11institutetext: LESIA UMR-8109, Observatoire de Paris, Université PSL, CNRS,
Sorbonne Université, Univ. Paris Diderot, Sorbonne Paris Cité, 5 place Jules
Janssen, 92195 Meudon, France 22institutetext: naXys, University of Namur, 8
Rempart de la Vierge, Namur, B-5000, Belgium 33institutetext: Université de
Lille, Observatoire de Lille, 1 impasse de l’Observatoire, 59000 Lille, France
44institutetext: IMCCE, Observatoire de Paris, CNRS UMR 8028, 77 Avenue
Denfert Rochereau, 75014 Paris, France 55institutetext: Univ. Lyon, Univ. Lyon
1, ENS de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574,
F-69230, Saint-Genis-Laval, France 66institutetext: Université de la Côte
d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France
77institutetext: Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France
88institutetext: University of Toronto Scarborough, Toronto, ON M1C 1A4,
Canada.
88email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Neptune’s ring arcs from VLT/SPHERE-IRDIS near-infrared
observations††thanks: This study is based on observations made with ESO
Telescopes at the La Silla Paranal Observatory, Chile, under programme SPHERE
ID - 097.C- 0336 (A).

D. Souami 11 2 2 S. Renner 33 4 4 B. Sicardy 11 M. Langlois 55 B. Carry 66 P.
Delorme 77 P. Golaszewska 88

(Received 20 June 2021; accepted 15 October 2021)

###### Abstract

Context. Neptune’s incomplete ring arcs have been stable since their discovery
in 1984 by stellar occultation. Although these structures should be destroyed
within a few months through differential Keplerian motion, imaging data over
the past couple of decades has shown that these structures are persistent.

Aims. We present here the first SPHERE near-infrared observations of Neptune’s
ring arcs taken at 2.2 $\mu$m (BB-Ks) with the IRDIS camera at the Very Large
Telescope in August 2016.

Methods. The images were aligned using the ephemerides of the satellite
Proteus and were suitably co-added to enhance ring and satellite signals.

Results. We analyse high-angular resolution near-infrared images of Neptune’s
ring arcs obtained in 2016 at the ESO VLT-UT3 with the adaptive-optics fed
camera SPHERE-IRDIS. We derive here accurate mean motion values for the arcs
and the nearby satellite Galatea. The trailing arcs Fraternité and Egalité are
stable since they were last observed in 2007. Furthermore, we confirm the
fading away of the leading arcs Courage and Liberté. Finally, we confirm the
mismatch between the arcs’ position and 42:43 inclined and eccentric
corotation resonances with Galatea; thus demonstrating that no 42:43
corotation model works to explain the azimuthal confinement of the arcs’
materiel.

###### Key Words.:

planets and satellites: individual: Neptune – planets and satellites: rings –
celestial mechanics – techniques: photometric.

##  1 Introduction

A stellar occultation campaign on July 22nd, 1984 yielded the first conclusive
evidence of an incomplete ring-like structure around Neptune, inside the
classical Roche limit (assuming a density of 1 for the ring material) (Roques
et al., 1984; Sicardy et al., 1985; Hubbard et al., 1986; Covault et al.,
1986). A summary of the discovery of the arcs is given in (Nicholson et al.,
1990); the authors compiled the results of five occultations (between April
18th, 1984 and August 20th, 1985), as well as the 1989 Voyager 2 data
confirming three distinct features.

Voyager 2 data revealed that the arcs are longitudinally confined over a 40∘
azimuthal range; they are embedded in the much fainter continuous Adams ring
around Neptune (Smith et al., 1989) which is the outermost ring of the
Neptunian system.

These incomplete rings have shown to be stable since their discovery in 1984
(Hubbard et al., 1986), while they would be expected to scatter in a few
months through differential Keplerian motion. The azimuthal confinement of the
arc system (Goldreich et al., 1986; Porco, 1991) was thought to be a
consequence of the arcs being within a 42:43 corotation inclination resonance
(hereafter CIR) forced by Galatea. However, observations in 1998 (Dumas et
al., 1999; Sicardy et al., 1999) showed a slight mismatch between the observed
arc mean motion and its expected value in the framework of the 42:43 CIR
model.

