# Operator quantum machine learning:
Navigating the chemical space of response properties

Anders S. Christensen  Institute of Physical Chemistry and National Center for
Computational Design and Discovery of Novel Materials (MARVEL), Department of
Chemistry, University of Basel, 4056 Basel, Switzerland  O. Anatole von
Lilienfeld<EMAIL_ADDRESS>Institute of Physical Chemistry
and National Center for Computational Design and Discovery of Novel Materials
(MARVEL), Department of Chemistry, University of Basel, 4056 Basel,
Switzerland

###### Abstract

The identification and use of structure property relationships lies at the
heart of the chemical sciences. Quantum mechanics forms the basis for the
unbiased virtual exploration of chemical compound space (CCS), imposing
substantial compute needs if chemical accuracy is to be reached. In order to
accelerate predictions of quantum properties without compromising accuracy,
our lab has been developing quantum machine learning (QML) based models which
can be applied throughout CCS. Here, we briefly explain, review, and discuss
the recently introduced operator formalism which substantially improves the
data efficiency for QML models of common response properties.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

Chemical compound space (CCS) comprises all the theoretically possible
molecules and materials one can conceive of. Just for medium sized organic
drug like molecules it was estimated to exceed 1060 ChemicalSpace , more than
atoms in our solar system mullard2017drug . Corresponding thermodynamic and
kinetic stability, as well as most other observable properties can be
calculated, often to a fair degree of accuracy, using well established
approximations and frameworks of quantum and statistical mechanics
JensenCompChem ; tuckerman_book_SM . Unfortunately, however, the resulting
numerical complexity is so dramatic that a first principles based exploration
of representative swaths of CCS remains prohibitive for all but its smallest
subsets. This computational burden can be reduced through the use of machine
learning (ML) which, after regressing on a sufficiently large training data
set, affords models with controlled test errors and milli-second prediction
speed.

While ML has had a long tradition in the chem- and bio-informatics communities
for a long time, its application to potential energy surface fitting for the
purpose of vibrational analysis, reaction dynamics, or molecular dynamics
already started in the nineties using neural networks (NN)
SumpterNoidNeuralNetworks1992  and kernel methods Rabitz1996 . By contrast,
inferring solutions of the electronic Schrödinger equation throughout compound
space was only made possible using Kernel Ridge Regression (KRR) and neural
networks (NN) in 2012 CM  and 2013 Montavon2013 , respectively. Corresponding
demonstrations for crystals followed shortly after schutt2014represent ;
MLcrystals_Felix2015 ; Elpasolite_2016 ; schmidt2017predicting , and ever
since, a substantial number of papers on this topic has been published in this
rapidly growing field. For more details, we refer interested readers to the
recent machine learning issues in the Int. J. Quantum Chem. RuppForeword2015
and J. Chem. Phys. rupp2018guest , as well as the overviews given in Refs.
anatole-ijqc2013 ; fpdesign2014anatole ; RaghusReview2016 ; huang2018quantum ;
QMLessayAnatole ; faber2019modeling . Due to the rigorous link to the
underlying physical laws of quantum and statistical mechanics, we have dubbed
this approach “Quantum Machine Learning” (QML), implying that we aim to model
quantum properties using classical ML algorithms. This usage of terms is in
complete analogy to Quantum Monte Carlo (using classical Monte Carlo to sample
electron configuration space), or Quantum Molecular Dynamics (integrating
Newton’s equations of motion with quantum chemically computed forces). Note,
however, that this is not to be confused with ML algorithms implemented on
quantum computers.

##  II Learning curves

For any QML model to be functional, it must learn, i.e. its predictive power
must improve with increasing training set size $N$. Vapnik and others showed
decades ago that ML models must, if set-up properly, exhibit this behaviour
and converge down to arbitrary accuracy in the limit of infinite training set
size vapnik2013book . In practice, however, arbitrary accuracy is neither
required, nor do we operate in large data set limits. Therefore the relevant
question for any distribution of data points consists of asking how much more
predictive power is to be gained for each additional training instance. This
dependence is manifested in learning curves which report prediction (or test)
error $E$ as a function of training set size, as discussed for KRR
vapnik1994learningcurves , as well as NNs StatError_Muller1996 .

In accordance with standard ML protocol AssessmentMLJCTC2013 , we stress the
importance of (i) prediction errors being obtained exclusively on hold-out
data, i.e. for input and output samples which have never been seen during
training, and (ii) hyper-parameters, noise-levels, and overfitting being
accounted for through cross-validation prior to prediction error evaluation.
Leading prediction error terms are known to be inversely proportional to
training set size, $E\propto a/N^{b}$, implying convenient linear behavior on
log-log scales vapnik1994learningcurves . Off-set $\log(a)$ and slope $b$
enable the assessment and comparison of different QML models, and imply that
prediction errors of newly developed QML models have to be reported for at
least three training set sizes in order to be (a) meaningful, and (b)
demonstrate linearity. Absence of linearity on log-log curves typically
indicates severe problems and can be due to at least one of any of the
following short-comings:

  * (i)

Not all input variables are being accounted for (e.g. due to overly coarse
representations), resulting in lack of uniqueness.

  * (ii)

The model is underdetermined due to lack of model complexity (too few
parameters, too rigid functional form).

  * (iii)

Noisy or inconsistent data.

Meeting the uniqueness criterion is a necessary condition for meaningful
models in chemistry (proof is given in Ref. FourierDescriptor ). This is is
not always obvious, however, since it also depends on subtleties in the nature
of the target property which defines which degrees of freedom are being
sampled. For example, molecular graph based representations, such as SMILES
strings, can be appropriate for generating QML models of free energies of
solvation for those temperature and pressure combinations for which all
conformational degrees of freedom are being averaged out while all covalent
bonds are being conserved. However, when it comes to properties which are
functions of instantaneous geometries (clamped nuclei), e.g. HOMO/LUMO
eigenvalues or atomic forces, SMILES are too coarse, and representations
encoding all degrees of freedom in a unique fashion are necessary.

Low QML model off-sets $\log(a)$ and steep slopes $b$ are highly desirable in
order to maximize data efficiency and thereby minimize training data needs in
order to maximize transferability through CCS. A considerable amount of recent
work has been devoted to this endeavour, demonstrating how to lower off-sets
through ever improving representations BOB ; BAML ; HDAD ; pronobis2018many ;
FCHL , the use of hierarchies among more approximate base line quantum
chemistry models DeltaPaper2015 ; Zaspel2018boosting , or optimal training
data-selection Nick2016GA . Steeper slopes have also been obtained through use
AMons (training on local building-blocks selected on the fly) Amons , as well
as through the operator QML approach discussed below.

##  III Operator approach

In 2015, we demonstrated that using the same representation and kernel, KRR
based QML models can account for any QM property SingleKernel2015 . By analogy
with basic quantum mechanics (knowledge about a single wavefunction enables
evaluation of expectation values for any property’s operator) this finding
suggest that the kernel assumes the role of the wavefunction, while KRR
regression coefficients carry the units of the property in question. These
results represent a meaningful baseline with all QML models being functional
for all properties (all learning curves indicated constant and negative slopes
$b$). We noticed, however, that slope and off-set were more favorable for some
properties than for others. In 2017, similar observations were also made for
other representations and regressors, including KRR, NNs, and random forests
HDAD .

It is known that learning curves improve upon explicit encoding of invariant
relationships between input features and labels. For example, it is crucial to
use rotational invariant features in order to predict the energy of molecules
accurately Montavon_NIPS2012 . Motivated by the apparent discrepancy of QML
models for atomization energies of small organic molecules reaching predictive
power after training on only thousands of examples, and the relatively worse
performance of the same models for predicting other properties,HDAD  we
developed and introduced the Operator QML (OQML) framework OQML . Within
kernel-based regression,MueMikRaeTsuSch01 ; scholkopf2002learning ; Vovk2013 ;
HasTibFri01  we decompose the total potential energy $U^{*}$ of a query
molecule in its electronic ground-state, into a sum of atomic contributions
$E_{I}$ for each atom $I$,

$\displaystyle U^{*}=\sum_{I\in C}E_{I}\left(q^{*}_{I}\right)=\sum_{I\in
i}\sum_{J}\alpha_{J}k(q_{J},q^{*}_{I})$ (1)

where $q$ is an atomic environment in some chosen representation basis, $J$ is
training atom $J$, and $\alpha_{J}$ is its regression weight. As a side note,
a comparative discussion of energies of atoms in molecules based on QML,
basis-set overlap, and alchemical perturbation DFT has just recently been
published AtomicAPDFT .

The OQML approach extends this KRR model while still ensuring that regression
coefficients are also obtained in closed-form. Within this framework, the
relationship between a property, typically the energy, and any of its
derivatives up to any order (typical response properties) is explicitly
enforces through the application of response operators to the learning
formalism. More specifically and using matrix notation, the response property
$\omega$, i.e. an observable which can be computed by applying a differential
operator $\mathcal{O}$ acting on the energy $U^{*}$, can be approximated by a
finite kernel as

$\displaystyle{\mathbf{\omega}}=\mathcal{O}[\mathbf{U}]\approx\mathcal{O}[\mathbf{K}]\bm{\alpha}.$
(2)

Here, we assume that the regression coefficients do not depend on the
perturbation (in complete analogy to force-field parameters, or the electron
density for first order perturbation theory within quantum mechanics).

Then, the set of regression coefficients, $\bm{\alpha}$ minimizes the
Lagrangian

$\displaystyle J(\bm{\alpha})$ $\displaystyle=$
$\displaystyle\sum_{\gamma}\beta_{\gamma}\|\mathcal{O}_{\gamma}[\mathbf{U}^{\rm
ref}]-\mathcal{O}_{\gamma}[\mathbf{K}\bm{\alpha}]\|^{2}_{L_{2}(\Omega_{\gamma})}\;\;\equiv\;\;\sum_{\gamma}\beta_{\gamma}\int_{\Omega_{\gamma}}\Big{[}\mathcal{O}_{\gamma}[\mathbf{U}^{\rm
ref}]-\mathcal{O}_{\gamma}[\mathbf{K}\bm{\alpha}]\Big{]}^{T}\Big{[}\mathcal{O}_{\gamma}[\mathbf{U}^{\rm
ref}]-\mathcal{O}_{\gamma}[\mathbf{K}\bm{\alpha}]\Big{]}$ (3)

over some training set of known derivative values of
$\mathcal{O}[\mathbf{U}^{\rm ref}]$. $\Omega_{\gamma}$ is the domain over
which the corresponding operator should be minimized, e.g. all rotational
degrees of freedom if the operator acts on a SO(3) group. $\gamma$ denotes the
specific perturbation, so that the model can be trained for multiple
properties simultaneously, for example energies, gradients, and dipole
moments. Assuming only one perturbation, the resulting expression for the
regression coefficients $\bm{\alpha}$ which solves the associated normal
equations reads,

$\displaystyle\bm{\alpha}$ $\displaystyle=$
$\displaystyle\left(\int_{\Omega}\mathcal{O}[\mathbf{K}]^{T}\mathcal{O}[\mathbf{K}]\right)^{-1}\left(\int_{\Omega}\mathcal{O}[\mathbf{U}^{\rm
ref}]^{T}\mathcal{O}[\mathbf{K}]\right)$ (4)

In comparison to the expression for regression coefficients within
conventional KRR, it is clear from these equations that kernel derivatives
must play a crucial role in the loss function and thereby constraining the
resulting regression solution towards a more meaningful manifold of solutions
from which the most probable is being obtained. For further details and
discussions, we refer the reader to the original work OQML ; FCHL19 .

