))\cap\delta(A)$, therefore
$\delta$ restricts to a coaction of $G$ on $I$ and the restricted homomorphism
in the second part of the above definition is well-defined. Second, since the
group coactions are assumed to be nondegenerate, the induced ideals are
automatically strongly induced in the group case.

######  Lemma 4.5.

For a $C^{*}$-algebra $A$ with a coaction $\delta$ of a discrete groupoid $G$,

$(i)$ the set $\sum_{g\in G}A^{\delta}_{g}$ consisting of finite sums of
elements in fibres is dense in $A$ and elements in distinct fibres are
linearly independent,

$(ii)$ if $I$ is an induced ideal of $A$ then $A/I$ is topologically graded,

$(iii)$ if $I$ is an induced ideal of $A$ then there is a coaction
$\delta^{I}$ of $G$ on $A/I$ such that $\delta^{I}\circ
q=(q\otimes\operatorname{id})\circ\delta$, where $q:A\to A/I$ is the quotient
map.

###### Proof.

$(i)$ In the notations of the proof of Lemma 4.3, since $B(G)$ separates the
points of $G$ and $C_{c}(G)\cap B(G)$ is dense in $B(G)$, the set
$\\{\delta_{f}(a):a\in A,f\in C_{c}(G)\cap B(G)\\}$ is a total set in $A$, and
so is the set $\bigcup_{g\in G}A^{\delta}_{g}=\\{\delta_{1_{g}}(a):g\in G,a\in
A\\}$. Also, if $a=\sum_{g\in F}a_{g}$ is a finite sum, then for $u\in r(F)$,
put $F_{u}:=\\{g\in F:r(g)=u\\}$ and observe that,
$\delta_{u}(aa^{*})=\sum_{g\in F_{u}}a_{g}a^{*}_{g}$. If $a=0$, then for each
$u\in r(F)$, $a_{g}^{*}a_{g}=0$ in the $C^{*}$-subalgebra
$\delta_{u}(A)=A^{\delta}_{u}$, hence $a_{g}=0$, for each $g\in F$,
establishing the linear independence.

$(ii)$ Let $\delta_{g}:=\delta_{1_{g}}$ be defined as in the proof of lemma
4.3. We claim that $I$ is fixed by each $\delta_{g}$. Given $u\in G^{(0)}$ and
a finite sum $a=\sum_{i}b_{g_{i}}a_{i}c_{h_{i}}$ in $A(I\cap
A^{\delta}_{u})A$, with $b_{g_{i}}\in A^{\delta}_{g_{i}},a_{i}\in
A^{\delta}_{u},c_{h_{i}}\in I\cap A^{\delta}_{h_{i}}$, with
$r(g_{i})=s(h_{i})=u$, for a given $g\in G$ we have,

$\displaystyle\delta_{g}(a)$ $\displaystyle=({\rm id}\otimes
1_{g})\delta(a)=({\rm id}\otimes
1_{g})\sum_{i}\delta(b_{g_{i}})\delta(a_{i})\delta(c_{h_{i}})$
$\displaystyle=({\rm id}\otimes 1_{g})\sum_{i}b_{g_{i}}a_{i}c_{h_{i}}\otimes
i_{G}(g_{i}h_{i})=\sum_{g_{i}h_{i}=g}b_{g_{i}}a_{i}c_{h_{i}},$

and the last term is in $I$, as $I$ is an ideal. This proves the claim, since
the set of all such finite sums are dense in $I$ by the assumption of $I$
being a reduced. Let $q:A\to A/I$ be the quotient map. We show that
$q(A^{\delta}_{g})$ is closed in $A/I$. For $a\in I$ and $b\in
A^{\delta}_{g}$,

$\|a+b\|\geq\|\delta_{g}(a+b)\|=\|\delta_{g}(a)+b\|,$

with $\delta_{g}(a)\in I\cap A^{\delta}_{g}$, thus $\|b+I\|=\|b+(I\cap
A^{\delta}_{g})\|,$ that is, the canonical map: $A^{\delta}_{g}/(I\cap
A^{\delta}_{g})\to A/I$ is isometric, and so
$q(A^{\delta}_{g})=A^{\delta}_{g}/(I\cap A^{\delta}_{g})$ is a Banach space,
as required. Finally, $\delta_{g}$ lifts to a projection
$\delta^{I}_{g}:A/I\to A^{\delta}_{g}/(I\cap A^{\delta}_{g})$, and for each
$u\in G^{(0)}$, the projection $\delta^{I}_{u}:A/I\to A^{\delta}_{u}/(I\cap
A^{\delta}_{u})$ is a contractive conditional expectation, vanishing on each
$A^{\delta}_{g}/(I\cap A^{\delta}_{g})$, for $g\in G\backslash G^{(0)}$.

$(iii)$ Since $q$ is surjective, the above equality defines a linear map
$\delta^{I}$. For $u\in G^{(0)}$ and $a\in A^{\delta}_{u}$,

$(q\otimes\operatorname{id})\circ\delta(a)=q(a)\otimes i_{G}(u)=0,$

if and only if $a\in I$, thus
$\ker((q\otimes\operatorname{id})\circ\delta)\cap A^{\delta}_{u}=I\cap
A^{\delta}_{u}$. It follows from the assumption on $I$ that
$\ker\big{(}(q\otimes\operatorname{id})\circ\delta\big{)}$ contains $I$, and
so $\delta^{I}$ is well-defined, and clearly a $*$-homomorphism.

To show nondegeneracy (mod $G^{(0)}$), given $n\geq 1$, let
$y_{1},\cdots,y_{n}\in A/I,$ $g_{1},\cdots g_{n}\in G$, and consider the basic
tensors $y_{i}\otimes 1_{g_{i}}\in A/I\otimes C^{*}(G)$. Choose $x_{i}\in A$
with $q(x_{i})=y_{i}$, for $1\leq i\leq n$, and choose units $u_{i}\in
G^{(0)}$ such that $\sum_{i}x_{i}\otimes(1_{u_{i}}*1_{g_{i}})$ is in the
closed span of $\delta(A)(A\otimes C^{*}(G))$, then since $q$ is norm
decreasing,
$\sum_{i}y_{i}\otimes(1_{u_{i}}*1_{g_{i}})=(q\otimes\operatorname{id})\big{(}\sum_{i}x_{i}\otimes(1_{u_{i}}*1_{g_{i}})\big{)}$
is in the closed span of

$(q\otimes\operatorname{id})\delta(A)(A\otimes
C^{*}(G))=\delta^{I}(A)(A/I\otimes C^{*}(G)),$

thus $\delta^{I}$ is nondegenerate (mod $G^{(0)}$).

Finally, for $g\in G$ and $a\in A^{\delta}_{g}$,

$(\operatorname{id}\otimes\delta_{G})\circ\delta^{I}(q(a))=q(a)\otimes
i_{G}(g)\otimes
i_{G}(g)=(\delta^{I}\otimes\operatorname{id})\circ\delta^{I}(q(a)),$

and again by a density argument, we get the coaction identity for
$\delta^{I}$. ∎

######  Definition 4.6.

Let $\delta$ be the coaction of a discrete groupoid $G$ on a $C^{*}$-algebra
$A$. Let $\lambda_{G}:C^{*}(G)\to C^{*}_{r}(G)$ be the regular representation.
The coaction $\delta$ is called normal if
$\operatorname{id}\otimes\lambda_{G}$ is injective on $\delta(A)$.

When $\delta$ is injective (something which is always assumed to be the case
for group coactions), this is equivalent to the condition that
$(\operatorname{id}\otimes\lambda_{G})\circ\delta$ is injective on $A$. If
$\delta$ is injective on a $C^{*}$-subalgebra $B$ of $A$, then normality of
$\delta$ implies that $(\operatorname{id}\otimes\lambda_{G})\circ\delta$ is
also injective on $B$.