Furthermore, adaptive-optics data obtained with the Keck telescope in 2002 and
2003 (de Pater et al., 2005) as well as VLT-UT3 NACO data of 2007 (Renner et
al., 2014) showed persisting two trailing arcs. They also showed that the
brightness and longitudes of the arcs (Fraternité, Egalité, Liberté, and
Courage) have significantly changed since Voyager (de Pater et al., 2005;
Renner et al., 2014; De Pater et al., 2018).

Several theoretical models addressed the question of the arcs’ stability and
the aforementioned mean motion mismatch. For example, Namouni & Porco (2002)
have shown that the 42:43 corotation eccentricity resonance (CER) could match
the current arcs’ semi-major axis and stabilise the system, if the arcs
contain a small fraction of Galatea’s mass ($m_{G}$). Assuming a mass
$m_{G}=2.1\times 10^{21}$g for Galatea, the model which is very sensitive to
the uncertainty in Galatea’s eccentricity ($e_{G}$) gave a ring mass of $\sim
0.23\,m_{G}$ ($0.002\,m_{G}$, resp.) for $e_{G}=10^{-4}$ ($e_{G}=10^{-6}$,
resp.).

Hereafter, we adopt the 42:43 nomenclature for simplicity reasons. While the
CER is a first order resonance (a true 42:43), the CIR is a second order
resonance (84:86). 111Their respective corotation critical arguments write as
follows $\Psi_{CER}=43\lambda-42\lambda_{G}-\varpi_{G}$ and
$\Psi_{CIR}=\leavevmode\nobreak\ {2\,[43\lambda-42\lambda_{G}-\Omega_{G}]}$.
Where $\lambda$ ($\lambda_{G}$) is the particle’s (Galatea’s) longitude, and
$\varpi_{G}$ and $\Omega_{G}$ are the longitudes of the periapse and the node
of the Galatea’s orbit, respectively.

Alternatively, multiple small co-orbital satellites in a stable stationary
configuration that generalises the Lagrangian points (Renner & Sicardy, 2004;
Renner et al., 2014) are able to efficiently confine the dusty arc material,
provided the right azimuthal spacings and masses of those co-orbital objects
are chosen. In the framework of that model, Giuliatti Winter et al. (2020)
investigated the arc orbital evolution under the effect of solar radiation.

Showalter et al. (2017) presented a possible three-body resonance mechanism
capable of confining ring material within the observed corotation sites. This
scenario constrains the orbital semi-major axis of the arcs to fall within
$\sim$10 meters of a three-body mean motion resonance, which involves the two
nearby inner satellites, Galatea and Larissa.

Finally, Showalter et al. (2019) recently discovered a seventh inner moon of
Neptune, Hippocamp, proving that the Neptunian system has not fully revealed
itself. Furthermore, in the context of the possible Trident mission
(preselected in NASA’s Discovery Programme) targeting the Neptune-Triton
system as well as a possible future ESA-NASA mission towards the Uranian and
Neptunian systems; the next couple of decades could revolutionise our
understanding of these systems.

In this paper, we report on astro-photometric measurements of the arcs are
obtained at the VLT- UT3 with the SPHERE-IRDIS instrument fed by its extreme
adaptive optics system (SAXO) obtained on August 23rd, 2016. The IRDIS
classical imaging mode has also been used for several asteroid studies,
leading to shape reconstruction (Viikinkoski et al., 2015; Marsset et al.,
2017). These studies largely benefit from the high resolution and high Strehl
providing much more detailed images than previous AO corrected images from
other instruments.

The paper is organised as follows. In section 2, we present our observational
data and the methods used for the analysis. Section 3 pertains to the
photometric profiles of the arcs and the mean motion of the arcs and the
satellite Galatea. Finally, we draw our final conclusions in Section 4.