Maybe the most frequent application concerned deals with derivatives with
respect to atomic displacement, i.e. forces. Much improved learning curves for
forces are obtained when compared to KRR based QML force models which were
trained directly on labels corresponding to each force vector elements after
rotating each atom into a common reference frame with unique handiness, as
done for atoms displaced along normal modes for the first time in 2015
MLatoms_2015 . Within the most recently revision of FCHL, FCHL19 FCHL19 , we
have provided a detailed analysis of the OQML based force model which can be
applied throughout CCS. Using the correct operators to generate appropriate
loss-functions ensures that important physics is automatically being enforced,
such as rotational covariance of forces, and that the resulting force models
conserve energy. A comparative discussion of other state of the art KRR force
models, such as SOAP BartokGabor_Descriptors2013 ; CeriottiScienceUnified2017
and GDML chmiela2017machine ; chmiela2019sgdml  has also been included in Ref.
FCHL19 .

Figure 1:  Improvement of learning curves of dipole moment predictions in QM9
data set QM9  with (blue) and without (red) response formalism. Both QML
models use the FCHL∗ representation and a Gaussian kernel function, and differ
only by their loss function and the resulting choice of regressor, i.e. KRR
(red) and OQML (blue). Figure reproduced from Ref. OQML , licensed under a
Creative Commons Attribution (CC BY) license.

Learning curves on display in Fig.1 exemplify the improvements gained when
exploiting the differential relationships through OQML. Learning curves are
shown for QML models of the dipole norm of small organic molecules with up to
nine atoms of the elemental types CNOF from the QM9 dataset, saturated with
hydrogen atoms QM9 . The corresponding response operator to the dipole is the
negative derivative with respect to a change in an external electric field.
Results for two kernel based models are shown, both using the FCHL
representationFCHL ; OQML  for the molecules and a Gaussian kernel function.
When training the conventional KRR model, the dipole norm is the only scalar
label and the loss function merely regresses its deviation from reference. By
contrast in the case of OQML, the operator is applied to the loss function of
the energy property before the regression (requiring the extension of the
representation to also include a simplified electric field model, denoted as
FCHL∗). This results in a loss function with very different terms which
trivially reproduces the orientational dependence of the applied field and
which yields QML models with much improved off-set and slope of learning
curve: The OQML-based machine learns the dipole norm to the same accuracy as
KRR with 20 times less data.

Similarly, in Ref. OQML  we have also demonstrated how OQML can be applied to
the second order derivatives defining Hessians which, together with dipole-
moments, has enabled the first QML based direct prediction of an infrared
spectrum from vibrational analysis.

##  IV Conclusions

We have briefly motivated the use of QML models, and highlighted the role of
learning curves. Thereafter, the OQML approach was described, and illustrated
for the case of dipole moment predictions. We stress that, in analogy to
perturbation theory going beyond the Hellmann-Feynman theorem, OQML is
sufficiently general to also go beyond first order derivatives. In fact, it is
straight-forward to apply the OQML-approach to any property which is a
derivative of the energy (or any other reference property), as long as the
representation of the molecule can be perturbed accordingly. If and how this
framework can also be used for operators other than differential operators
remains to be studied.

## Acknowledgement

We acknowledge support by the Swiss National Science foundation (No.
PP00P2_138932, 407540_167186 NFP 75 Big Data, 200021_175747), and from the
European Research Council (ERC-CoG grant QML). This work was partly supported
by the NCCR MARVEL, funded by the Swiss National Science Foundation.

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#  Cuntz-Nica-Pimsner algebras associated to product systems over quasi-
lattice ordered groupoids

Feifei Miao, Liguang Wang, Wei Yuan School of Mathematical Sciences, Qufu
Normal University, Qufu, Shandong, 273165, China<EMAIL_ADDRESS>School of
Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, China
<EMAIL_ADDRESS>AMSS, Chinese Academy of Sciences, Beijing, 100190,
China
and
School of Mathematical Sciences
University of Chinese Academy of Sciences, Beijing 100049, China
<EMAIL_ADDRESS>
###### Abstract.

We characterize Cuntz-Nica-Pimsner algebras for compactly aligned product
systems over quasi-lattice ordered groupoids. We show that the full cross
sectional $C^{*}$-algebras of Fell bundles of Morita equivalence bimodules are
isomorphic to the related Cuntz-Nica-Pimsner algebras under certain
conditions.

###### Key words and phrases:

Cuntz-Nica-Pimsner algebras; Product systems; Quasi-lattice ordered groupoids.

###### 2010 Mathematics Subject Classification:

46L05

Wang was supported in part by NSF of China (No. 11871303, No. 11971463, No.
11671133) and NSF of Shandong Province (No. ZR2019MA039, No. ZR2020MA008).
Yuan was supported in part by NSF of China (No. 11871303, No. 11871127, No.
11971463).

##  1\. Introduction

A class of $C^{*}$-algebras later called Cuntz-Krieger algebras were
introduced by Cuntz and Krieger in [2] and [3]. Since then, there have been
many generalizations of Cuntz-Krieger algebras. In [17], Pimsner studied a
class of $C^{*}$-algebras associated to C∗-correspondences which also
generalized the Cuntz-Krieger algebras. Pimsner’s work was refined by Katsura
[10]. The C∗-algebras constructed by Katsura were later called Cuntz-Pimsner
algebras.

From another perspective, Fowler generalized Pimsner’s work and studied
$C^{*}$-algebras for regular product systems of C∗-correspondences over
directed quasi-lattice ordered groups [7]. Based on the work of Nica [16] and
of Laca and Raeburn [12], Folwer introduced the notions of Nica covariant
Toeplitz representations of product systems and Cuntz-Pimsner covariant
Toeplitz representations of compactly aligned product systems. Moreover, he
described Nica-Toeplitz algebras which are universal for Nica covariant
Toeplitz representations, and Cuntz-Pimsner algebras which are universal for
Cuntz-Pimsner covariant Toeplitz representations respectively. However, it was
unclear what the right analogue of the Cuntz algebras for product systems of
non-injective $C^{*}$-correspondences should be and this was left open for
quite some time. Eventually, Sims and Yeend described Cuntz-Nica-Pimsner
algebras for some product systems over quasi-lattice ordered groups under
certain conditions [20]. Cuntz-Nica-Pimsner algebras are universal for Cuntz-
Nica-Pimsner covariant representations which are Nica covariant in the sense
of Fowler, and satisfy Cuntz-Pimsner covariant conditions which are
generalizations of Fowler’s Cuntz-Pimsner covariant conditions.

There is also another version of Cuntz-Pimsner algebras which are co-
universal. This class of C∗-algebras were introduced by Carlsen, Larsen, Sims
and Vittadello in [1] and are co-universal for injective, Nica covariant,
gauge-compatible representations of product systems under certain conditions.
It was shown in [1] that the co-universal Cuntz-Pimsner algebras and the
Cuntz-Nica-Pimsner algebras of some product systems are generalizations of
reduced and full crossed products of $C^{*}$-algebras by groups respectively.
In further work, Dor-On, Kakariadis, Katsoulis, Laca, and Li [5] generalized
Dor-On and Katsoulis’ results [4] and showed that there exist co-universal
algebras of compactly aligned product systems over group-embeddable right LCM
semigroups. Motivated by the above results, we introduced the notion of
product systems over quasi-lattice ordered groupoids in [14]. And we showed
that the co-universal algebras of compactly aligned product systems over
quasi-lattice ordered groupoids do exist.

In this paper, we will describe Cuntz-Nica-Pimsner algebras for some compactly
aligned product systems over quasi-lattice ordered groupoids which are
universal for injective Cuntz-Nica-Pimsner covariant representations.

This paper is organized as follows. In section 2, we recall the terminologies
and basic results which will be used later. In section 3, we introduce the
Cuntz-Pimsner condition for Toeplitz representations and study the Cuntz-Nica-
Pimsner representations for compactly aligned product systems over quasi-
lattice ordered groupoids. In section 4, we show that the full cross sectional
$C^{*}$-algebra of the Fell bundle is isomorphic to the Cuntz-Nica-Pimsner
algebra of the associated product system when the fibers of a Fell bundle over
a quasi-lattice ordered groupoid are all Morita equivalence bimodules.

##  2\. Preliminaries

###  2.1. $C^{\ast}$-correspondences

Let $\mathcal{B}$ be a C∗-algebra and $X$ be a right Hilbert
$\mathcal{B}$-module (see [13]). We use $\mathcal{L}(X)$ (resp.
$\mathcal{K}(X)$) to denote the C∗-algebra of all adjointable (resp. compact)
operators on $X$.

######  Definition 2.1.

For $C^{\ast}$-algebras $\mathcal{A}$ and $\mathcal{B}$, a
$\mathcal{A}$-$\mathcal{B}$ $C^{\ast}$-correspondence is a right Hilbert
$\mathcal{B}$-module $X$ together with a left $\mathcal{A}$-action induced by
a non-degenerate $*$-homomorphism $\phi:\mathcal{A}\to\mathcal{L}(X)$, i.e.
$\mathcal{A}\cdot X:=\overline{\rm{span}}\\{\phi(a)\xi:a\in\mathcal{A},\xi\in
X\\}=X$.

To simplify notation, we will write $\phi(a)\xi$ as $a\xi$ in the following.

######  Definition 2.2.

A bimodule map $F:X\to Y$ between two $\mathcal{A}$-$\mathcal{B}$
C∗-correspondences $X$ and $Y$ is called a C∗-correspondence map if

$\langle\xi,\beta\rangle_{{\mathcal{B}}{}}=\langle
F(\xi),F(\beta)\rangle_{{\mathcal{B}}{}}$

for all $\xi$, $\beta\in X$. A $C^{*}$-correspondence map is called a
C∗-correspondence isomorphism if it is a unitary.

######  Remark 2.1.

A C∗-correspondence map between two $\mathcal{A}$-$\mathcal{B}$
C∗-correspondences $X$ and $Y$ is not adjointable in general. For example, let
$\Omega$ be a compact Hausdorff space and $\Omega_{0}$ be a non-empty closed
subset of $\Omega$ whose complement is dense in $\Omega$. Then

$X:=\\{f\in C(\Omega):f(\Omega_{0})=0\\}$

is a non-unital C∗-subalgebra of $C(\Omega)$. It is clear that $X$ and
$C(\Omega)$ are both $C(\Omega)$-$C(\Omega)$ C∗-correspondences with the
canonical structure. And the inclusion map $F:X\to Y$ is a C∗-correspondence
map. But $F$ is not adjointable since $F^{*}$, if exists, must map
$1_{C(\Omega)}$ to the unit of $X$.