For the sake of brevity, let us write
$\tilde{\delta}:=(\operatorname{id}\otimes\lambda_{G})\circ\delta$. For $u\in
G^{(0)}$, we put,

$A^{\delta}_{u}:=\\{a\in A:\delta(a)=a\otimes i_{G}(u)\\},\
A^{\tilde{\delta}}_{u}:=\\{a\in
A:\tilde{\delta}(a)=a\otimes\lambda_{G}(u)\\}.$

Next, first note that in the notations of [55, page 18], for $u\in G^{(0)}$
and $f\in C_{c}(G)$,

$\displaystyle|\langle 1_{u},f\rangle|:=|f(u)|$
$\displaystyle\leq\sum_{s(g)=u}|f(g)|=\|{\rm
Ind}\delta_{u}(f)1_{u}\|_{2}^{2}\leq\|{\rm Ind}\delta_{u}(f)\|\leq\|f\|_{r},$

that is, $1_{u}$ could be regarded as a continuous linear functional on
$C^{*}_{r}(G)$. We denote this functional by $\tilde{1}_{u}$ to distinguish it
from the functional $1_{u}$ on $C^{*}(G)$ constructed in the proof of Lemma
4.14 (indeed, $1_{u}=\tilde{1}_{u}\circ\lambda_{G}$). We also consider the
bounded linear maps on $A$, defined by
$\delta_{u}:=\delta_{1_{u}}=(\operatorname{id}\otimes 1_{u})\circ\delta$.
Recall that $\delta_{u}(A)=A^{\delta}_{u}$, and we have a conditional
expectation,

$\Phi_{0}:=\oplus\delta_{u}:A\to\bigoplus_{u\in G^{(0)}}A^{\delta}_{u}.$

We also have a conditional expectation $\Phi:C^{*}(G)\to C_{0}(G^{(0)})$,
mapping $f\in C_{c}(G)$ to $\sum_{u\in G^{(0)}}f(u)$, or alternatively, if we
identify $C_{0}(G^{(0)})$ with the $c_{0}$-direct sum $\bigoplus_{u\in
G^{(0)}}\mathbb{C}$, $\Phi$ is the map sending $f\in C_{c}(G)$ to
$(f(u))_{u\in G^{(0)}}$. Under this identification, $\Phi=\oplus 1_{u}$, where
$1_{u}:f\mapsto f(u)$ is the above state on $C^{*}(G)$. This means that
$\Phi_{0}=(\operatorname{id}\otimes\Phi)\circ\delta$. In the group case, where
coactions are assumed to be injective, $\Phi_{0}$ is faithful on positive
elements iff $\Phi$ is so, but in our setting faithfulness of $\Phi$ on
positive elements is in general weaker than that of $\Phi_{0}$. Indeed, we
have the following result.

######  Lemma 4.7.

Let $G$ be a discrete groupoid, then the set $\\{\tilde{1}_{u}:u\in
G^{(0)}\\}$ is a faithful family of states on $C^{*}_{r}(G)$.

###### Proof.

. First note that

$\langle\tilde{1}_{u},f\rangle=f(u)=\langle{\rm
Ind}(\delta_{u})(f)1_{u},1_{u}\rangle,\ \ (u\in G^{(0)},f\in C_{c}(G)),$

where $1_{u}$ is regarded as an element in $\ell^{2}(G_{u})$. This in
particular shows that $\tilde{1}_{u}$ extends to a state on $C^{*}_{r}(G)$.

We just need to show that if $\langle\tilde{1}_{u},b\rangle=0$, for $b\in
C^{*}_{r}(G)_{+}$ and each $u\in G^{(0)}$, then $b=0$. Without loss of
generality, we may assume that $b$ is of norm at most one. Choose element $c$
with $b=c^{*}c$, then $c$ is of norm at most one. Next, for a given
$\varepsilon>0$, choose $f\in C_{c}(G)$ within $\varepsilon$ of $c$, and
observe that $f^{*}*f$ is within $3\varepsilon$ of $b$. Now,

$\sum_{s(\gamma)=u}|f(\gamma)|^{2}=\sum_{r(\gamma)=u}\bar{f}(\gamma^{-1})f(\gamma^{-1}u)=f^{*}*f(u)=\langle\tilde{1}_{u},f^{*}*f-b\rangle\leq
3\varepsilon,$

for each $u\in G^{(0)}$, which in turn implies that
$\|f\|_{\infty}\leq\sqrt{3\varepsilon}$, and so $f=0$, as $\varepsilon$ was
arbitrary, thus $b=0$. ∎

######  Lemma 4.8.

Let a discrete groupoid $G$ coact on a $C^{*}$-algebra $A$ with $\delta$.

  1. $(i)$

The coaction $\delta$ is normal iff the conditional expectation

$\operatorname{id}\otimes\Phi:A\otimes C^{*}(G)\to A\otimes C_{0}(G)$

is faithful on positive cone of $\delta(A)$. When $\delta$ is injective, this
is also equivalent to faithfulness of the conditional expectation

$\Phi_{0}:=\oplus\delta_{u}:A\to\bigoplus_{u\in G^{(0)}}A^{\delta}_{u},$

on positive cone of $A$.

  2. $(ii)$

Let $I$ be an induced ideal of $A$ with quotient map $q:A\to A/I$. The
coaction $\delta^{I}$ is normal iff
$q^{-1}\big{(}\ker(\delta^{I})\big{)}=\ker((q\otimes\lambda_{G})\circ\delta)$.
When $\delta^{I}$ is injective, this is also equivalent to
$I=\ker((q\otimes\lambda_{G})\circ\delta)$.

###### Proof.

$(i)$ If $\operatorname{id}\otimes\Phi$ is faithful on $\delta(A_{+})$, given
$a\in A$, if $(\operatorname{id}\otimes\lambda_{G})\circ\delta(a)=0$, then,

$(\operatorname{id}\otimes\lambda_{G})\circ\delta(a^{*}a)=[(\operatorname{id}\otimes\lambda_{G})\circ\delta(a^{*})][(\operatorname{id}\otimes\lambda_{G})\circ\delta(a)]=0,$

and for $u\in G^{(0)}$,

$(\operatorname{id}\otimes
1_{u})\circ\delta(a^{*}a)=(\operatorname{id}\otimes\tilde{1}_{u}\circ\lambda_{G})\circ\delta(a^{*}a)=(\operatorname{id}\otimes\tilde{1}_{u})\circ(\operatorname{id}\otimes\lambda_{G})\circ\delta(a^{*}a)=0,$

thus, $(\operatorname{id}\otimes\Phi)\big{(}\delta(a^{*}a)\big{)}=0$, thus
$\delta(a^{*}a)=0$, and so $\delta(a)=0$, i.e.,
$\operatorname{id}\otimes\lambda_{G}$ is injective on $\delta(A)$, that is,
$\delta$ is normal. Conversely, arguing by contradiction, if
$(\operatorname{id}\otimes\Phi)\big{(}\delta(a^{*}a)\big{)}=0$, but
$(\operatorname{id}\otimes\lambda_{G})\circ\delta(a^{*}a)$ is a nonzero
positive element of $A\otimes C^{*}_{r}(G)$. Choose a state $\omega$ of $A$
such that $b:=(\omega\otimes\lambda_{G})\circ\delta(a^{*}a)$ is a nonzero
positive element of $C^{*}_{r}(G)$. For each $u\in G^{(0)}$,

$\tilde{1}_{u}(b)=\tilde{1}_{u}\circ(\omega\otimes\lambda_{G})\circ\delta(a^{*}a)=\omega[(\operatorname{id}\otimes(\tilde{1}_{u}\circ\lambda_{G}))(\delta(a^{*}a))]=\omega[(\operatorname{id}\otimes
1_{u})(\delta(a^{*}a))]=0,$

which is a contradiction, by Lemma 4.7. When $\delta$ is injective, by
equality $\Phi_{0}=(\operatorname{id}\otimes\Phi)\circ\delta$, faithfulness of
$\Phi_{0}=(\operatorname{id}\otimes\Phi)$ on positive elements of $\delta(A)$
is equivalent to faithfulness of $\Phi_{0}$ on positive elements of $A$.