##  2 Observations

We used the high-angular resolution adaptive-optics Spectro-Polarimetric High-
contrast Exoplanet REsearch (SPHERE) instrument installed on the Very Large
Telescope VLT-UT3 at the European Southern Observatory (ESO), with the
InfraRed Dual-band Imager and Spectrograph (IRDIS) (Dohlen et al., 2008;
Beuzit et al., 2019), to image the Neptunian system (ring arcs and moons) on
August 23rd, 2016.

IRDIS Classical imaging mode (Langlois et al., 2010) provides simultaneous
images recorded in the same broadband filter on two distinct detector areas.
The data were acquired using this classical mode of SPHERE, in the Ks broad-
band filter centred at 2.2 $\mu$m, which corresponds to a strong absorption in
the methane spectrum, hence reducing the otherwise overwhelming scattered
light from Neptune’s atmosphere.

In broad-band-Ks (BB-Ks), the instrument exhibits a fairly high thermal
background from thermal emission reaching 60 to 200 photon s-1 pixel-1 (Beuzit
et al., 2019), which renders nearly impossible the classical reduction
procedures to identify faint objects such as small Neptunian satellites as
well as the rings and arcs features.

IRDIS Dual-band Imager in classical imaging mode provides a FOV of $11"\times
12.5"$ with a pixel size of $(12.255\leavevmode\nobreak\
\pm\leavevmode\nobreak\ 0.009)$ mas/pixel on the sky (Maire et al., 2016);
this corresponds to $\sim 257$ km/pixel at Neptune’s geocentric distance of
28.959 AU (cf. Table 1).

We acquired field-stabilised data between 03:09:30.9281 and 07:58:36.2025 UT,
on 2016-08-23. This acquisition mode uses a specific derotator rotation law to
maintain the same field of view orientation during the observing sequence.

Figure 1 shows the details on the atmospheric conditions (seeing, airmass, and
coherence time $\tau_{0}$) during the entire observational run, together with
the time window of frames selected for the photometric analysis (green shaded
area). Although the seeing remains fairly stable throughout the observation,
about 27.4% of our data (57 frames) were acquired at times associated with
non-negligible atmospheric turbulence and thus not used for our photometric
analysis. The threshold defined by ESO as associated with fast wind and/or
fast turbulence conditions (i.e. turbulence evolving on timescales smaller
than 3.5 ms) is defined by low coherence times, i.e. $\tau\leq 3.5\,$ms. These
constraints combined with the instability of the AO tracking (see section 2.2
for details), the highest quality data that we have selected and retained for
the photometric analysis are obtained at the smallest airmass ($<1.1$).

| August 23rd, 2016
---|---
Ref. frame UT time (hr:min:sec) | 05:33:51.9581
MJD epoch of ref. frame | 57623.23185137
Geocentric distance (AU) | 28.959
Heliocentric distance (AU) | 29.954
Phase angle (deg) | 0.35138
$B$ (deg) | -26.133
$P$ (deg) | 326.237
$U$ (deg) | 318.808
Number of used images | 68
Individual exposure time (sec) | 64
Scale (mas/pixel) | $12.255\pm 0.009$
Table 1: Circumstances of observation: The epoch given is the time of the
reference frame used to produce the equivalent width profile (cf. section
3.1). The geocentric and heliocentric distances, as well as the phase angle
are retrieved from the Rings Node of NASA’s Planetary Data System (http://pds-
rings.seti.org). The angles $B$, $P$, and $U$ are the ring’s opening angle to
Earth, Neptune’s pole orientation, and the longitude of Earth measured in the
ring plane from the J2000.0 ascending node of Neptune’s equatorial plane,
respectively. The pixel scale value is that given by Maire et al. (2016).
Figure 1:  The observing conditions (airmass, seeing, and coherence time) at
Paranal during the observation run (between 03:09:30.9281 and 07:58:36.2025 UT
on Aug. 23rd, 2016). The large fluctuations in coherence time $\tau_{0}$ of
atmospheric turbulence show that some images cannot be used, because of small
$\tau_{0}$ values that is associated with fast atmospheric turbulence (the
shaded light red horizontal band for which $\tau_{0}<3.5\,$ms). The green
shaded area represents the time window from which the final images used in our
analysis were extracted (see text for explanation). These images correspond to
the lowest airmass observing window.