Let $X$ be a $\mathcal{A}$-$\mathcal{B}$ C∗-correspondence and $Y$ be a
$\mathcal{B}$-$\mathcal{C}$ $C^{*}$-correspondence, where $\mathcal{A}$,
$\mathcal{B}$, and $\mathcal{C}$ are C∗-algebras. The $\mathcal{B}$-balanced
tensor product $X\odot_{{\mathcal{B}}{}}Y$ together with right
$\mathcal{C}$-action

$\displaystyle(\xi\odot\eta)\cdot c=\xi\odot\eta\cdot c$

and $\mathcal{C}$-valued inner product

$\displaystyle\langle\xi_{1}\odot\eta_{1},\xi_{2}\odot\eta_{2}\rangle_{{\mathcal{C}}{}}:=\langle\eta_{1},\langle\xi_{1},\xi_{2}\rangle_{{\mathcal{B}}{}}\eta_{2}\rangle_{{\mathcal{C}}{}}$

for $\xi,\xi_{1},\xi_{2}\in X$, $\eta,\eta_{1},\eta_{2}\in Y$, is a pre-
Hilbert right $\mathcal{C}$-module. We use $X\otimes_{{\mathcal{B}}{}}Y$ to
denote the completion of $X\odot_{{\mathcal{B}}{}}Y$. Then
$X\otimes_{{\mathcal{B}}{}}Y$ is a $\mathcal{A}$-$\mathcal{C}$
C∗-correspondence.

######  Definition 2.3.

A $\mathcal{A}$-$\mathcal{B}$ $C^{*}$-correspondence $X$ is called a Hilbert
$C^{*}$ $\mathcal{A}$-$\mathcal{B}$-bimodule if $X$ equipped with a left
$\mathcal{A}$-valued inner product
${}_{{\mathcal{A}}{}}\langle\cdot,\cdot\rangle$ such that the left
$\mathcal{A}$-module $X$ together with
${}_{{\mathcal{A}}{}}\langle\cdot,\cdot\rangle$ is a left Hilbert
$\mathcal{A}$-module and

$\displaystyle{}_{{\mathcal{A}}{}}\langle\xi,\eta\rangle\zeta=\xi\langle\eta,\zeta\rangle_{{\mathcal{B}}{}},\quad\forall\xi,\eta,\zeta\in
X.$

If, moreover, ${}_{{\mathcal{A}}{}}\langle X,X\rangle$ and $\langle
X,X\rangle_{{\mathcal{B}}{}}$ are dense in $\mathcal{A}$ and $\mathcal{B}$
respectively, then $X$ is called a Morita equivalence bimodule.

###  2.2. Product systems over quasi-lattice ordered groupoids

Let $G$ be a small category. We use $\operatorname{Ob}(G)$ to denote the set
of objects in $G$ and $G(x,y)$ to denote the set of morphisms in $G$ with
source $x$ and target $y$. Let $G(-,-)$ be the set of morphisms in $G$. For
every $x,y\in\operatorname{Ob}(G)$, we use $G(x,-)$ (resp. $G(-,y)$) to denote
the set of morphisms in $G$ with source $x$ (resp. target $y$). For $f\in
G(-,-)$, we use $s(f)$ (resp. $t(f)$) to denote the source (resp. target) of
$f$. For $x\in\operatorname{Ob}(G)$, we use $1_{x}$ to denote the identity
morphism of $x$. A discrete groupoid is a category in which every morphism is
invertible (We refer the reader to [18] for a basic introduction of category
theory).

######  Definition 2.4 ([14]).

A quasi-lattice ordered groupoid is a pair $(G,P)$, where $G$ is a discrete
groupoid and $P$ is a wide subcategory of $G$, i.e.,
$\operatorname{Ob}(P)=\operatorname{Ob}(G)$, such that

  1. (1)

for every $x,y\in\operatorname{Ob}(P)$,

$\displaystyle P(x,y)\cap P^{-1}(x,y)=\begin{cases}\emptyset,&\mbox{if $x\neq
y$;}\\\ 1_{x},&\mbox{if $x=y$,}\end{cases}$

where $P^{-1}(x,y):=\\{f^{-1}:f\in P(y,x)\\}$;

  2. (2)

under the partial order $f\leq g\iff f^{-1}g\in P$, $f,g\in G(-,-)$, every
finite set of morphisms in $G$ with an upper bound in $P$ has a least upper
bound in $P$.

######  Example 2.2 ([14]).

Let $(G^{\prime},P^{\prime})$ be a quasi-lattice ordered group.

  1. (1)

Note that $G^{\prime}$ is a groupoid with only one object and
$(G^{\prime},P^{\prime})$ is a quasi-lattice ordered groupoid.

  2. (2)

Let $X$ be a $G^{\prime}$-set, $G^{\prime}$ acts on $X$. The transformation
groupoid $G$ is the category with $\operatorname{Ob}(G)=X$ and $G(x,y)=\\{g\in
G^{\prime}:g\cdot x=y\\}$ (see example 8.1.15 in [19]). Let $P$ be the wide
subcategory of $G$ such that $P(x,y)=\\{g\in P^{\prime}:g\cdot x=y\\}$. Recall
that $P^{\prime}\cap P^{\prime-1}=\\{e\\}$, where $e$ is the unit of
$G^{\prime}$. Then $(G,P)$ is a quasi-lattice ordered groupoid.

For the rest of this paper, we assume that $(G,P)$ is a quasi-lattice ordered
groupoid. For $f,g\in G(-,-)$, if they have a common upper bound in $P$, we
use $f\vee g$ or $\vee\\{f,g\\}$ to denote their least upper bound. Otherwise,
we write $f\lor g=\infty$.

######  Definition 2.5.

A product system over (G, P) is a triple

$(\\{\mathcal{A}_{x}\\}_{x\in Ob(P)},\\{X_{f}\\}_{f\in
P(-,-)},\\{m_{g,f}\\}_{\\{(g,f)\in P(-,-):t(f)=s(g)\\}}),$

where $\mathcal{A}_{x}$ is a C∗-algebra, $X_{f}$ is a
$\mathcal{A}_{t(f)}$-$\mathcal{A}_{s(f)}$ C∗-correspondence, and $m_{g,f}$ is
a $C^{*}$-correspondence isomorphism from
$X_{g}\otimes_{{\mathcal{A}_{t(f)}}{}}X_{f}$ to $X_{gf}$, subject to the
following axioms:

  1. (1)

$X_{1_{x}}=\mathcal{A}_{x}$ for every $x\in\operatorname{Ob}(P)$;

  2. (2)

$m_{1_{x},f}(a\otimes_{\mathcal{A}_{x}}\xi)=a\xi$ and
$m_{f,1_{w}}(\xi\otimes_{\mathcal{A}_{w}}b)=\xi b$ for $f\in P(w,x)$,
$a\in\mathcal{A}_{x}$, $b\in\mathcal{A}_{w}$, $\xi\in X_{f}$;

  3. (3)

for every $f\in P(w,x)$, $g\in P(x,y)$, $h\in P(y,z)$, the following diagram

commutes.

In the following, we will write$(\\{\mathcal{A}_{x}\\}_{x\in
Ob(P)},\\{X_{f}\\}_{f\in P(-,-)},\\{m_{g,f}\\})$ as
$(\\{\mathcal{A}_{x}\\},\\{X_{f}\\})$ and write
$m_{g,f}(\xi_{g}\otimes_{\mathcal{A}_{s}(f)}\eta_{f})$ as $\xi_{g}\eta_{f}$.

For $f,g\in P(-,-)$, there is a $*$-homomorphism
$l^{g}_{f}:\mathcal{L}(X_{f})\to\mathcal{L}(X_{g})$ defined by

$\displaystyle
l^{g}_{f}(A):=\begin{cases}m_{f,f^{-1}g}(A\otimes_{\mathcal{A}_{s(f)}}I)m^{*}_{f,f^{-1}g},&\text{
if}\quad f\leq g;\\\ 0,&\text{otherwise},\end{cases}$

where $A\in\mathcal{L}(X_{f})$. By identifying $\mathcal{K}(X_{1_{t(g)}})$
with $\mathcal{A}_{t(g)}$, we have $l^{g}_{1_{t(g)}}(a)(\xi_{g})=a\xi_{g}$ by
condition (2) in Definition 2.5, for $\xi_{g}\in X_{g}$,
$a\in\mathcal{A}_{t(g)}$.

######  Definition 2.6.

A product system $X=(\\{\mathcal{A}_{x}\\},\\{X_{f}\\})$ over $(G,P)$ is
called compactly aligned if $S\vee T:=l^{f\vee g}_{f}(S)l^{f\vee g}_{g}(T)$ is
compact, i.e., $S\vee T\in\mathcal{K}(X_{f\vee g})$, for every
$S\in\mathcal{K}(X_{f})$, $T\in\mathcal{K}(X_{g})$, and $f,g\in P(-,-)$ such
that $f\vee g<\infty$.

###  2.3. Representations of a product system over a quasi-lattice ordered
groupoid

######  Definition 2.7.

Let $(\\{\mathcal{A}_{x}\\},\\{X_{f}\\})$ be a product system over $(G,P)$. A
Toeplitz representation of $(\\{\mathcal{A}_{x}\\},\\{X_{f}\\})$ is a family
of linear maps $\\{\psi_{f}:X_{f}\to\mathcal{B}\\}_{f\in P(-,-)}$, where
$\mathcal{B}$ is a C∗-algebra, such that

  1. (1)

$\psi_{1_{x}}:\mathcal{A}_{x}\to\mathcal{B}$ is a $*$-homomorphism for every
$x\in\operatorname{Ob}(P)$;

  2. (2)

$\psi_{g}(\beta)\psi_{f}(\xi)=\psi_{gf}(\beta\xi)$ for every $\beta\in X_{g}$,
$\xi\in X_{f}$, $g$, $f\in P(-,-)$, $s(g)=t(f)$;

  3. (3)

$\psi_{f}(\eta)^{*}\psi_{f}(\xi)=\psi_{1_{s(f)}}(\langle\eta,\xi\rangle_{{\mathcal{A}_{s(f)}}{}})$
for $\xi,\eta\in X_{f}$, $f\in P(-,-)$.

Let $\psi=\\{\psi_{f}:X_{f}\to\mathcal{B}\\}_{f\in P(-,-)}$ be a Toeplitz
representation of $(\\{\mathcal{A}_{x}\\},\\{X_{f}\\})$. The condition (3) in
Definition 2.7 implies that $\psi_{f}$ is a contraction, for every $f\in
P(-,-)$. And $\psi_{f}$ is isometric if $\psi_{1_{s(f)}}$ is injective. We say
$\psi$ is injective if $\psi_{1_{x}}$ is injective for all
$x\in\operatorname{Ob}(P)$.