$(ii)$ First observe that,

$(q\otimes\lambda_{G})\circ\delta=(\operatorname{id}\otimes\lambda_{G})\circ(q\otimes\operatorname{id})\circ\delta=(\operatorname{id}\otimes\lambda_{G})\circ\delta^{I}\circ
q.$

If $\delta^{I}$ is normal, by the above equality
$x\in\ker((q\otimes\lambda_{G})\circ\delta)$ iff
$(\operatorname{id}\otimes\lambda_{G})(\delta^{I}(q(x))=0$ iff
$\delta^{I}(q(x))=0$, that is, $x\in q^{-1}\big{(}\ker(\delta^{I})\big{)},$
since $\operatorname{id}\otimes\lambda_{G}$ is injective on range of
$\delta^{I}$. Conversely if
$q^{-1}\big{(}\ker(\delta^{I})\big{)}=\ker((q\otimes\lambda_{G})\circ\delta)$
and $(\operatorname{id}\otimes\lambda_{G})(\delta^{I}(q(x))=0$, then
$(q\otimes\lambda_{G})\circ\delta(x)=0,$ thus $q(x)\in\ker(\delta^{I})$, that
is, $\delta^{I}(q(x))=0$, which means that
$\operatorname{id}\otimes\lambda_{G}$ is injective on the range of
$\delta^{I}$, that is, $\delta^{I}$ is normal. When $\delta^{I}$ is injective,
$q^{-1}\big{(}\ker(\delta^{I})\big{)}=q^{-1}(0)=I$. ∎

Recall that a discrete group $G$ is called exact if its reduced
$C^{*}$-algebra $C^{*}_{r}(G)$ is exact.

######  Lemma 4.9.

Let $G$ be a discrete groupoid. Consider the following statements:

  1. ($i$)

For every normal injective coaction $\delta$ of $G$ on a $C^{*}$-algebra $A$,
and every induced ideal $I$, the induced coaction $\delta^{I}$ of $G$ on $A/I$
is normal and injective.

  2. ($ii$)

$G$ is exact.

  3. ($iii$)

For every normal coaction $\delta$ of $G$ on a $C^{*}$-algebra $A$, and every
strongly induced ideal $I$, the induced coaction $\delta^{I}$ of $G$ on $A/I$
is normal.

Then $(i)\Rightarrow(ii)\Rightarrow(iii).$

###### Proof.

$(i)\Rightarrow(ii).$ Assume that $G$ is not exact. Then there exists a
$C^{*}$-algebra $B$ with an ideal $J$ such that, min-tensoring from right with
$C^{*}_{r}(G)$ does not preserve exactness of the short exact sequence

$0\to J\to B\to B/J\to 0,$

that is, $J\otimes C^{*}_{r}(G)\subsetneq\ker(q\otimes\operatorname{id})$,
where $q\colon B\to B/J$ is the quotient map. Consider the coaction
$\delta:=\operatorname{id}_{B}\otimes\delta_{G}$ of $G$ on $A:=B\otimes
C^{*}_{r}(G)$, where $\delta_{G}$ is the canonical coaction of $G$ on
$C^{*}_{r}(G)$ defined by $\delta_{G}(\lambda_{G}(g)):=\lambda_{G}(g)\otimes
i_{G}(g)$, for $g\in G$. Let

$\Delta\colon C^{*}_{r}(G)\to C^{*}_{r}(G)\otimes C^{*}_{r}(G);\ x\mapsto
x\otimes x,\ \ (x\in C^{*}_{r}(G))$

be the co-multiplication map, and observe that,
$(\operatorname{id}\otimes\lambda_{G})\circ\delta=\operatorname{id}\otimes\Delta$.
In particular, $(\operatorname{id}\otimes\lambda_{G})\circ\delta$ is
injective, that is, $\delta$ is normal and injective. We claim that, the ideal
$I:=J\otimes C^{*}_{r}(G)$ of $A=B\otimes C^{*}_{r}(G)$ is induced. Let $u\in
G^{(0)}$, then the projection $1_{u}$ in $C^{*}_{r}(G)$ generates a one
dimensional subspace $\mathbb{C}1_{u}$ and $(B\otimes
C^{*}_{r}(G))^{\delta}_{u}=B\otimes\mathbb{C}1_{u}$. By the identity
$1_{g}*1_{s(g)}=1_{g}$, for $g\in G$, the union of intersections of $I$ with
fibres $(B\otimes C^{*}_{r}(G))^{\delta}_{u}$ contains the algebraic tensor
product $J\odot C_{c}(G)$, which is dense in $I$, establishing the claim.
Finally, for the quotient map $\tilde{q}:A\to A/I$, we have
$(\tilde{q}\otimes\lambda_{G})\circ\delta=q\otimes\Delta$,

$I=J\otimes
C^{*}_{r}(G)\subsetneq\ker(q\otimes\operatorname{id})\subseteq\ker(q\otimes\Delta)=\ker((\tilde{q}\otimes\lambda_{G})\circ\delta),$

hence by Lemma 4.8$(ii)$, $\delta^{I}$ is either not normal or not injective.

$(ii)\Rightarrow(iii).$ Assume now that $G$ is exact and let $\delta$ be a
normal coaction of $G$ on a $C^{*}$-algebra $A$, and $I$ be a strongly induced
ideal of $A$. We show that $\delta^{I}$ is also normal. If $I$ is zero, there
is nothing to prove, so let us assume that $I$ is nonzero. Let $q:A\to A/I$ be
the quotient map. Let $x\in\ker((q\otimes\lambda_{G})\circ\delta)$, then
$(\operatorname{id}_{A}\otimes\lambda_{G})\circ\delta(x)\in\ker(q\otimes\operatorname{id})=I\otimes
C^{*}_{r}(G)$, where the equality follow from exactness of $G$. Since $I$ is a
strongly induced ideal, $\delta$ restricts to a nondegenerate $*$-homomorphism

$\delta:I\to(I\otimes C^{*}(G))\cap\delta(A)=:D.$

Since $\operatorname{id}\otimes\lambda_{G}:A\otimes C^{*}(G)\to A\otimes
C^{*}_{r}(G)$ is surjective, it is nondegenerate [54, Proposition 2.2.16], and
so the $*$-homomorphism

$(\operatorname{id}\otimes\lambda_{G})\circ\delta:I\to(I\otimes
C^{*}_{r}(G))\cap(\operatorname{id}\otimes\lambda_{G})\delta(A)=:C$

is nondegenerate. By Lemma 3.2,
$(\operatorname{id}\otimes\lambda_{G})\circ\delta$ maps an approximate
identity $(e_{i})$ of $I$ to an approximate identity of $C$. Now, since
$(\operatorname{id}\otimes\lambda_{G})\circ\delta(x)\in C$,

$(\operatorname{id}\otimes\lambda_{G})\circ\delta(e_{i}x)=(\operatorname{id}\otimes\lambda_{G})\circ\delta(e_{i})(\operatorname{id}\otimes\lambda_{G})\circ\delta(x)\to(\operatorname{id}\otimes\lambda_{G})\circ\delta(x),$

as $i\to\infty$. By normality of $\delta$,
$\operatorname{id}\otimes\lambda_{G}$ is injective and so isometric on the
range of $\delta$, hence it follows that the net $\delta(e_{i}x)$ is
convergent in $I\otimes C^{*}(G)$, say to $y$. Since $\delta(I)$ is closed,
there is $z\in I$ with $y=\delta(z)$, which in turn gives,
$(\operatorname{id}\otimes\lambda_{G})\circ\delta(x)=(\operatorname{id}\otimes\lambda_{G})\circ\delta(z)$,
and so $\delta(x)=\delta(z)$, again by normality of $\delta$. Thus,
$\delta(x)\in\delta(I)\subseteq I\otimes C^{*}(G)$. Therefore,

$\delta^{I}(q(x))=(q\otimes\operatorname{id})(\delta(x))=0,$

that is, $x\in q^{-1}(\ker(\delta^{I}))$, as required. ∎

Since the ideal $I:=J\otimes C^{*}_{r}(G)$ constructed in the proof of the
implication $(i)\Rightarrow(ii)$ of the above lemma is not only induced, but
also strongly induced, we immediately get the following corollary.

######  Corollary 4.10.

Let $G$ be a discrete groupoid. Then the following statements are equivalent:

  1. ($i$)

For every normal injective coaction $\delta$ of $G$ on a $C^{*}$-algebra $A$,
and every strongly induced ideal $I$, the induced coaction $\delta^{I}$ of $G$
on $A/I$ is normal and injective.

  2. ($ii$)

$G$ is exact.

  3. ($iii$)

For every normal coaction $\delta$ of $G$ on a $C^{*}$-algebra $A$, and every
strongly induced ideal $I$, the induced coaction $\delta^{I}$ of $G$ on $A/I$
is normal.