###  2.1 Data reduction

We used the SPHERE pipeline package to correct for flat field, bad pixels and
thermal background due to instrument and sky. Because the thermal background
varies with the instrument temperature and with the sky background, we
selected background calibration images acquired using the same integration
time and the same filter (BB-Ks, of 300 nm width) as the science frames, but
acquired at a different time, so they match better the conditions of the
observations. Some uncorrected background leaks persist in our final K images
due to thermal background fluctuations. The astrometrisation (centring,
correction for anamorphism, scale, and true north) of the final data cubes are
corrected following the SPHERE Data Center procedures (Delorme et al., 2017).

We have also corrected the true-north offset of the images, by rotating the
images by the angle $P_{cor}=(-1.57\,\pm\,0.08)$ deg (Maire et al., 2016),
where $P_{cor}$ is the position angle of the celestial north direction with
respect to the frame columns and the position of Neptune’s centre.

###  2.2 Astro-photometric analysis of the data

Using the IRDIS HAWAII-2-RG near-infrared detector, we initially acquired 208
exposures of 64 seconds exposure time each. However, because of the
aforementioned thermal background, passing clouds, and varying winds, only the
best 68 images of the right panel were deemed of high enough SNR and used for
our analysis (cf. Table 1).

In Renner et al. (2014), we were able to refine the determination of the pixel
scale and the orientation on sky by comparing the positions of Triton and
Proteus to their expected relative positions. In this paper, given the fact
that Proteus is the only satellite with sufficiently accurate astrometry and
ephemerides, we had to operate under the following assumptions: (i) perfect
ephemerides of the satellite Proteus which are used to retrieve Neptune’s
centre position, (ii) the scale and orientation provided by the SPHERE-IRDIS
consortium are correct. The instability of the AO correction due to the
observation of such extended object as Neptune (2.4” in diameter) used as the
guide star, results in image motion that required a re-alignment. Such re-
alignment is obtained by using Proteus’ ephemerides (Jacobson, 2009) because
the photo-centre of Neptune cannot be accurately determined (due to its non-
uniformity and its extended nature). To centre the images, we first determined
an approximate position (photo-centre) of Proteus on each image. From the
ephemerides and the measured position of the photo-centre, we were able to
retrieve a ’first order’ position for Neptune’s centre. We then iterated the
analysis on the first order centred images to which we subtracted the median
of the first centred cube files in which Neptune only is stationary, thus
refining our determination of Proteus’ photo-centre along its orbit, and
therefore the position of Neptune’s centre. This last step was repeated on
this new cube of centred images, as this process allows the removal of the
diffuse scattered light around Neptune, thus improving the determination of
the satellite’s photo-centre. Two iterations of the procedures are
indispensable to achieve a sub-pixel accuracy for the satellite’s and
Neptune’s positions. Examples of an individual frame as well as a final stack
of the centred clean images are presented in Figure 2.

Figure 2: (left) Single 64 seconds exposure of Neptune in the BB-Ks band
(2.182 $\mu$m), 2016-08-23T05:12:52.2635 UT August 23rd, 2016, revealing the
satellites Proteus (P), Galatea (G), and Larissa (L). The frame is $5.39\times
5.92$ arcsec2 wide. The point source is due to a bad pixel cluster which is
difficult to properly interpolate. (right) Image of Neptune’s system obtained
by co-adding 68 individual frames of the August 23rd, 2016 data (72 minutes
and 32 seconds total exposure time, between 04:24:50.5202 and 06:43:23.0627
UT). The frame is $6.99\times 6.99$ arcsec2 wide. We can see Proteus (P) and
Larissa (L) along their respective orbits during this time interval. We can
also see the trail of Galatea (G) and the arcs, which evolve along the much
fainter Adams ring. The bright arc on the left is due to thermal electronic
noise.