From now on, we always assume that $X=(\\{\mathcal{A}_{x}\\},\\{X_{f}\\})$ is
a compactly aligned product system over $(G,P)$ unless otherwise specified.
For a Toeplitz representation $\psi=\\{\psi_{f}:X_{f}\to\mathcal{B}\\}_{f\in
P(-,-)}$ of $X$, there are $*$-homomorphisms
$\psi^{(f)}:\mathcal{K}(X_{f})\to\mathcal{B}$ such that
$\psi^{(f)}(\theta_{\xi,\eta})=\psi_{f}(\xi)\psi_{f}(\eta)^{*}$, where
$\theta_{\xi,\eta}(-)$ is the rank one operator
$\xi\langle\eta,-\rangle_{{\mathcal{A}_{s(f)}}{}}$ in $\mathcal{K}(X_{f})$(see
[17, p. 202], [9, Lemma 2.2] and [8, Remark 1.7]). In the following, we adopt
the convention that $\psi^{\infty}\equiv 0$.

######  Definition 2.8.

A Toeplitz representation $\psi=\\{\psi_{f}\\}_{f\in P(-,-)}$ of
$(\\{\mathcal{A}_{x}\\},\\{X_{f}\\})$ is called Nica covariant if

$\displaystyle\psi^{(f)}(S)\psi^{(g)}(T)=\psi^{(f\vee g)}(S\vee T),$

for every $f,g\in P(-,-)$, $S\in\mathcal{K}(X_{f})$ and
$T\in\mathcal{K}(X_{g})$.

######  Remark 2.3 (see [14]).

Let $\psi$ be a Nica covariant Toeplitz representation of
$(\\{\mathcal{A}_{x}\\},\\{X_{f}\\})$. For $f,g\in P(-,-)$, $\xi_{f}\in
X_{f}$, $\eta_{g}\in X_{g}$,

  1. (1)

if $s(f)\neq t(g)$, then $\psi_{f}(\xi_{f})\psi_{g}(\eta_{g})=0$;

  2. (2)

if $f\vee g<\infty$, then
$\psi_{f}(\xi_{f})^{*}\psi_{g}(\eta_{g})\in\overline{\rm{span}}\\{\psi_{f^{-1}(f\vee
g)}(\xi^{\prime})\psi_{g^{-1}(f\vee g)}(\eta^{\prime})^{*}:\xi^{\prime}\in
X_{f^{-1}(f\vee g)},\eta^{\prime}\in X_{g^{-1}(f\vee g)}\\}$. Otherwise,
$\psi_{f}(\xi_{f})^{*}\psi_{g}(\eta_{g})=0$.

Let $\psi=\\{\psi_{f}\\}_{f\in P(-,-)}$ be a Nica covariant Toeplitz
representation of $X$. In the following, we use $C^{*}(\psi)$ to denote the
$C^{*}$-algebra generated by the image of $\psi$. It follows from Remark 2.3
that

(1) $\displaystyle
C^{*}(\psi)=\overline{\mbox{span}}\\{\psi_{f}(\xi_{f})\psi_{g}(\eta_{g})^{*}:f,g\in
P(-,-),\xi_{f}\in X_{f},\eta_{g}\in X_{g}\\}.$

######  Example 2.4 (The Fock representation).

Let $X=(\\{\mathcal{A}_{x}\\},\\{X_{f}\\})$ be a compactly aligned product
system over $(G,P)$. We use $C_{c}(X)$ to denote the set of maps

$\xi:P(-,-)\to\cup_{f\in P(-,-)}X_{f}$

with finite support such that $\xi(f)\in X_{f}$. Similarly,
$C_{c}(\\{\mathcal{A}_{x}\\}_{x\in\operatorname{Ob}(P)})$ is the set of maps
$\beta:\operatorname{Ob}(P)\to\cup_{x\in\operatorname{Ob}(P)}\mathcal{A}_{x}$
with finite support such that $\beta(x)\in\mathcal{A}_{x}$. Let
$C_{0}(\\{\mathcal{A}_{x}\\})$ be the completion of
$C_{c}(\\{\mathcal{A}_{x}\\})$ with respect to the supremum norm. Note that
$C_{0}(\\{\mathcal{A}_{x}\\})$ is a C∗-algebra, and $C_{c}(X)$ together with
the right $C_{0}(\\{\mathcal{A}_{x}\\})$-action and the
$C_{0}(\\{\mathcal{A}_{x}\\})$-valued inner product

$\displaystyle[\xi\cdot\beta](f):=\xi(f)\cdot\beta(s(f)),\quad\langle\xi,\eta\rangle_{{C_{0}(\\{\mathcal{A}_{x}\\})}{}}(x):=\sum_{f\in
P(x,-)}\langle\xi(f),\eta(f)\rangle_{{\mathcal{A}_{x}}{}},$

is a pre-Hilbert right module over $C_{0}(\\{\mathcal{A}_{x}\\})$. The Fock
space $\mathscr{F}(X)$ is the completion of $C_{c}(X)$. The Fock
representation $\\{\eta_{g}\mapsto
L_{\eta_{g}}:X_{g}\to\mathcal{L}(\mathscr{F}(X))\\}_{g\in P(-,-)}$ of $X$ is
defined by

$\displaystyle(L_{\eta_{g}}\xi)(f)=\begin{cases}\eta_{g}\xi(g^{-1}f),&g\leq
f;\\\ 0,&\mbox{otherwise}.\end{cases}$

The Fock representation is a Nica covariant Toeplitz representation of $X$.
And the Fock algebra $\mathcal{T}^{r}_{X}$ is the $C^{*}$-algebra generated by
the image of the Fock representation of $X$ (see [14] for more details).

The Nica-Toeplitz algebra for $X$ is a C∗-algebra $\mathcal{N}\mathcal{T}_{X}$
equipped with a Nica covariant Toeplitz representation
$i=\\{i_{f}:X_{f}\to\mathcal{N}\mathcal{T}_{X}\\}_{f\in P(-,-)}$ such that

  1. (1)

$\mathcal{N}\mathcal{T}_{X}=C^{*}(i)$,

  2. (2)

$i$ is universal in the sense that for any other Nica covariant Toeplitz
representation $\psi:X\to\mathcal{B}$, there is a unique $*$-homomorphism
$\psi^{\prime}:\mathcal{N}\mathcal{T}_{X}\to\mathcal{B}$ rendering the
following diagram

commutative.

###  2.4. The full cross sectional $C^{*}$-algebra $C^{*}(\mathfrak{B})$ of a
Fell bundle $\mathfrak{B}$

Let $\\{\mathfrak{B}_{f}\\}_{f\in G(-,-)}$ be a Banach bundle over discrete
groupoid $G$ (see [6, Definition 13.4]). We write $\mathfrak{B}_{1_{x}}$ as
$\mathfrak{B}_{x}$ for convenience. A multiplication on
$\\{\mathfrak{B}_{f}\\}_{f\in G(-,-)}$ is a family of continuous bilinear maps

$\displaystyle\\{\mathfrak{B}_{f}\times\mathfrak{B}_{g}\to\mathfrak{B}_{fg}\\}_{(f,g)\in
G(-,-)^{2},s(f)=t(g)}$

such that

  1. (1)

$(\xi_{f}\eta_{g})\zeta_{h}=\xi_{f}(\eta_{g}\zeta_{h})$ whenever the
multiplication is defined;

  2. (2)

$\|\xi_{f}\eta_{g}\|\leq\|\xi_{f}\|\|\eta_{g}\|$ for all
$\xi_{f}\in\mathfrak{B}_{f}$, $\eta_{g}\in\mathfrak{B}_{g}$, $f,g\in G(-,-)$,
$s(f)=t(g)$.

An involution on $\\{\mathfrak{B}_{f}\\}_{f\in G(-,-)}$ is a family of
continuous involutive conjugate linear maps

$\displaystyle\xi_{f}\mapsto\xi_{f}^{*}:\mathfrak{B}_{f}\to\mathfrak{B}_{f^{-1}},\quad
f\in G(-,-),\xi_{f}\in\mathfrak{B}_{f}.$

######  Definition 2.9 ([11]).

A Fell bundle over a discrete groupoid $G$ is a Banach bundle
$\\{\mathfrak{B}_{g}\\}_{g\in G(-,-)}$ equipped with a multiplication and an
involution such that for $f,g\in G(-,-)$, $\xi_{f}\in\mathfrak{B}_{f}$,
$\eta_{g}\in\mathfrak{B}_{g}$ with $s(f)=t(g)$,

$\displaystyle(\xi_{f}\eta_{g})^{*}=\eta^{*}_{g}\xi^{*}_{f},\quad\|\xi_{f}^{*}\xi_{f}\|=\|\xi_{f}\|^{2},$

and $\xi_{f}^{*}\xi_{f}$ is a positive element of the C∗-algebra
$\mathfrak{B}_{s(f)}$.

For the rest of this subsection, we assume that
$\mathfrak{B}=\\{\mathfrak{B}_{f}\\}_{f\in G(-,-)}$ is a Fell bundle over a
discrete groupoid $G$.

######  Remark 2.5.

For each $f\in G(-,-)$, $\mathfrak{B}_{f}$ together with inner products

$\displaystyle\langle\xi_{f},\eta_{f}\rangle_{{\mathfrak{B}_{s(f)}}{}}:=\xi^{*}_{f}\eta_{f},\quad{}_{{\mathfrak{B}_{t(f)}}{}}\langle\xi_{f},\eta_{f}\rangle:=\xi_{f}\eta^{*}_{f}$

is a Hilbert $C^{*}$ $\mathfrak{B}_{t(f)}$-$\mathfrak{B}_{s(f)}$-bimodule. And
$\mathfrak{B}^{*}_{f}=\mathfrak{B}_{f^{-1}}$.

######  Definition 2.10 ([14]).

A representation of $\mathfrak{B}$ in a $C^{*}$-algebra $\mathcal{C}$ is a
collection of linear maps
$\pi=\\{\pi_{g}:\mathfrak{B}_{g}\to\mathcal{C}\\}_{g\in G(-,-)}$ such that

  1. (1)

$\pi_{f}(\xi_{f})\pi_{g}(\eta_{g})=\delta_{s(f),t(g)}\pi_{fg}(\xi_{f}\eta_{g})$;

  2. (2)

$\pi_{g}(\eta_{g})^{*}=\pi_{g^{-1}}(\eta^{*}_{g})$,

for all$f,g\in G(-,-)$, $\xi_{f}\in\mathfrak{B}_{f}$,
$\eta_{g}\in\mathfrak{B}_{g}$.