This last result urges finding examples of strongly induced ideals of
$\mathcal{T}_{\rm cov}(X)$ and $\operatorname{\mathcal{NO}}_{\\!X}$, for the
canonical coactions of $G$. Using Lemma 3.15, this is not hard when $G$ has
uniformely bounded fibres.

######  Lemma 4.11.

If $G$ has uniformly bounded fibres, the closed ideal
$\langle\mathcal{F}\rangle$ of $\mathcal{T}_{\rm cov}(X)$ generated by the
core $\mathcal{F}$ is a strongly induced ideal with respect to the gauge
action. Similarly, closed ideal $\langle q_{{\rm CNP}}(\mathcal{F})\rangle$ of
$\operatorname{\mathcal{NO}}_{\\!X}$, generated by the image of the core
$\mathcal{F}$ under the quotient map $q_{\rm CNP}:\mathcal{T}_{\rm
cov}(X)\to\operatorname{\mathcal{NO}}_{\\!X}$, is a strongly induced ideal
with respect to the induced gauge coaction.

###### Proof.

Since $\mathcal{F}$ is generated by $\bigcup_{u\in G^{(0)}}\mathcal{T}_{\rm
cov}(X)^{\delta}_{u}$, $\langle\mathcal{F}\rangle$ is an induced ideal. Since
$\mathcal{F}$ contains an approximate identity of $\mathcal{T}_{\rm cov}(X)$,
we have $M(\mathcal{F})=M(\langle\mathcal{F}\rangle)=M(\mathcal{T}_{\rm
cov}(X))$. Therefore, $D:=(\mathcal{F}\otimes
C^{*}(G))\cap\delta(\mathcal{T}_{\rm cov}(X))$ and
$D^{{}^{\prime}}:=(\langle\mathcal{F}\rangle\otimes
C^{*}(G))\cap\delta(\mathcal{T}_{\rm cov}(X))$ have the same multiplier
algebra. Now the first statement follows from Lemma 3.15. The second statement
is proved similarly. ∎

In the notations of Lemma 4.3, let $A^{r}:=C^{*}_{r}(E_{A})$, and let
$\lambda_{r}:A\to A^{r}$ be the corresponding regular representation. Consider
the canonical regular representation $\lambda_{G}:C^{*}(G)\to C^{*}_{r}(G)$.
If $\delta$ is normal, $\delta^{n}$ is a reduced coaction of $G$ on $A$. If
not, then $G$ coacts normally on $A^{n}:=A/{\ker(\delta^{n})}$. As in Lemma
4.2$(ii)$, there is a normal coaction $\delta^{n}$ of $G$ on $A^{r}$ with
$\delta^{n}\bigl{(}\lambda_{r}(a)\bigr{)}=\lambda_{r}(a)\otimes\lambda_{G}(g)$,
for $g\in G$ and $a\in A^{\delta}_{g},$ and
$\ker(\delta^{n})=\ker(\lambda_{r})$, that is, $A^{r}$ is isomorphic to
$A^{n}$.

Let $(G,P)$ be a quasi-lattice ordered groupoid, and $X$ be a compactly
aligned product system of Hilbert bimodules over $P$. By Lemma 3.6,
$\mathcal{T}_{\rm cov}(X)$ admits a coaction $\delta$, giving rise to a Fell
bundle $E_{X}:=(\mathcal{T}_{\rm cov}(X)^{\delta}_{g}\times\\{g\\})_{g\in G}$
over $G$, and by Lemma 3.7$(i)$, $C^{*}(E_{X}|_{G^{(0)}})\simeq\mathcal{F}$.
We identify $\mathcal{T}_{\rm cov}(X)^{\delta}_{g}$, with $\mathcal{T}_{\rm
cov}(X)^{\delta}_{g}\times\\{g\\}$.

Let $G$ be discrete and consider the map $\iota_{X}:\bigcup_{g\in
G}\mathcal{T}_{\rm cov}(X)^{\delta}_{g}\hookrightarrow C_{c}(E_{X})$, defined
as the canonical embedding on each fibre. This combined with the canonical
inclusion $\eta_{X}:C_{c}(E_{X})\hookrightarrow\mathcal{T}_{\rm cov}(X)$ gives
and embedding of $\bigcup_{g\in G}\mathcal{T}_{\rm cov}(X)^{\delta}_{g}$ in
$\mathcal{T}_{X}$.

######  Lemma 4.12.

Let $\iota_{\rm max}:C_{c}(E_{X})\hookrightarrow C^{*}(E_{X})$ be the
canonical embedding, then

$(i)$ the map $\iota_{\rm max}\circ\iota_{X}\circ i_{X}:X\to C^{*}(E_{X})$ is
a Nica covariant Toeplitz representation.

$(ii)$ $C^{*}(E_{X})$ is equivariantly isomorphic to $\mathcal{T}_{\rm
cov}(X)$.

###### Proof.

$(i)$ Just note that $i_{X}$ is a Nica covariant representation and
$\iota_{\rm max}\circ\iota_{X}$ restricts to a linear map on each fiber
$\mathcal{T}_{\rm cov}(X)^{\delta}_{g}$ and to a $*$-homomorphism on the fiber
$\mathcal{T}_{\rm cov}(X)^{\delta}_{s(g)}$.

$(ii)$ By the universal property of $\mathcal{T}_{\rm cov}(X)$, there is a
$*$-homomorphism $\psi_{X}:\mathcal{T}_{\rm cov}(X)\to C^{*}(\mathcal{B})$
such that $\psi_{X}\circ i_{X}=\iota_{\rm max}\circ\iota_{X}\circ i_{X}$.
Choose an epimorphism $\phi_{X}:C^{*}(E_{X})\to\mathcal{T}_{\rm cov}(X)$ such
that $\phi_{X}\circ\iota_{\rm max}=\eta_{X}$. We have

$\phi_{X}\circ\psi_{X}\circ i_{X}=\phi_{X}\circ\iota_{\rm
max}\circ\iota_{X}\circ i_{X}=\eta_{X}\circ\iota_{X}\circ i_{X}=i_{X},$

and

$\psi_{X}\circ\phi_{X}\circ\iota_{\rm max}\circ\iota_{X}\circ
i_{X}=\psi_{X}\circ\eta_{X}\circ\iota_{X}\circ i_{X}=\psi_{X}\circ
i_{X}=\iota_{\rm max}\circ\iota_{X}\circ i_{X},$

that is, $\psi_{X}$ is the inverse of $\phi_{X}$, and $\phi_{X}$ is an
equivariant isomorphism of $C^{*}(E_{X})$ and $\mathcal{T}_{\rm cov}(X)$. ∎

Next, suppose that $G$ is discrete and $X$ satisfies conditions of Theorem
4.1. By Lemmas 3.7 and 3.10, the kernel of the canonical homomorphism
$q_{\operatorname{CNP}}\colon\mathcal{T}_{\rm
cov}(X)\to\operatorname{\mathcal{NO}}_{\\!X}$ is generated by its intersection
with $\mathcal{F}$. Therefore Lemma 4.5 applied to the coaction $\delta$ on
$\mathcal{T}_{\rm cov}(X)$, yields a coaction on
$\operatorname{\mathcal{NO}}_{\\!X}$, called the gauge coaction, and again
denoted by $\delta$. The spectral subspaces

$(\operatorname{\mathcal{NO}}_{\\!X})^{\delta}_{g}:=\\{\,c\in\operatorname{\mathcal{NO}}_{\\!X}\mid\delta(c)=c\otimes
i_{G}(g)\,\\}$

give rise to a Fell bundle $E_{\mathcal{O}}$ over $G$, and by an argument as
in that of Lemma 4.12$(ii)$, $C^{*}(E_{\mathcal{O}})$ is equivariantly
isomorphic to $\operatorname{\mathcal{NO}}_{\\!X}$ for coactions
$\delta_{E_{\mathcal{O}}}$ and $\delta$.