The individual images were then projected onto Neptune’s equatorial plane. The
single projected frames were rotated taking into account the respective
orbital motion and co-added to increase the signal from the ring arcs or a
given satellite.

##  3 Results

###  3.1 Photometry

In what follows, we use only the best 68 selected projected frames, taken
between 04:24:50.5202 and 06:43:23.0627 on August 23rd, 2016 UT. These are co-
added after correction for the arcs’ mean motion, $820.11213$ deg day-1
(Renner et al., 2014), to produce the panel on figure 3(a) (total exposure
time 72 minutes and 32 seconds).

Unlike the SPHERE data for which the field is depleted of stars, the 2007 NACO
data (Renner et al., 2014) included a reference star which could be used for
photometric calibration. We had to adapt to this additional constraint.

Using the same subset of SPHERE images with a good enough SNR for both Proteus
and Galatea, we derotated these images at the respective mean motion values
(320.765625$\pm$0.000001) deg day-1 for Proteus and (839.661311$\pm$0.000005)
deg day-1 for Galatea (Showalter et al., 2019). On these images, we measured
the fluxes of the satellites using classical aperture photometry. We then
compared the ratio of fluxes (Proteus/Galatea) for the NACO data (Renner et
al., 2014) to that of the SPHERE data used in this paper, and found ratios of
$0.12\pm 0.01$ and $0.13\pm 0.01$, respectively. The flux ratio of Proteus to
Galatea is thus unchanged (at the $1\sigma$-error level).

Similarly to what was done in (Renner et al., 2014), the brightness
longitudinal profiles are given in equivalent width, i.e. the width of a
perfect Lambert diffuser that would reflect sunlight at the distance of
Neptune. Following earlier studies of Neptune’s rings (Smith et al. (1989);
Porco et al. (1995)), the equivalent width is defined by
$E(\lambda)\leavevmode\nobreak\ =\leavevmode\nobreak\ \mu\int
I(\lambda)/F(\lambda)dr,$ where $I(\lambda)$ denotes the observed flux
reflected from the arcs, $\pi F(\lambda)$ the incident solar flux, $\lambda$
the wavelength at which observations were done, and $\mu$ denotes the cosine
of the emission angle with respect to the ring-plane normal.

For our photometric analysis of the arcs’ profile, we proceed as follows:

  1. i.

We obtained the profiles of the arcs in ADU (Analog to Digital Units) by
subtracting the sky background and Neptune’s scattered light from the arcs’
signal. To perform this operation, we integrated the ring region in the radial
direction over 13 pixels. We selected circular annuli on both sides of the
ring arcs, summed the pixels in the radial direction, and fitted a two-degree
polynomial as a function of the longitude. The radial width of these inner and
outer circular rings are $\sim$11 and 13 pixels, respectively. The average of
the two polynomial fits was then subtracted from the arcs’ flux.

  2. ii.

Using the same 68 individual projected frames as in Fig. 2 which we rotate at
Proteus’ mean motion, we perform aperture photometry and extract the flux in
ADU for the satellite.

  3. iii.

We then normalise the rings’ flux profile obtained in (i.) by Proteus’ flux
obtained in (ii.).

  4. iv.

Steps (i.) to (iii.) are then repeated on the NACO 2007 data taken at 2.2
$\mu$m (Ks band) (Renner et al., 2014).

  5. v.

We use the normalised (NACO 2007) ADU flux profile of the rings obtained in
(iv.) and the equivalent width profile for these same rings (Renner et al.,
2014) to derive the conversion factor between the normalised flux and the
profile in equivalent width. We then apply this factor to the SPHERE 2016
data. This is permitted since both NACO and SPHERE data were acquired in Ks-BB
at 2.2 $\mu$m, and under the assumption of a fixed magnitude for Proteus (cf.
above). This approach is justified by the absence of a reference (calibration)
star in the SPHERE 2016 data.