Recall that $C_{c}(\mathfrak{B})$ is the set of finitely supported sections of
$\mathfrak{B}$, i.e., a family of maps $\xi:G(-,-)\to\cup_{g\in
G(-,-)}\mathfrak{B}_{g}$ with finite support such that
$\xi(g)\in\mathfrak{B}_{g}$. Then $C_{c}(\mathfrak{B})$ together with
multiplication and involution

$\displaystyle\xi\eta(h):=\sum_{h=fg}\xi(f)\eta(g),\quad\xi^{*}(f):=\xi(f^{-1})^{*}$

is a $*$-algebra (see [14] for more detials).

######  Definition 2.11 ([14]).

The full cross sectional $C^{*}$-algebra $C^{*}(\mathfrak{B})$ of
$\mathfrak{B}$ is the completion of $C_{c}(\mathfrak{B})$ with respect to the
norm

$\displaystyle\|\xi\|_{u}:=sup_{\pi}\\{\|\pi(\xi)\|:\pi\ \ \textit{is the
$*$-representation of $C_{c}(\mathfrak{B})$}\\},$

where $\xi\in C_{c}(\mathfrak{B})$.

The set of representations of $\mathfrak{B}$ in C∗-algebras corresponds to the
set of $*$-homomorphisms from $C_{c}(\mathfrak{B})$ into C∗-algebras one-to-
one. Then $C^{*}(\mathfrak{B})$ is a $C^{*}$-algebra generated by the image of
the universal representation $\pi^{u}=\\{\pi^{u}_{g}\\}$ of $\mathfrak{B}$ in
the sense that if $\pi=\\{\pi_{g}\\}_{g\in G(-,-)}$ is a representation of
$\mathfrak{B}$ in a $C^{*}$-algebra $\mathcal{B}$, then there exists a unique
$*$-homomorphism $\widehat{\pi}:C^{*}(\mathfrak{B})\to\mathcal{B}$ such that
$\widehat{\pi}\circ\pi^{u}_{g}(\xi_{g})=\pi_{g}(\xi_{g})$ for all $g\in
G(-,-)$ and $\xi_{g}\in\mathfrak{B}_{g}$.

##  3\. Cuntz-Nica-Pimsner algebras of product systems over quasi-lattice
ordered groupoids

Let $X=(\\{\mathcal{A}_{x}\\},\\{X_{f}\\})$ be a compactly aligned product
system over $(G,P)$. For $g\in P(x,y)$, let

$\displaystyle I_{g}:=\begin{cases}\mathcal{A}_{y}&x=y\mbox{ and }g=1_{y};\\\
\cap_{1_{y}<f\leqslant g}\operatorname{Ker}\phi_{f}&x=y\mbox{ and }g\neq
1_{y};\\\ 0&x\neq y,\end{cases}$

where $\phi_{f}$ is the left action of $\mathcal{A}_{y}$ on $X_{f}$. We use
$C^{g}_{c}(X)$ to denote the set of maps

$\xi:\\{f\in P(x,y):f\leq g\\}\to\cup_{f\in P(x,y),f\leq g}X_{f}I_{f^{-1}g}$

with finite support such that $\xi(f)\in X_{f}I_{f^{-1}g}$. Note that
$C^{g}_{c}(X)$ together with the right $\mathcal{A}_{x}$-action and right
$\mathcal{A}_{x}$-valued inner product

$\displaystyle[\xi\cdot a_{x}](f):=\xi(f)a_{x}\ \ \mbox{and}\ \
\langle\xi,\eta\rangle_{{\mathcal{A}_{x}}{}}=\sum_{f\in P(x,y),f\leq
g}\langle\xi(f),\eta(f)\rangle_{{\mathcal{A}_{x}}{}},$

for all $\xi,\eta\in C^{g}_{c}(X)$ and $a_{x}\in\mathcal{A}_{x}$, is a pre-
Hilbert right $\mathcal{A}_{x}$-module. Let $\widetilde{X}_{g}$ be the
completion of $C^{g}_{c}(X)$. We will use $\widetilde{\phi}_{g}$ to denote the
left action of $\mathcal{A}_{y}$ on $\widetilde{X}_{g}$ induced by $\phi_{g}$.
We say $X$ is $\widetilde{\phi}$-injective if every $\widetilde{\phi}_{g}$ is
injective.

For $f,g\in P(-,y)$, let
$\widetilde{l}_{f}^{g}:\mathcal{L}(X_{f})\to\mathcal{L}(\widetilde{X}_{g})$ be
the $*$-homomorphism defined by

$\displaystyle\widetilde{l}^{g}_{f}(A)(\xi)(h):=l^{h}_{f}(A)(\xi(h)),$

for $A\in L(X_{f})$, $\xi\in\widetilde{X}_{g}$, $h\in P(s(g),y)$ and
$h\leqslant g$. In particular,
$\widetilde{l}_{1_{y}}^{g}(a)(\xi)=\widetilde{\phi}_{g}(a)\xi$, for all
$a\in\mathcal{A}_{y}$.

Similar to the definition 3.8 in [20], we say a statement $\mathcal{P}(g)$ is
true for large $g$ if for every $f\in P(-,-)$, there exists $h\in P(-,-)$,
$h\geqslant f$ such that $\mathcal{P}(g)$ is true whenever $g\geq h$.

######  Definition 3.1.

Assume that $X$ is $\widetilde{\phi}$-injective. A Toeplitz representation
$\psi=\\{\psi_{f}\\}_{f\in P(-,-)}$ of $X$ is Cuntz-Pimsner covariant if
$\sum_{f\in\mathscr{F}}\psi^{(f)}(T_{f})=0$ for every
$y\in\operatorname{Ob}(P)$, every finite subset $\mathscr{F}$ of $P(-,y)$, and
every choice of compact operators
$\\{T_{f}\in\mathcal{K}(X_{f}):f\in\mathscr{F}\\}$ such that
$\sum_{f\in\mathscr{F}}\widetilde{l}^{g}_{f}(T_{f})=0$ for large $g$.

A Nica covariant Toeplitz representation is called a Cuntz-Nica-Pimsner
covariant (or CNP-covariant) representation if it is Cuntz-Pimsner covariant.

A nonempty set $\mathcal{S}\subseteq P(-,y)$ is said to be bounded if there
exists $g\in P(-,y)$ such that $k\leqslant g$ for every $k\in\mathcal{S}$. An
element $f\in\mathcal{S}$ is called maximal (resp. minimal) if $f\nleqslant h$
(resp. $h\nleq f$) for every $h\in\mathcal{S}\backslash\\{f\\}$.

We next show that $X$ is $\widetilde{\phi}$-injective under some conditions.

######  Proposition 3.1.

  1. (1)

If $\phi_{f}$ is injective for all $f\in P(-,-)$, then $X$ is
$\widetilde{\phi}$-injective.

  2. (2)

Suppose every nonempty bounded subset of $P(x,y)$ contains a maximal element
and for every $k\in P(x,y)$, $\\{f\in P(x,y):f\leq k\\}$ has a minimal element
$f^{\prime}$ such that $\phi_{f^{\prime}}$ is injective. Then $X$ is
$\widetilde{\phi}$-injective.

###### Proof.

(1) If $\phi_{g}$ is injective for every $g\in P(-,-)$, then we can identify
$\widetilde{X}_{g}$ with $X_{g}$. So $\widetilde{\phi}_{g}$ is injective.

(2) Let $g\in P(x,y)$ and $0\neq a\in\mathcal{A}_{y}$. It is enough to show
that there exist some $f\leqslant g$, $f\in P(x,y)$, such that
$\phi_{f}(a)|_{X_{f}\cdot I_{f^{-1}g}}\neq 0$. Let

$\mathcal{S}=\\{f\in P(x,y):f\leqslant g,\phi_{f}(a)\neq 0\\}.$

Since $\\{f\in P(x,y):f\leqslant g\\}$ has a minimal element $f^{\prime}$ such
that $\phi_{f^{\prime}}$ is injective, $\mathcal{S}$ is nonempty. Let $f_{0}$
be a maximal element of $\mathcal{S}$. Then there is a $\xi\in X_{f_{0}}$ such
that $\phi_{f_{0}}(a)\xi\neq 0$. If $1_{x}<h\leqslant f_{0}^{-1}g$, we have
$\phi_{f_{0}h}(a)=0$. For $\eta\in X_{h}$, we have

$\displaystyle\langle\eta,\langle\phi_{f_{0}}(a)\xi,\phi_{f_{0}}(a)\xi\rangle_{{\mathcal{A}_{x}}{}}\eta\rangle_{{\mathcal{A}_{s(h)}}{}}=\langle(\phi_{f_{0}}(a)\xi)\eta,(\phi_{f_{0}}(a)\xi)\eta\rangle_{{\mathcal{A}_{s(h)}}{}}=0.$

So
$\langle\phi_{f_{0}}(a)\xi,\phi_{f_{0}}(a)\xi\rangle_{{\mathcal{A}_{x}}{}}\in\ker\phi_{h}$.
It follows that
$\langle\phi_{f_{0}}(a)\xi,\phi_{f_{0}}(a)\xi\rangle_{{\mathcal{A}_{x}}{}}\in
I_{f_{0}^{-1}g}$. Recall that if $X$ is a right Hilbert $\mathcal{A}$-module
and $I$ is an ideal of $\mathcal{A}$, then

$X\cdot I=\\{\xi\in X:\langle\xi,\xi\rangle_{{\mathcal{A}}{}}\in I\\}$

(see [20]). Then $\phi_{f_{0}}(a)\xi\in X_{f_{0}}I_{f_{0}^{-1}g}$. Suppose
$\\{e_{i}\\}_{i\in\mathcal{I}}$ is an increasing positive approximate identity
for $I_{f_{0}^{-1}g}$. Then

$\displaystyle
0\neq\phi_{f_{0}}(a)\xi=\lim_{i\in\mathcal{I}}\phi_{f_{0}}(a)\xi\cdot e_{i}.$

Then there exists $i_{0}$ such that $\phi_{f_{0}}(a)\xi\cdot e_{i_{0}}\neq 0$.
We have $\phi_{f_{0}}(a)|_{X_{f_{0}}I_{f_{0}^{-1}g}}\neq 0$ since $\xi\cdot
e_{i_{0}}\in X_{f_{0}}I_{f_{0}^{-1}g}$. This completes the proof. ∎

For a Nica covariant Toeplitz representation $\psi=\\{\psi_{f}\\}_{f\in
P(-,-)}$ of product system $X=(\\{\mathcal{A}_{x}\\},\\{X_{f}\\})$, let
$I_{C^{*}(\psi)}\subset C^{*}(\psi)$ be the closed ideal generated by
$\cup_{y\in\operatorname{Ob}(P)}I_{C^{*}(\psi),y}$, where

$\displaystyle I_{C^{*}(\psi),y}=$
$\displaystyle\\{\sum_{f\in\mathscr{F}}\psi^{(f)}(T_{f}):\textit{$\mathscr{F}$
is a finite set in $P(-,y)$},$
$\displaystyle\text{$T_{f}\in\mathcal{K}(X_{f})$ and
$\sum_{f\in\mathscr{F}}\widetilde{l}^{g}_{f}(T_{f})=0$ for large $g\in
P(-,y)$}\\}.$

Let $q_{C^{*}(\psi)}$ be the quotient map from $C^{*}(\psi)$ to
$C^{*}(\psi)/I_{C^{*}(\psi)}$.