###### Proof of Theorem 4.1.

Let $\operatorname{\mathcal{NO}}_{\\!X}^{r}$ be the reduced cross-sectional
algebra of the Fell bundle $E_{\mathcal{O}}$, and
$\lambda_{r}\colon\operatorname{\mathcal{NO}}_{\\!X}\to\operatorname{\mathcal{NO}}_{\\!X}^{r}$
be the corresponding regular representation. Put $j^{r}_{X}:=\lambda_{r}\circ
j_{X}$, and let $\delta^{n}$ to be the induced normal coaction on
$\operatorname{\mathcal{NO}}_{\\!X}^{r}$.

To prove (1), let us note that $\operatorname{\mathcal{NO}}_{\\!X}$ is
generated by the CNP-covariant representation $j_{X}$, which is injective by
Proposition 2.6, thus $j_{X}^{r}$ is CNP-covariant. Also, since $\lambda_{r}$
is surjective, $\operatorname{\mathcal{NO}}_{\\!X}^{r}$ is generated by
$j_{X}^{r}$. Now, for $u\in G^{(0)}$, $\lambda_{r}$ restricts to a bijection
from $(\operatorname{\mathcal{NO}}_{\\!X})^{\delta^{n}}_{u}$ to
$(\operatorname{\mathcal{NO}}_{\\!X}^{r})^{\delta}_{u}$, and since
$j_{X}(A_{u})\subset(\operatorname{\mathcal{NO}}_{\\!X})^{\delta}_{u}$, the
representation $j_{X}^{r}$ is also injective, and we have
$\delta^{n}(j^{r}_{X}(x))=j^{r}_{X}(x)\otimes i_{G}(d(x))$, for $x\in X$.

Next to prove (2), take $\psi\colon X\to D$ as in (2), and let $\beta$ be a
coaction on $D$ with the prescribed gauge compatibility. For $g\in G$, let
$D^{\beta}_{g}=\\{\,b\in D\mid\beta(b)=b\otimes i_{G}(g)\,\\}$. Then
$\\{D^{\beta}_{g}\\}_{g\in G}$ is a topological grading of $D$, by Lemma 4.3.
By the universal property of $\mathcal{T}_{\rm cov}(X)$, there is a
$*$-homomorphism $\psi_{*}\colon\mathcal{T}_{\rm cov}(X)\to D$ with
$\psi=\psi_{*}\circ i_{X}$. Consider the ideal
$I_{u}:=\psi_{*}(\ker(q_{\operatorname{CNP}}))\cap D^{\beta}_{u}$ of
$D_{u}^{\beta}$, for $u\in G^{(0)}$, and let $I$ be the ideal of $D$ generated
by $\bigcup_{u\in G^{(0)}}I_{u}$. Then by Lemma 3.8 and the fact that the
image of $\psi$ generates $D$, $I$ is the ideal of $D$ generated by
$\psi_{*}(\ker(q_{\operatorname{CNP}})\cap\mathcal{F})$. By construction, $I$
is an induced ideal. Let $\pi\colon D\to D/I$ be the quotient map, then by
Lemma 4.5$(ii)$, $\\{\pi(D^{\beta}_{g})\\}_{g\in G}$ is a topological grading
of $D/I$. Since the image of $\psi$ generates $D$, we also have
$\pi(\psi_{*}(\mathcal{T}_{\rm cov}(X)^{\delta}_{g}))=\pi(D^{\beta}_{g})$, for
each $g\in G$. Next, let us show that, for every $g\in G$,

$\ker(\pi\circ\psi_{*})\cap\mathcal{T}_{\rm
cov}(X)^{\delta}_{g}=\ker(q_{\operatorname{CNP}})\cap\mathcal{T}_{\rm
cov}(X)^{\delta}_{g}.$

Let us first handle the case $g=:u\in G^{(0)}$. For the first case, by Lemma
3.7$(i)$, we only need to show that
$\ker(\pi\circ\psi_{*})\cap\mathcal{F}=\ker(q_{\operatorname{CNP}})\cap\mathcal{F}$.
If $c\in\ker(q_{\operatorname{CNP}})\cap\mathcal{F}$, then $\psi_{*}(c)\in I$,
thus $\pi(\psi_{*}(c))=0$, and so $c\in\ker(\pi\circ\psi_{*})\cap\mathcal{F}$.
Conversely, since $\pi\circ\psi_{*}=(\pi\circ\psi)_{*}$, by Lemma 3.8 it
suffices to show that the Toeplitz representation $\pi\circ\psi$ is injective.
If $a\in A_{u}$ and $\pi(\psi_{u}(a))=0$, we have,

$\psi_{u}(a)\in I\cap
D_{u}^{\beta}\subseteq\psi_{*}(\ker(q_{\operatorname{CNP}}))\cap
D_{u}^{\beta}\subseteq\psi_{*}\big{(}\ker(q_{\operatorname{CNP}})\cap\mathcal{F}\big{)},$

thus there exists $b\in\ker(q_{\operatorname{CNP}})\cap\mathcal{F}$ such that
$\psi_{*}(b)=\psi(a)$. Hence,

$b-i_{X}(a)\in\ker(\psi_{*})\cap\mathcal{F}\subseteq\ker(q_{\operatorname{CNP}}),$

where the inclusion follows from Lemma 3.8. Therefore,
$i_{X}(a)\in\ker(q_{\operatorname{CNP}})\cap i_{X}(A_{u})=0$, hence $a=0$, as
$i_{X}$ is an injective Toeplitz representation, i.e., $\pi\circ\psi$ is
injective, as required. The claimed equality now follows for each $g\in G$,
since $c\in\ker(\pi\circ\psi_{*})\cap\mathcal{T}_{\rm cov}(X)^{\delta}_{g}$ is
equivalent to $c^{*}c\in\ker(\pi\circ\psi_{*})\cap\mathcal{T}_{\rm
cov}(X)^{\delta}_{s(g)}=\ker(q_{\operatorname{CNP}})\cap\mathcal{T}_{\rm
cov}(X)^{\delta}_{s(g)}$, which is in turn equivalent to
$c\in\ker(q_{\operatorname{CNP}})\cap\mathcal{T}_{\rm cov}(X)^{\delta}_{g}$.

The above equality now implies that for $g\in G$, the map,

$\tilde{\phi}_{g}:(\operatorname{\mathcal{NO}}_{\\!X})^{\delta}_{g}\to\pi(D^{\beta}_{g});\
q_{CNP}(a)\mapsto\pi\circ\psi_{*}(a),\ \ (a\in\mathcal{T}_{\rm
cov}(X)^{\delta}_{g}),$

is well defined. This gives a covariant isomorphism of Fell bundles with
fibres $\pi(D^{\beta}_{g})\times\\{g\\}$ and
$(\operatorname{\mathcal{NO}}_{\\!X})^{\delta}_{g}\times\\{g\\}$. By the
couniversal properties of $\operatorname{\mathcal{NO}}_{\\!X}^{r}$ and
$C_{r}^{*}(E_{\pi(D)})=:\pi(D)^{r}$, there is an isomorphism
$\tilde{\phi}\colon\operatorname{\mathcal{NO}}_{\\!X}^{r}\to\pi(D)^{r}$ such
that $\lambda^{I}_{r}\circ\tilde{\phi}_{g}=\tilde{\phi}\circ\lambda_{r}$ on
each $(\operatorname{\mathcal{NO}}_{\\!X})^{\nu}_{g}$, where
$\lambda_{r}:\operatorname{\mathcal{NO}}_{\\!X}\to\operatorname{\mathcal{NO}}_{\\!X}^{r}$
and $\lambda^{I}_{r}:(D/I)\to(D/I)^{r}$ are the corresponding regular
representations. Put $\phi:=\tilde{\phi}^{-1}\circ\lambda^{I}_{r}\circ\pi$.
Then $\phi:D\to\operatorname{\mathcal{NO}}_{\\!X}^{r}$ is a $*$-homomorphism,
and for each $x\in X$,

$\displaystyle\phi(\psi(x))$
$\displaystyle=\tilde{\phi}^{-1}(\lambda^{I}_{r}(\pi(\psi_{*}(i_{X}(x)))))=\tilde{\phi}^{-1}(\lambda^{I}_{r}(\phi_{d(x)}(q_{\operatorname{CNP}}(i_{X}(x)))))$
$\displaystyle=\lambda_{r}(q_{\operatorname{CNP}}(i_{X}(x)))=\lambda_{r}(j_{X}(x))=j_{X}^{r}(x),$

as claimed in (2).

Finally, to see the uniqueness, let us suppose that $(C,\rho,\gamma)$ also
satisfies (1) and (2). By the property (2) for
$\operatorname{\mathcal{NO}}_{\\!X}^{r}$ and $C$, there are equivariant
homomorphisms $\phi:C\to\operatorname{\mathcal{NO}}_{\\!X}^{r}$ and
$\phi^{{}^{\prime}}:\operatorname{\mathcal{NO}}_{\\!X}^{r}\to C$, which are
then clearly inverse of each other. ∎

######  Definition 4.13.

For a quasi-lattice ordered groupoid $(G,P)$ and a $\tilde{\phi}$-injective
compactly aligned product system $X$ over $P$. We say that
$\operatorname{\mathcal{NO}}_{\\!X}$ has the gauge-invariant uniqueness
property provided that a surjective $*$-homomorphism
$\phi:\operatorname{\mathcal{NO}}_{\\!X}\to B$ is injective if and only if the
following hold:

(GI1) there is a coaction $\beta$ of $G$ on $D$ such that
$\beta\circ\phi=(\phi\otimes\operatorname{id})\circ\delta$,

(GI2) the homomorphism $\phi|_{j_{X}(A)}$ is injective.

######  Corollary 4.14.

Let $G$ be discrete and $X$ be compactly aligned. Suppose that either the left
action on each fibre is injective, or $P$ is directed and $X$ is
$\tilde{\phi}$-injective.

  1. (1)

If $\phi\colon\operatorname{\mathcal{NO}}_{\\!X}^{r}\to D$ is a surjective
$*$-homomorphism, then $\phi$ is injective if and only if
$\phi{|j_{X}^{r}(A)}$ is injective and there is a coaction $\beta$ of $G$ on
$D$ such that $\beta\circ\phi=(\phi\otimes\operatorname{id})\circ\delta^{n}$.

  2. (2)

Suppose that $(C,\rho)$ is a couniversal pair for injective gauge-compatible
(not necessarily Nica covariant) Toeplitz representations of $X$. Then there
is a $*$-isomorphism $\phi\colon\operatorname{\mathcal{NO}}_{\\!X}^{r}\to C$
such that $\phi(j_{X}^{r}(x))=\rho(x)$ for all $x\in X$.

  3. (3)

Assume that both coactions $\beta$ and $\delta^{n}$ are injective. If
$\psi\colon X\to D$ is an injective Nica covariant gauge-compatible
representation whose image generates $D$, then the surjective $*$-homomorphism
$\phi\colon D\to\operatorname{\mathcal{NO}}_{\\!X}^{r}$ of Theorem 4.1(2) is
injective if and only if $\psi$ is Cuntz-Pimsner covariant and $\beta$ is
normal.

###### Proof.

(1) The necessity is trivial. For the sufficiency, note that $\phi\circ
j_{X}^{r}$ is an injective gauge-compatible Nica covariant representation of
$X$ in $D$, whose image generates $D$. Theorem 4.1 guarantees the existence of
an epimorphism which is a left inverse for $\phi$.

(2) Since $\operatorname{\mathcal{NO}}_{\\!X}^{r}$ is generated by an
injective gauge-compatible Toeplitz representation of $X$, by the couniversal
property of $(C,\rho)$, there is an epimomorphism
$\phi\colon\operatorname{\mathcal{NO}}_{\\!X}^{r}\to C$ with
$\phi(j_{X}^{r}(x))=\rho(x)$, for $x\in X$. By part (1), $\phi$ is injective
and so an isomorphism.

(3) If $\phi$ is injective, then $\psi$ is CNP-covariant and $\beta$ is normal
as $j_{X}^{r}$ is CNP-covariant and $\delta^{n}$ is normal. Conversely, if
$\psi$ is CNP-covariant and $\beta$ is normal, by the universal property of
$\operatorname{\mathcal{NO}}_{\\!X}$, there is a homomorphism
$\Pi\psi\colon\operatorname{\mathcal{NO}}_{\\!X}\to D$ satisfying
$\lambda_{r}=\phi\circ\Pi\psi$, for the regular representation
$\lambda_{r}\colon\operatorname{\mathcal{NO}}_{\\!X}\to\operatorname{\mathcal{NO}}_{\\!X}^{r}$.
For each $u\in G^{(0)}$, the representation $\lambda_{r}$ restricts to an
isomorphism of $(\operatorname{\mathcal{NO}}_{\\!X})^{\delta}_{u}$ onto
$(\operatorname{\mathcal{NO}}_{\\!X}^{r})^{\delta^{n}}_{u}$, hence $\phi$
restricts to an isomorphism of $D^{\beta}_{u}$ onto
$(\operatorname{\mathcal{NO}}_{\\!X}^{r})^{\delta^{n}}_{u}$. Since $\beta$ is
normal, by Lemma 4.8 there is a faithful conditional expectation
$\Phi^{\beta}:=\oplus_{u}\beta_{u}\colon D\to\bigoplus_{u}D^{\beta}_{u}$, and
$\phi$ intertwines $\Phi^{\beta}$ and the conditional expectation
$\Phi^{\delta^{n}}$ from $\operatorname{\mathcal{NO}}_{\\!X}^{r}$ onto
$\bigoplus_{u}(\operatorname{\mathcal{NO}}_{\\!X}^{r})^{\delta^{n}}_{u}$, and
injectivity of $\phi$ follows by an standard argument. ∎

For the next result, we first need to define and relate the notions of
amenability and approximation property for Fell bundles over groupoids. The
second notion is defined in the case of bundles over discrete groups by Ruy
Exel [18, Definition 4.4].

######  Definition 4.15.

Let $G$ be a discrete groupoid and $E$ is a Fell bundle over $G$. We say that

$(i)$ $E$ is amenable if the regular representation $\lambda_{r}:C^{*}(E)\to
C^{*}_{r}(E)$ is injective,

$(ii)$ $E$ has approximation property if for each $u\in G^{(0)}$ there exists
a net $(a^{u}_{i})$ of functions $a^{u}_{i}:G\to E_{u}$ such that,

$\sup_{i}\big{\|}\sum_{r(h)=u}a^{u}_{i}(h)^{*}a^{u}_{i}(h)\big{\|}<\infty,\ \
(u\in G^{(0)})$

and,

$\lim_{i}\big{\|}\sum_{r(h)=u}a^{u}_{i}(gh)^{*}ba^{u}_{i}(h)-b\big{\|}=0,\ \
(u\in G^{(0)},g\in G_{u},b\in E_{g}).$

Let us quickly observe that if $E$ has approximation property, then the
functions $a^{u}_{i}$ could be chesen to be of finite support. Indeed, given
$u\in G^{(0)}$, there exists a constant $M>0$ such that for each
$\varepsilon>0$, each finite subset $F\subseteq G$, and elements $b_{k}\in
E_{k}$, for $k\in F$, there exists a function $a^{u}:G\to E_{u}$ such that

$\big{\|}\sum_{r(h)=u}a^{u}(h)^{*}a^{u}(h)\big{\|}<M,\ \
\big{\|}\sum_{r(h)=u}a^{u}(gh)^{*}b_{k}a^{u}(h)-b_{k}\big{\|}<\varepsilon,\ \
(k\in F,g\in G_{u}).$

Choose a finite subset $F_{\varepsilon}\supseteq F$ with

$\sup_{k\in F}\big{\|}\sum_{h\in G^{u}\backslash
F_{\varepsilon}}a^{u}(gh)^{*}b_{k}a^{u}(h)\|<\varepsilon,$

for each $g\in G_{u}$. Let $a^{u}_{\varepsilon,F}$ be the restriction of
$a^{u}$ to $F_{\varepsilon}$, and observe that,

$\big{\|}\sum_{r(h)=u}a^{u}_{\varepsilon,F}(h)^{*}a^{u}_{\varepsilon,F}(h)\big{\|}<M,\
\
\big{\|}\sum_{r(h)=u}a^{u}_{\varepsilon,F}(gh)^{*}b_{k}a^{u}_{\varepsilon,F}(h)-b_{k}\big{\|}<2\varepsilon,\
\ (k\in F,g\in G_{u}),$

which establishes the claim.