######  Lemma 3.2.

With notations as above.

$q_{C^{*}(\psi)}\circ\psi:=\\{q_{C^{*}(\psi)}\circ\psi_{f}:X_{f}\to
C^{*}(\psi)/I_{C^{*}(\psi)}\\}_{f\in P(-,-)}$

is a CNP-covariant Toeplitz representation.

###### Proof.

Since $\psi$ is a Nica covariant Toeplitz representation of $X$ and
$q_{C^{*}(\psi)}$ is a $*$-homomorphism, $q_{C^{*}(\psi)}\circ\psi$ is a Nica
covariant Toeplitz representation of $X$ and hence $q_{C^{*}(\psi)}\circ\psi$
is CNP-covariant. ∎

######  Lemma 3.3.

If $X$ is $\widetilde{\phi}$-injective, then $q_{\mathcal{T}^{r}_{X}}\circ L$
is injective, where $L$ is the Fock representation of $X$ and
$\mathcal{T}^{r}_{X}=C^{*}(L)$ (see Example 2.4).

###### Proof.

It suffices to show that for every $y\in\operatorname{Ob}(P)$, we have
$I_{\mathcal{T}^{r}_{X}}\cap L_{\mathcal{A}_{y}}=\\{0\\}$. Note that
$\widetilde{X}_{g}$ is a submodule of $\mathscr{F}(X)$, and
$L_{a}\xi=\widetilde{\phi}_{g}(a)\xi$ for $a\in\mathcal{A}_{y}$,
$\xi\in\widetilde{X}_{g}$, $g\in P(-,y)$. Since $\widetilde{\phi}_{g}$ is
injective, $\|L_{a}|_{\widetilde{X}_{g}}\|=\|\widetilde{\phi}_{g}(a)\|=\|a\|$.

We use $\overline{I}_{\mathcal{T}^{r}_{X},z}$ to denote the closure of
$I_{\mathcal{T}^{r}_{X},z}$. Let $S\in\overline{I}_{\mathcal{T}^{r}_{X},z}$
and $\xi_{f}\in X_{f}$, $\eta_{k}\in X_{k}$, $\xi^{\prime}_{f^{\prime}}\in
X_{f^{\prime}}$, $\eta^{\prime}_{k^{\prime}}\in X_{k^{\prime}}$. We claim that
for every $\epsilon\geq 0$, and $g_{0}\in P(-,y)$, there exists
$\gamma_{0}\geqslant g_{0}$ such that
$\|L_{\xi_{f}}L_{\eta_{k}}^{*}SL_{\xi^{\prime}_{f^{\prime}}}L_{\eta^{\prime}_{k^{\prime}}}^{*}|_{\widetilde{X}_{h}}\|\leq\epsilon$
for every $h\geqslant\gamma_{0}$.

If $k^{\prime}\vee g_{0}=\infty$, then for $h\geqslant g_{0}$, we have
$h\ngeqslant k^{\prime}$. Therefore
$L_{\eta^{\prime}_{k^{\prime}}}^{*}|_{\widetilde{X}_{h}}=0$. And

$L_{\xi_{f}}L_{\eta_{k}}^{*}SL_{\xi^{\prime}_{f^{\prime}}}L_{\eta^{\prime}_{k^{\prime}}}^{*}|_{\widetilde{X}_{h}}=0,\forall
h\geqslant g_{0}.$

Assume now that $k^{\prime}\vee g_{0}<\infty$. If $f^{\prime}\notin
P(s(k^{\prime}),z)$, then
$L_{\xi_{f}}L_{\eta_{k}}^{*}SL_{\xi^{\prime}_{f^{\prime}}}L_{\eta^{\prime}_{k^{\prime}}}^{*}|_{\widetilde{X}_{h}}=0$
for every $h\in P(-,-)$. If $f^{\prime}\in P(s(k^{\prime}),z)$, then there’s
$\gamma^{\prime}\geqslant f^{\prime}k^{\prime-1}(k^{\prime}\vee g_{0})$ such
that

(2)
$\displaystyle\|S|_{\widetilde{X}_{h^{\prime}}}\|\leq\epsilon/(\|\xi_{f}\|\|\eta_{k}\|\|\xi^{\prime}_{f^{\prime}}\|\|\eta^{\prime}_{k^{\prime}}\|)$

for every $h^{\prime}\geqslant\gamma^{\prime}$, since
$S\in\overline{I}_{\mathcal{T}^{r}_{X},z}$. Let
$\gamma_{0}:=k^{\prime}f^{\prime-1}\gamma^{\prime}(\geqslant g_{0})$. Then for
$h\geqslant\gamma_{0}$ we have
$f^{\prime}k^{\prime-1}h\geqslant\gamma^{\prime}$. Thus eq. 2 implies that

$\displaystyle\|L_{\xi_{f}}L_{\eta_{k}}^{*}SL_{\xi^{\prime}_{f^{\prime}}}L_{\eta^{\prime}_{k^{\prime}}}^{*}|_{\widetilde{X}_{h}}\|\leq\|\xi_{f}\|\|\eta_{k}\|\|\xi^{\prime}_{f^{\prime}}\|\|\eta^{\prime}_{k^{\prime}}\|\|S|_{\widetilde{X}_{f^{\prime}k^{\prime-1}h}}\|\
\leq\epsilon,\quad\forall h\geqslant\gamma_{0}.$

Since $L$ is Nica covariant (see eq. 1), we have

$I_{\mathcal{T}^{r}_{X}}=\overline{\rm{span}}\\{L_{\xi_{f}}L_{\eta_{k}}^{*}SL_{\xi^{\prime}_{f^{\prime}}}L_{\eta^{\prime}_{k^{\prime}}}^{*}:S\in\cup_{z\in\operatorname{Ob}(P)}\overline{I}_{\mathcal{T}^{r}_{X},z},\xi_{f},\eta_{k},\xi^{\prime}_{f^{\prime}},\eta^{\prime}_{k^{\prime}}\in
X\\}.$

Then the claim above implies that $I_{\mathcal{T}^{r}_{X}}\cap
L_{\mathcal{A}_{y}}=\\{0\\}$. ∎

Next we will describe the universal $C^{*}$-algebras for CNP-covariant
representations of compactly aligned product systems over quasi-lattice
ordered groupoids.

######  Theorem 3.4.

If $X$ is $\widetilde{\phi}$-injective, then there is a pair
$(\mathcal{N}\mathcal{O}_{X},j)$ such that

  1. (1)

$j$ is an injective CNP-covariant Toeplitz representation of $X$ and
$C^{*}(j)=\mathcal{N}\mathcal{O}_{X}$;

  2. (2)

$j$ is universal, i.e., for any CNP-covariant representation
$\psi=\\{\psi_{f}\\}_{f\in P(-,-)}$ of $X$, there is a unique $*$-homomorphism
$\psi_{*}:\mathcal{N}\mathcal{O}_{X}\to C^{*}(\psi)$ such that the following
diagram

commutes.

Moreover, $(\mathcal{N}\mathcal{O}_{X},j)$ is the unique pair (up to canonical
isomorphism) satisfying conditions (1) and (2).

###### Proof.

Let $(\mathcal{N}\mathcal{T}_{X},i)$ be the Nica-Toeplitz algebra of $X$. Let
$\mathcal{N}\mathcal{O}_{X}:=\mathcal{N}\mathcal{T}_{X}/I_{\mathcal{N}\mathcal{T}_{X}}$.
By Lemma 3.2, $j:=q_{\mathcal{N}\mathcal{T}_{X}}\circ i$ is a CNP-covariant
representation of $X$, where
$q_{\mathcal{N}\mathcal{T}_{X}}:\mathcal{N}\mathcal{T}_{X}\to\mathcal{N}\mathcal{O}_{X}$
is the quotient map. Let $\psi=\\{\psi_{f}\\}$ be a CNP-covariant
representation of $X$. By the universal property of $i$, there is a
$*$-homomorphism $\psi^{\prime}:\mathcal{N}\mathcal{T}_{X}\to C^{*}(\psi)$
such that $\psi=\psi^{\prime}\circ i$. Since $\psi$ is Cuntz-Pimsner
covariant, $I_{\mathcal{N}\mathcal{T}_{X}}\subset\ker(\psi^{\prime})$. Thus
$\phi^{\prime}$ induces a $*$-homomorphism
$\psi_{*}:\mathcal{N}\mathcal{O}_{X}\to C^{*}(\psi)$ such that
$\psi=\psi_{*}\circ j$. In particular, Lemma 3.3 implies that $j$ is
injective. By the universal property of $j$, it is clear that
$(\mathcal{N}\mathcal{O}_{X},j)$ is the unique pair (up to canonical
isomorphism) satisfying (1) and (2). ∎

The pair $(\mathcal{N}\mathcal{O}_{X},j)$ is called the Cuntz-Nica-Pimsner
algebra of $X$. These algebras generalize Cuntz-Nica-Pimsner algebras of
product systems over quasi-lattice ordered group (see [20, Proposition 3.12]).

##  4\.  Cuntz-Nica-Pimsner algebras are generalizations of full cross
sectional $C^{*}$-algebras of Fell bundles

In this section, we assume that $X=(\\{\mathcal{A}_{x}\\},\\{X_{f}\\})$ is a
compactly aligned product system over quasi-lattice ordered groupoid $(G,P)$.
We will show that the full cross sectional $C^{*}$-algebras of Fell bundles of
Morita equivalence bimodules are isomorphic to the related Cuntz-Nica-Pimsner
algebras. In order to do this, we first provide a characterization of the
Cuntz-Pimsner covariant representation.

######  Lemma 4.1.

Let $\psi=\\{\psi_{f}\\}_{f\in P(-,-)}$ be a Toeplitz representation of
$X=(\\{\mathcal{A}_{x}\\},\\{X_{f}\\})$. If
$\phi_{f}(\mathcal{A}_{t(f)})\subseteq\mathcal{K}(X_{f})$ and
$\psi^{(f)}(\phi_{f}(-))=\psi_{1_{t(f)}}(-)$ for every $f\in P(-,-)$, then
$l^{g}_{f}(T_{f})\in\mathcal{K}(X_{g})$ and

$\displaystyle\psi^{(g)}(l^{g}_{f}(T_{f}))=\psi^{(f)}(T_{f}),$

for $f\leqslant g\in P(-,-)$, $T_{f}\in\mathcal{K}(X_{f})$.