Let $G$ be an étale groupoid and $E$ be a Fell bundle over $G$. A
representation of $E$ on a Hilbert space $H$ in the sense of Fell and Doran
[20, VIII. 9.1] is nothing but a morphism $\pi:E\to\mathbb{B}(H)$ in the sense
of the paragraph before Lemma 4.2, where $\mathbb{B}(H)$ is regarded as a
trivial $G$-bundle.

######  Lemma 4.16.

Let $G$ be a discrete groupoid and $E$ be a Fell bundle over $G$. Let $u\in
G^{(0)}$ and $a:G\to E_{u}$ be a map such that the sum
$a_{u}:=\sum_{s(h)=u}a(h)^{*}a(h)$ converges unconditionally in $E_{u}$. Then
there exists a bounded c.p. map $\psi_{a}:C^{*}_{r}(E)\to C^{*}(E)$ such that
$\|\psi_{a}\|\leq\|a_{u}\|$ and
$\psi_{a}\circ\lambda_{E}(b)=\sum_{r(h)=u}a(gh)^{*}ba(h)$, for $g\in G_{u}$
and $b\in E_{g}$.

###### Proof.

Think of $C^{*}(E)$ as a subalgebra of $\mathbb{B}(H)$, for some Hilbert space
$H$ and consider the corresponding representation $\pi$ of $E$ on $H$. For
$u\in G^{(0)}$, let ${\rm Ind}\delta_{u}$ be the regular representation of $G$
on $\ell^{2}(G_{u})$ [55, page 18], where $\delta_{u}$ is the Dirac measure at
$u$ on $G^{(0)}$. Consider the representation $\pi$ of $E$ on the Hilbert
bundle $(H\otimes\ell^{2}(G_{u}))_{u\in G^{(0)}}$, defined by

$\pi(b_{g})=b_{g}\otimes{\rm Ind}\delta_{s(g)}(1_{g}),\ \ (g\in G,b_{g}\in
E_{g}).$

Let $B$ be the $C^{*}$-algebra generated by the range of $\pi$. Consider the
isometry

$j_{g}:H\to H\otimes\ell^{2}(G_{s(g)});\ \ \xi\mapsto\xi\otimes 1_{g},\ \
(g\in G).$

For $b=\sum_{k}b_{k}$ be an element in $E$ with finitely many nonzero terms
$b_{k}\in E_{k}$, $k\in G$, and observe that for $\xi\in H$, $g\in G$ and
$h\in G_{s(g)}$,

$\displaystyle j^{*}_{g}\pi(b)j_{h}(\xi)$
$\displaystyle=j^{*}_{g}\pi(b)(\xi\otimes 1_{h})$
$\displaystyle=\sum_{k}j^{*}_{g}\big{(}b_{k}\xi\otimes{\rm
Ind}\delta_{s(k)}(1_{k})1_{h}\big{)}$
$\displaystyle=\sum_{s(k)=r(h)}j^{*}_{g}\big{(}b_{k}\xi\otimes
1_{kh}\big{)}=b_{gh^{-1}}\xi.$

In particular, if $b=0$ then $b_{k}=0$, for each $k$. Since the canonical
regular representation $\lambda_{G}:C^{*}(G)\to C^{*}_{r}(G)$ is the direct
sum of representations ${\rm Ind}\delta_{u}$, for $u\in G^{(0)}$, the above
observation means that we have a graded Fell bundle $F$ with fibres
$F_{g}:=\pi(E_{g})\otimes C^{*}_{r}(G)^{\delta_{G}^{n}}_{g}$, for $g\in G$.
Let us observe that there is a conditional expectation $\mathbb{E}_{u}:B\to
B_{u}$, vanishing on each $B_{g}$, for $g\in G\backslash G^{(0)}$, that is, we
have a topological grading. Indeed, if we define,
$\mathbb{E}_{u}(\pi(b)):=j_{u}^{*}\pi(b)j_{u}\otimes{\rm
Ind}\delta_{u}(1_{u})$, for $b\in B$, then, for $g\in G\backslash G^{(0)}$,
$j_{u}^{*}\pi(b_{g})j_{u}=0$, by the above calculation, as required. Next, let
us observe that $\mathbb{E}:=\oplus\mathbb{E}_{u}$ is faithful. If
$\mathbb{E}_{u}(\pi(b^{*}b))=0$, for each $u\in G^{(0)}$, then
$\pi(b)j_{u}=0$, for each $u\in G^{(0)}$. Consider the representation of $G$
given by,

$\rho_{t}:H\otimes\ell^{2}(G_{s(t)})\to H\otimes\ell^{2}(G_{r(t)});\ \
\xi\otimes 1_{s}\mapsto\xi\otimes 1_{st^{-1}},\ \ (t\in G),$

then, since $\pi$ acts on the second leg by a left multiplication, where as
$\rho$ acts by a right multiplication, we have,
$\pi(b)\rho_{t}=\sum_{k}b_{k}\xi\otimes 1_{kst^{-1}}=\rho_{t}\pi(b),$ for each
$b=\sum_{k}b_{k}$ and $t\in G$. Also, $\rho_{t}j_{s}=j_{st^{-1}},$ whenever
$s(s)=s(t)$. These together imply that,

$\pi(b)j_{t^{-1}}=\pi(b)\rho_{t}j_{s(t)}=\rho_{t}\pi(b)j_{s(t)}=0,$

thus, $b_{t}=j_{r(t)}\pi(b)j_{t^{-1}}=0$, for each $t$, that is, $b=0$, as
claimed. We conclude, by Lemma 4.3, that $B$ is isomorphic with
$C^{*}_{r}(E)$, with a canonical isomorphism $\Phi:C^{*}_{r}(E)\to B$, mapping
$\lambda_{E}(b_{g})$ to $\pi(b_{g})$, for each $g\in G$ and $b_{g}\in E_{g}$.

Next, let $u\in G^{(0)}$ and $a:G\to E_{u}$ be the map satisfying our
unconditional convergence assumption. Consider the operator $V_{u}:H\to
H\otimes\ell^{2}(G_{u});\ \xi\mapsto\sum_{r(h)=u}a(h)\xi\otimes 1_{h}$. The
sum on the right hand side is convergent since,

$\sum_{r(h)=u}\|a(h)\xi\|^{2}\leq\sum_{r(h)=u}\langle
a(h)^{*}a(h)\xi,\xi\rangle\leq\big{\|}\sum_{r(h)=u}a(h)^{*}a(h)\big{\|}\|\xi\|^{2},$

which also shows that $\|V_{u}\|^{2}\leq\|a_{u}\|.$ Put $\psi_{u}={\rm
ad}_{V^{*}_{u}}:\mathbb{B}(H\otimes\ell^{2}(G_{u}))\to\mathbb{B}(H)$. Then,

$\displaystyle\psi_{u}\circ\pi(b_{g})\xi$
$\displaystyle=V^{*}_{u}\big{(}b_{g}\otimes{\rm
Ind}\delta_{u}(1_{g})\big{)}V_{u}\xi$
$\displaystyle=V^{*}_{u}\big{(}b_{g}\otimes{\rm
Ind}\delta_{u}(1_{g})\big{)}\big{(}\sum_{r(h)=u}a(h)\xi\otimes 1_{h}\big{)}$
$\displaystyle=V^{*}_{u}\big{(}\sum_{r(h)=u}b_{g}a(h)\xi\otimes 1_{gh}\big{)}$
$\displaystyle=\sum_{r(h)=u}a(gh)^{*}b_{g}a(h)\xi,$

for each $g\in G_{u}$. In particular, $\psi_{u}$ maps $B$ into $C^{*}(E)$. Put
$\psi_{a}:=\psi_{u}\circ\Phi$. This is a c.p. map, and for $b_{g}\in E_{g}$,

$\displaystyle\psi_{a}(\lambda_{E}(b_{g}))$
$\displaystyle=\psi_{u}(\pi(b_{g}))=\sum_{r(h)=u}a(gh)^{*}b_{g}a(h).$

Finally, $\|\psi_{a}\|\leq\|\psi_{u}\|\leq\|V_{u}\|^{2}\leq\|a_{u}\|,$ as
required. ∎

Recall that a discrete groupoid $G$ is amenable if there is a bounded net
$(f_{i})$ in $C_{c}(G)$ with

$\sup_{u\in G^{(0)}}\sum_{r(h)=u}|f_{i}(h)|^{2}<\infty,\ \
\big{|}1-\sum_{r(h)=u}f_{i}(h)\bar{f}_{i}(gh)\big{|}\to 0,\ \ (u\in
G^{(0)},g\in G_{u}),$

as $i\to\infty$ [55, Definition 9.3].

######  Lemma 4.17.

For a discrete groupoid $G$,

$(i)$ each Fell bundle $E$ over $G$ with approximation property is amenable,

$(ii)$ if $G$ is amenable, each Fell bundle $E$ over $G$ has the approximation
property.

###### Proof.

$(i)$ Let $(a^{u}_{i})$ be the net of maps as in Definition 4.15, which could
be assumed to be of finite support by the calculations in paragraph after that
definition. Let $\psi_{i}:=\psi_{a^{u}_{i}}:C^{*}_{r}(E)\to C^{*}(E)$ be the
map constructed in Lemma 4.16. Put $\phi_{i}:=\psi_{i}\circ\lambda_{E}$. Then
for $g\in G$ and $b_{g}\in E_{g}$, then identifying $b_{g}$ with $\pi(b_{g})$,
we have,

$\phi_{i}(b_{g})=\sum_{r(h)=u}a^{u}_{i}(gh)^{*}b_{g}a^{u}_{i}(h)\to b_{g},$

as $i\to\infty$. Since the net $(\phi_{i})$ is uniformly bounded by Lemma
4.16, it follows that $\phi_{i}(x)\to x$, as $i\to\infty$, for each $x\in
C^{*}(E)$. We conclude that $\lambda_{E}$ is injective, since if
$\lambda_{E}(x)=0$, then $\psi_{i}(\lambda_{E}(x))=\phi_{i}(x)\to x$, as
$i\to\infty$, thus $x=0$.

$(ii)$ Let $G$ be amenable and choose a net $(f_{i})$ in $C_{c}(G)$ as above.
Let $u\in G^{(0)}$ and choose an approximate identity $(e^{u}_{j})$ in $E_{u}$
consisting of positive elements of norm at most one, and consider the map,

$a^{u}_{i,j}:G\to E_{u};\ \ a^{u}_{i,j}(g):=f_{i}(g)e_{j},\ (g\in G).$

Then

$\big{\|}\sum_{r(h)=u}a^{u}_{i,j}(h)^{*}a^{u}_{i,j}(h)\big{\|}\leq\sup_{u\in
G^{(0)}}\sum_{r(h)=u}|f_{i}(h)|^{2}<\infty,$

and,

$\displaystyle\big{\|}\sum_{r(h)=u}a^{u}_{i,j}(gh)^{*}ba^{u}_{i,j}(h)-b\big{\|}$
$\displaystyle=\big{\|}\sum_{r(h)=u}f_{i}(h)\bar{f}_{i}(gh)e_{j}be_{j}-b\big{\|}$
$\displaystyle\leq\big{|}1-\sum_{r(h)=u}f_{i}(h)\bar{f}_{i}(gh)\big{|}\|b\|+\|e_{j}be_{j}-b\|\to
0,$

as $(i,j)\to\infty$, for each $g\in G_{u},b\in E_{g}.$

∎

######  Corollary 4.18 (The gauge-invariant uniqueness theorem).

Let $G$ be discrete and $X$ be compactly aligned. Suppose that either the left
action on each fibre is injective, or $P$ is directed and $X$ is
$\tilde{\phi}$-injective. The following statements are equivalent:

  1. (1)

The coaction $\delta$ is normal and injective.

  2. (2)

The Fell bundle
$\bigl{(}(\operatorname{\mathcal{NO}}_{\\!X})^{\delta}_{g}\times\\{g\\}\bigr{)}_{g\in
G}$ is amenable.

  3. (3)

The $*$-homomorphism
$\lambda_{r}\colon\operatorname{\mathcal{NO}}_{\\!X}\to\operatorname{\mathcal{NO}}_{\\!X}^{r}$
is an isomorphism.

  4. (4)

$\operatorname{\mathcal{NO}}_{\\!X}$ has the gauge-invariant uniqueness
property.

  5. (5)

If $\psi_{1}:X\to D_{1}$ and $\psi_{2}:X\to D_{2}$ are two injective gauge-
compatible CNP-covariant representations of $X$ whose images generate $D_{1}$
and $D_{2}$ respectively, then there exists a covariant $*$-isomorphism
$\phi:D_{1}\to D_{2}$ such that $\phi\circ\psi_{1}=\psi_{2}$.

###### Proof.

(1)$\Leftrightarrow$(3). This follows from the fact that
$\ker(\delta^{n})=\ker(\lambda_{r})$, as seen in the paragraph after Lemma
4.11.

(2)$\Leftrightarrow$(3). This follows from the definition of amenability for
Fell bundles over groupoids and the fact that
$\operatorname{\mathcal{NO}}_{\\!X}$ is isomorphic to
$C^{*}(E_{\mathcal{O}})$, as observed in the paragraph after Lemma 4.12.

(3)$\Rightarrow$(4) Suppose first that (3) holds. Let
$\phi:\operatorname{\mathcal{NO}}_{\\!X}\to D$ be a $*$-epimorphism. We must
show that $\phi$ is injective if and only if conditions (GI1) and (GI2) hold.
If $\phi$ is injective, then one could define $\beta$ by
$\beta:=(\phi\otimes\operatorname{id})\circ\delta\circ\phi^{-1}$, and observe
that condition (GI1) is satisfied. Now by Theorem 4.1, $(\phi\circ
j_{X})|_{A}$ is injective and condition (GI2) is also satisfied. Conversely,
suppose that conditions (GI1) and (GI2) hold. Then $\phi\circ j_{X}$ is a
gauge-compatible Nica covariant representation whose image generates $D$, and
again by Theorem 4.1, there is a $*$-homomorphism
$\tilde{\phi}:D\to\operatorname{\mathcal{NO}}_{\\!X}^{r}$ satisfying
$\tilde{\phi}\circ(\phi\circ j_{X})=j_{X}^{r}=\lambda_{r}\circ j_{X}$. Since
$\operatorname{\mathcal{NO}}_{\\!X}$ is generated by the elements
$\\{\,j_{X}(x)\mid x\in X\,\\}$ and $\lambda_{r}$ is an isomorphism by
assumption, $\lambda_{r}^{-1}\circ\tilde{\phi}$ is an inverse for $\phi$, and
$\phi$ is injective, as required.

(4)$\Rightarrow$(5). If $\psi:X\to D$ is an injective gauge-compatible CNP-
covariant representation of $X$ whose image generates $D$, then
$\Pi\psi:\operatorname{\mathcal{NO}}_{\\!X}\to D$ is an epimorphism such that
$\Pi\psi\circ j_{X}=\psi$ and (GI1) and (GI2) are satisfied, hence $\Pi\psi$
is an isomorphism.

(5)$\Rightarrow$(3). The canonical maps
$j_{X}^{r}:X\to\operatorname{\mathcal{NO}}_{\\!X}^{r}$ are two injective
gauge-compatible CNP-covariant representations whose images generate
$\operatorname{\mathcal{NO}}_{\\!X}$ and
$\operatorname{\mathcal{NO}}_{\\!X}^{r}$, respectively. Thus there exists a
$*$-isomorphism
$\phi:\operatorname{\mathcal{NO}}_{\\!X}^{r}\to\operatorname{\mathcal{NO}}_{\\!X}$
such that $\phi\circ j_{X}^{r}=j_{X}$, therefore, $\phi$ is a right inverse of
$\lambda_{r}$. Since $\phi$ is surjective, $\lambda_{r}$ is injective, as
required. ∎

We give sufficient conditions under which $\operatorname{\mathcal{NO}}_{\\!X}$
has the gauge-invariant uniqueness property. In the following result, the
coactions of $G$ on $\mathcal{T}_{X}$ and $\operatorname{\mathcal{NO}}_{\\!X}$
are different, but with a slight abuse of notation, we have denoted both by
$\delta$.

######  Corollary 4.19.

Let $G$ be discrete and $X$ be compactly aligned. Suppose that either the left
action on each fibre is injective, or $P$ is directed and $X$ is