###### Proof.

Let $f\in P(-,-)$. Since
$\phi_{f}(\mathcal{A}_{t(f)})\subseteq\mathcal{K}(X_{f})$, we have
$l^{g}_{f}(T_{f})\in\mathcal{K}(X_{g})$ for all $g\geq f$ and
$T_{f}\in\mathcal{K}(X_{f})$ by [13, Proposition 4.7]. Note that

$\psi^{(f^{-1}g)}(\phi_{f^{-1}g}(-))=\psi_{1_{s(f)}}(-)$

for $g\geq f$. For $\xi$, $\eta\in X_{f}$, $a,b\in\mathcal{A}_{s(f)}$, we have

$\displaystyle\psi^{(f)}(\theta_{\xi a,\eta b})$
$\displaystyle=\psi_{f}(\xi)\psi_{1_{s(f)}}(ab^{*})\psi_{f}(\eta)^{*}$
$\displaystyle=\psi_{f}(\xi)\psi^{(f^{-1}g)}(\phi_{f^{-1}g}(ab^{*}))\psi_{f}(\eta)^{*}.$

By [17, corollary 3.7], there eixst $\xi^{n}_{i},\eta^{n}_{i}\in X_{f^{-1}g}$
such that

$\displaystyle\phi_{f^{-1}g}(ab^{*})=\lim_{n}\sum_{i}\theta_{\xi^{n}_{i},\eta^{n}_{i}},\mbox{
and }l^{g}_{f}(\theta_{\xi a,\eta
b})=\lim_{n}\sum_{i}\theta_{\xi\xi^{n}_{i},\eta\eta^{n}_{i}}.$

Therefore, $\psi^{(f)}(\theta_{\xi a,\eta b})=\psi^{(g)}(l^{g}_{f}(\theta_{\xi
a,\eta b}))$. Since $\mathcal{K}(X_{f})$ is a $C^{*}$-algebra generated by
rank one compact operators and $X_{f}\mathcal{A}_{s(f)}=X_{f}$, we have

$\psi^{(g)}(l^{g}_{f}(T_{f}))=\psi^{(f)}(T_{f})$

for every $T_{f}\in\mathcal{K}(X_{f})$. ∎

######  Proposition 4.2.

Let $\psi:X\to\mathcal{B}$ be a Toeplitz representation of
$X=(\\{\mathcal{A}_{x}\\},\\{X_{f}\\})$. Assume that any two elements in
$P(-,y)$ has a least upper bound for every $y\in\operatorname{Ob}(P)$, and the
left actions $\phi_{f}:\mathcal{A}_{t(f)}\to\mathcal{L}(X_{f})$ is injective
for every $f\in P(-,-)$.

(1) If $\psi$ is Cuntz-Pimsner covariant, then
$\psi^{(f)}(\phi_{f}(a))=\psi_{1_{t(f)}}(a)$ for
$a\in\phi_{f}^{-1}(\mathcal{K}(X_{f}))$, $f\in P(-,-)$.

(2) If $\phi_{f}(\mathcal{A}_{t(f)})\subseteq\mathcal{K}(X_{f})$ and
$\psi^{(f)}(\phi_{f}(-))=\psi_{1_{t(f)}}(-)$ for every $f\in P(-,-)$, then
$\psi$ is Cuntz-Pimsner covariant.

###### Proof.

Since $\phi_{f}$ is injective for $f\in P(-,-)$, we have
$\widetilde{X}_{f}=X_{f}$ and $\widetilde{l}^{g}_{f}(-)=l^{g}_{f}(-)$ for all
$g\in P(-,-)$.

(1) Let $f\in P(-,y)$ and $a\in\phi^{-1}_{f}(\mathcal{K}(X_{f}))$. For every
$h\in P(-,y)$ we have

$\displaystyle
l^{g}_{1_{y}}(a)-l^{g}_{f}(\phi_{f}(a))=l^{g}_{f}(l^{f}_{1_{y}}(a)-\phi_{f}(a))=0$

for every $g\geqslant f\vee h$, since $l^{f}_{1_{y}}(a)=\phi_{f}(a)$.
Therefore

$\displaystyle\psi_{1_{y}}(a)-\psi^{(f)}(\phi_{f}(a))=\psi^{(1_{y})}(a)-\psi^{(f)}(\phi_{f}(a))=0,$

since $\psi$ is Cuntz-Pimsner covariant.

(2) Suppose $l^{g}_{h}(T)=0$ for $T\in\mathcal{L}(X_{h})$, $g\geqslant h\in
P(-,-)$. Then for $\xi\in X_{h}$, $\eta\in X_{h^{-1}g}$,

$\displaystyle 0$ $\displaystyle=\langle
l^{g}_{h}(T)(\xi\eta),l^{g}_{h}(T)(\xi\eta)\rangle_{{\mathcal{A}_{s(g)}}{}}$
$\displaystyle=\langle(T\xi)\eta,(T\xi)\eta\rangle_{{\mathcal{A}_{s(g)}}{}}$
$\displaystyle=\langle\eta,\langle
T\xi,T\xi\rangle_{{\mathcal{A}_{s(h)}}{}}\eta\rangle_{{\mathcal{A}_{s(g)}}{}}.$

Then $\phi_{h}(\langle T\xi,T\xi\rangle_{{\mathcal{A}_{s(h)}}{}})=0$. Since
$\phi_{h}$ is injective, $T=0$. Thus $l^{g}_{h}$ is injective. Let
$\mathscr{F}$ be a finite subset of $P(-,y)$, and $T_{f}\in\mathcal{K}(X_{f})$
such that

$\displaystyle\sum_{f\in\mathscr{F}}l^{g}_{f}(T_{f})=0$

for large $g\in P(-,y)$. Then

$\displaystyle l^{g}_{h_{0}}(\sum_{f\in\mathscr{F}}l^{h_{0}}_{f}(T_{f}))=0$

for $g\geqslant h_{0}=\vee\mathscr{F}$. Since $l^{g}_{h_{0}}$ is injective, we
have $\sum_{f\in\mathscr{F}}l^{h_{0}}_{f}(T_{f})=0$. Then by Lemma 4.1, we
have

$\displaystyle\sum_{f\in\mathscr{F}}\psi^{(f)}(T_{f})=\psi^{(h_{0})}(\sum_{f\in\mathscr{F}}l^{h_{0}}_{f}(T_{f}))=0.$

Thus $\psi$ is Cuntz-Pimsner covariant. ∎

Let $\psi=\\{\psi_{f}:X_{f}\to\mathcal{B}(\mathcal{H})\\}_{f\in P(-,-)}$ be a
Toeplitz representation of $X$, where $\mathcal{B}(\mathcal{H})$ is the
$C^{*}$-algebra of all bounded operators on a Hilbert space $\mathcal{H}$.
Proposition 2.5 in [13] implies that the $*$-homomorphism $\psi^{(f)}$ admits
a strict-SOT continuous extension
$\psi^{(f)}:\mathcal{L}(X_{f})\to\mathcal{B}(\mathcal{H})$. It is obvious that
$\psi:X\to\mathcal{B}(\mathcal{H})$ is Nica covariant if and only if

$\displaystyle\psi^{(f)}(I)\psi^{(g)}(I)=\psi^{(f\vee g)}(I)$

for $f,g\in P(-,-)$.

######  Corollary 4.3.

Suppose that each pair in $P(-,y)$ has a least upper bound for all
$y\in\operatorname{Ob}(P)$ and left actions
$\phi_{f}:\mathcal{A}_{t(f)}\to\mathcal{L}(X_{f})$ are all injective with
$\phi_{f}(\mathcal{A}_{t(f)})\subseteq\mathcal{K}(X_{f})$ for $f\in P(-,-)$.
Let $\psi:X\to\mathcal{B}(\mathcal{H})$ be a Toeplitz representation of $X$
such that

$\psi_{1_{x}}(\mathcal{A}_{x})\psi_{1_{y}}(\mathcal{A}_{y})=0$

for all $x\neq y\in\operatorname{Ob}(P)$. Then $\psi$ is CNP-covariant if and
only if

$\psi^{(f)}\circ\phi_{f}(-)=\psi_{1_{t(f)}}(-).$

###### Proof.

If $\psi$ is CNP-covariant, then
$\psi^{(f)}\circ\phi_{f}(-)=\psi_{1_{t(f)}}(-)$ by Proposition 4.2 (1).
Suppose $\psi^{(f)}\circ\phi_{f}(-)=\psi_{1_{y}}(-)$ for all $f\in P(x,y)$.
Let

$\psi_{f}(X_{f})\mathcal{H}:=\overline{\rm{span}}\\{\psi_{f}(\xi_{f})\eta:\xi_{f}\in
X_{f},\eta\in\mathcal{H}\\}.$

Since $\phi_{f}(\mathcal{A}_{y})\subseteq\mathcal{K}(X_{f})$, we have
$\psi_{1_{y}}(\mathcal{A}_{y})\mathcal{H}\subseteq\psi_{f}(X_{f})\mathcal{H}$
by [15, Lemma 1.9]. Note that $\mathcal{A}_{y}\cdot X_{f}=X_{f}$. By the
condition (2) in Definition 2.7, we have

$\displaystyle\psi_{f}(X_{f})\mathcal{H}=\psi_{f}(\mathcal{A}_{y}\cdot
X_{f})\mathcal{H}=\psi_{1_{y}}(\mathcal{A}_{y})\psi_{f}(X_{f})\mathcal{H}\subseteq\psi_{1_{y}}(\mathcal{A}_{y})\mathcal{H}.$

Therefore
$\psi_{1_{y}}(\mathcal{A}_{y})\mathcal{H}=\psi_{f}(X_{f})\mathcal{H}$ and
$\psi^{(f)}(I)=\psi^{(1_{y})}(I)$. Let $g\in P(-,-)$. Note $f\vee g<\infty$ if
and only if $t(g)=y$. If $f\vee g<\infty$, then

$\psi^{(f)}(I)\psi^{(g)}(I)=\psi^{(1_{y})}(I)=\psi^{(f\vee g)}(I).$

If $t(g)\neq y$, then $\psi^{(f)}(I)\psi^{(g)}(I)=0$ since
$\psi_{1_{y}}(\mathcal{A}_{y})\psi_{1_{t}(g)}(\mathcal{A}_{t(g)})=0$. Thus
$\psi$ is Nica covariant. By Proposition 4.2 (2), $\psi$ is CNP-covariant. ∎

Suppose $(G,P)$ is a quasi-lattice ordered groupoid such that each pair in
$P(-,y)$ has a least upper bound for all $y\in\operatorname{Ob}(P)$ and $G$ is
generated by $P$. Let $\mathfrak{B}=\\{\mathfrak{B}_{g}\\}_{g\in G(-,-)}$ be a
Fell bundle over $G$ with each $\mathfrak{B}_{g}$ is a Morita equivalence
bimodule. Note that $\mathfrak{B}_{g}\mathfrak{B}_{k}=\mathfrak{B}_{gk}$ for
$g,k\in G(-,-)$, $s(g)=t(k)$. Then

$\widehat{\mathfrak{B}}:=(\\{\mathfrak{B}_{x}\\},\\{\mathfrak{B}_{f}\\}_{f\in
P(-,-)})$

is a product system over $(G,P)$ with isomorphisms
$\mathfrak{B}_{f}\otimes_{\mathfrak{B}_{s(f)}}\mathfrak{B}_{h}\cong\mathfrak{B}_{fh}$
corresponding to the multiplication in the Fell bundle $\mathfrak{B}$. It is
obvious that $\widehat{\mathfrak{B}}$ is a compactly aligned product system.

######  Theorem 4.4.

With notations as above. We have $C^{*}(\\{\mathfrak{B}_{g}\\}_{g\in
G(-,-)})\cong\mathcal{N}\mathcal{O}_{\widehat{\mathfrak{B}}}$.

###### Proof.

Let $\pi^{u}=\\{\pi^{u}_{g}\\}_{g\in G(-,-)}$ be the universal representation
of the Fell bundle $\mathfrak{B}$. By the definition of representations of
Fell bundles, we see that $\\{\pi^{u}_{f}:\mathfrak{B}_{f}\to
C^{*}(\\{\mathfrak{B}_{g}\\}_{g\in G(-,-)})\\}_{f\in P(-,-)}$ is a Toeplitz
representation of the compactly aligned product system
$\widehat{\mathfrak{B}}$ in $C^{*}(\\{\mathfrak{B}_{g}\\}_{g\in G(-,-)})$.

For each $f\in P(-,-)$, we see that
$\phi_{f}:\mathfrak{B}_{t(f)}\to\mathcal{K}(\mathfrak{B}_{f})$ is an
isomorphism which sends
${}_{{\mathfrak{B}_{t(f)}}{}}\langle\xi_{f},\eta_{f}\rangle$ to
$\theta_{\xi_{f},\eta_{f}}$ for $\xi_{f},\eta_{f}\in\mathfrak{B}_{f}$. Then by
the definition of representations of the Fell bundles, we have

$\displaystyle(\pi^{u})^{(f)}(\phi_{f}(\sum_{i}{}_{{\mathfrak{B}_{t(f)}}{}}\langle\xi_{i},\eta_{i}\rangle))$
$\displaystyle=(\pi^{u})^{(f)}(\sum_{i}\theta_{\xi_{i},\eta_{i}})$
$\displaystyle=\sum_{i}\pi^{u}_{f}(\xi_{i})\pi^{u}_{f}(\eta_{i})^{*}$
$\displaystyle=\sum_{i}\pi^{u}_{1_{t(f)}}(\xi_{i}\eta^{*}_{i})$
$\displaystyle=\pi^{u}_{1_{t(f)}}(\sum_{i}{}_{{\mathfrak{B}_{t(f)}}{}}\langle\xi_{i},\eta_{i}\rangle),$

where $\sum_{i}{}_{{\mathfrak{B}_{t(f)}}{}}\langle\xi_{i},\eta_{i}\rangle$ is
a finite sum of $\mathfrak{B}_{t(f)}$ with
$\xi_{i},\eta_{i}\in\mathfrak{B}_{f}$. Since
${}_{{\mathfrak{B}_{t(f)}}{}}\langle\mathfrak{B}_{f},\mathfrak{B}_{f}\rangle$
is dense in $\mathfrak{B}_{t(f)}$, we have

$(\pi^{u})^{(f)}\circ\phi_{f}(-)=\pi^{u}_{1_{t(f)}}(-).$

It is easy to see that

$\displaystyle\pi^{u}_{1_{x}}(\mathfrak{B}_{x})\pi^{u}_{1_{y}}(\mathfrak{B}_{y})=0$

for $x\neq y\in\operatorname{Ob}(P)$. By Corollary 4.3,
$\\{\pi^{u}_{f}\\}_{f\in P(-,-)}$ is a CNP-covariant Toeplitz representation
of $\widehat{\mathfrak{B}}$. Recall that
$(\mathcal{N}\mathcal{O}_{\widehat{\mathfrak{B}}},j)$ is the Cuntz-Nica-
Pimsner algebra of $\widehat{\mathfrak{B}}$. By the universal property of $j$,
there is a $*$-homomorphism $\phi_{1}$ such that $\phi_{1}\circ
j_{f}=\pi^{u}_{f}$ for $f\in P(-,-)$.

Since $G$ is generated by $P$, for each $g\in G(-,-)$ there exist $f,h\in
P(-,-)$ such that $\mathfrak{B}_{g}=\mathfrak{B}_{f}\mathfrak{B}_{h}^{*}$. Let
$\pi_{g}:\mathfrak{B}_{g}\to\mathcal{N}\mathcal{O}_{\widehat{\mathfrak{B}}}$
be a linear map defined by

$\pi_{g}(\xi_{f}\eta_{h}^{*}):=j_{f}(\xi_{f})j_{h}(\eta_{h})^{*}$

for $\xi_{f}\in\mathfrak{B}_{f}$, $\eta_{h}\in\mathfrak{B}_{h}$, $f,h\in
P(-,-)$ with $fh^{-1}=g$. We see that

$\pi_{g}(\xi_{f}\eta_{h}^{*})^{*}=j_{h}(\eta_{h})j_{f}(\xi_{f})^{*}=\pi_{g^{-1}}(\eta_{h}\xi_{f}^{*}).$

Then $\pi_{g}(\xi)^{*}=\pi_{g^{-1}}(\xi^{*})$ for $\xi\in\mathfrak{B}_{g}$.

If $f\leq h\in P(-,-)$, since $j$ is a Toeplitz representation of
$\widehat{\mathfrak{B}}$, we have

$\displaystyle j_{f}(\xi_{f})^{*}j_{h}(\eta_{f}\zeta_{f^{-1}h})$
$\displaystyle=j_{f}(\xi_{f})^{*}j_{f}(\eta_{f})j_{f^{-1}h}(\zeta_{f^{-1}h})$
$\displaystyle=j_{1_{s(f)}}(\langle\xi_{f},\eta_{f}\rangle_{{\mathfrak{B}_{s(f)}}{}})j_{f^{-1}h}(\zeta_{f^{-1}h})$
$\displaystyle=j_{f^{-1}h}(\xi_{f}^{*}\eta_{f}\zeta_{f^{-1}h})$

for all $\xi_{f},\eta_{f}\in\mathfrak{B}_{f}$,
$\zeta_{f^{-1}h}\in\mathfrak{B}_{f^{-1}h}$. It follows that
$j_{f}(\xi_{f})^{*}j_{h}(\eta_{h})=j_{f^{-1}h}(\xi_{f}^{*}\eta_{h})$ for
$\eta_{h}\in\mathfrak{B}_{h}$. For $g,k\in G(-,-)$ with $s(g)=t(k)$, we have

$\displaystyle\pi_{g}(\xi_{f_{1}}\eta^{*}_{h_{1}})\pi_{k}({}_{{\mathfrak{B}_{t(k)}}{}}\langle\beta_{h_{1}\vee
f_{2}},\zeta_{h_{1}\vee
f_{2}}\rangle\xi^{\prime}_{f_{2}}\eta^{\prime*}_{h_{2}})$
$\displaystyle=j_{f_{1}}(\xi_{f_{1}})j_{h_{1}}(\eta_{h_{1}})^{*}j_{1_{t(k)}}({}_{{t(k)}{}}\langle\beta_{h_{1}\vee
f_{2}},\zeta_{h_{1}\vee
f_{2}}\rangle)j_{f_{2}}(\xi^{\prime}_{f_{2}})j_{h_{2}}(\eta^{\prime}_{h_{2}})^{*}$
$\displaystyle=j_{f_{1}}(\xi_{f_{1}})j_{h_{1}}(\eta_{h_{1}})^{*}j_{h_{1}\vee
f_{2}}(\beta_{h_{1}\vee f_{2}})j_{h_{1}\vee f_{2}}(\zeta_{h_{1}\vee
f_{2}})^{*}j_{f_{2}}(\xi^{\prime}_{f_{2}})j_{h_{2}}(\eta^{\prime}_{h_{2}})^{*}$
$\displaystyle=j_{f_{1}h^{-1}_{1}(h_{1}\vee
f_{2})}(\xi_{f_{1}}\eta^{*}_{h_{1}}\beta_{h_{1}\vee
f_{2}})j_{h_{2}f^{-1}_{2}(h_{1}\vee
f_{2})}((\eta^{\prime}_{h_{2}}\xi^{\prime*}_{f_{2}}\zeta_{h_{1}\vee
f_{2}})^{*}$
$\displaystyle=\pi_{gk}(\xi_{f_{1}}\eta^{*}_{h_{1}}\beta_{h_{1}\vee
f_{2}}\zeta_{h_{1}\vee f_{2}}^{*}\xi^{\prime}_{f_{2}}\eta^{\prime*}_{h_{2}})$

where $f_{1},h_{1},f_{2},h_{2}\in P(-,-)$ with $f_{1}h^{-1}_{1}=g$,
$f_{2}h^{-1}_{2}=k$, $\xi_{f_{1}}\in\mathfrak{B}_{f_{1}}$,
$\eta_{h_{1}}\in\mathfrak{B}_{h_{1}}$, $\beta_{h_{1}\vee
f_{2}},\zeta_{h_{1}\vee f_{2}}\in\mathfrak{B}_{h_{1}\vee f_{2}}$,
$\xi^{\prime}_{f_{2}}\in\mathfrak{B}_{f_{2}}$,
$\eta^{\prime}_{h_{2}}\in\mathfrak{B}_{h_{2}}$. Since
${}_{{\mathfrak{B}_{t(k)}}{}}\langle\mathfrak{B}_{h_{1}\vee
f_{2}},\mathfrak{B}_{h_{1}\vee f_{2}}\rangle\mathfrak{B}_{k}$ is dense in
$\mathfrak{B}_{k}$, we have $\pi_{g}(-)\pi_{k}(-)=\pi_{gk}(-)$. If $s(g)\neq
t(k)$, by condition (1) in Remark 2.3, we have $\pi_{g}(-)\pi_{k}(-)=0$ since
$j$ is a Nica covariant Toeplitz representation of $\widehat{\mathfrak{B}}$.
Therefore, $\pi=\\{\pi_{g}\\}_{g\in G(-,-)}$ is a representation of the Fell
bundle $\mathfrak{B}$. Then there is a $*$-homomorphism

$\phi_{2}:C^{*}(\\{\mathfrak{B}_{g}\\}_{g\in
G(-,-)})\to\mathcal{N}\mathcal{O}_{\widehat{\mathfrak{B}}}$

such that $\phi_{2}\circ\pi^{u}_{g}(-)=\pi_{g}(-)$. It is obvious that
$\phi_{1}$ and $\phi_{2}$ are mutually reversible. Then

$C^{*}(\\{\mathfrak{B}_{g}\\}_{g\in
G(-,-)})\cong\mathcal{N}\mathcal{O}_{\widehat{\mathfrak{B}}}.$

This completes the proof. ∎

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