(a)

(b)

Figure 3:  (a) The 68 projected and co-added images of Neptune’s equatorial
plane revealing material along the arcs Fraternité and Egalité, as well as the
satellites Proteus (P) and Galatea (G). The frame is $7.1\times 10.7\,$arcsec2
wide. (b) The equivalent width of the arcs (Fraternité and Egalité) vs.
longitude extracted from Figure 3(b), at an angular resolution of 2∘. The
X-axis origin is the longitude $L_{Fr}$ of the centre of the Fraternité arc
measured from J2000.0 ascending node, where $L_{Fr}=217.43\,$deg, at the
reference epoch (cf. Table 1).

The equivalent width profiles of the ring arcs are presented in Fig. 3(b). We
detected the arcs Fraternité and Egalité, with flux densities and longitudinal
extensions similar to those of the previous VLT-NACO observations (Renner et
al., 2014) and Keck observations (de Pater et al., 2005).

The trailing arcs Fraternité and Egalité are clearly separated in our data
(see Figure 3(a)), which was not the case in the NACO data (Renner et al.,
2014). Despite the high thermal noise, we were able to draw conclusions on the
stability of the trailing arcs Fraternité and Égalité, and extract equivalent
width profiles. Finally, this new data confirms that the leading arcs Courage
and Liberté have faded away, as previous studies were leaning in that
direction (Dumas et al., 1999; de Pater et al., 2005). Accounting for the
geocentric distance given in Table 1, we derived the equivalent width of the
arcs Fraternité and Égalité at the epoch of observation at the 1$\sigma$
uncertainty level, and find that $E_{Fr}=(84\pm 18)\,$m and $E_{Eg}=(76\pm
18)$ m, respectively.

For comparison, Renner et al. (2014) and de Pater et al. (2005) found
$E_{Fr}=(71\pm 10)\,$m and $\sim$65 m, respectively. In (Renner et al., 2014),
we reported that Égalité was about $\sim$20% fainter than Fraternité from our
2007 VLT-NACO data (keeping in mind the pollution by Galatea of the arcs). On
the other hand, de Pater et al. (2005) report that Égalité was $\sim$17%
brighter than Fraternité ($\sim 3\sigma$) in their 2002 Keck II data, while
its intensity had decreased to $\sim 7\%$ below that of Fraternité ($\sim
2\sigma$) in 2003.

To summarise, given the uncertainties on the equivalent width measurements,
the arcs Fraternité and Égalité appear to be stable when compared to previous
studies (Dumas et al., 1999; de Pater et al., 2005; Renner et al., 2014).

###  3.2 Mean motions

We improved the average mean motion of the satellite Galatea: (i) at epoch
2016-08-23T04:25:22.5202 UT, we derive the satellite’s longitude
$L=\leavevmode\nobreak\ 259^{\circ}.58\leavevmode\nobreak\
\pm\leavevmode\nobreak\ 0^{\circ}.30$, measured in the ring plane from the
J2000.0 ascending node of Neptune’s equatorial plane, (ii) using the reference
longitudes measured by Voyager (Owen et al., 1991) at epoch JD 2447757.0 (i.e.
1989 August 18th, 12 h) at Neptune, we derive the average mean motion of
Galatea, $n_{G}\leavevmode\nobreak\ =\leavevmode\nobreak\
(839.66134\,\pm\,0.00003)$ deg day-1. This value is in agreement with that
given in (Showalter et al., 2019), $(839.661311\leavevmode\nobreak\
\pm\leavevmode\nobreak\ 0.000005)\,$deg day-1.

Furthermore, from the stacked image (Fig. 3(a)) with respect to the reference
frame (see Table 1), we derived a longitude $L_{Fr}\leavevmode\nobreak\
=\leavevmode\nobreak\ (217.43\pm 0.30)$ deg for the trailing arc Fraternité at
this same reference epoch. We used the position of the middle of this arc
measured from Voyager data (251.88 deg at epoch JD 2447757.0, i.e. 1989 August
18, 12h UT) as given in (Porco et al., 1995), and derived the following
average mean motion for the arcs (Fraternité and Égalité):
$n_{arcs}\leavevmode\nobreak\ =(820.11178\leavevmode\nobreak\
\pm\leavevmode\nobreak\ 0.00003)$ deg day-1 which is consistent with, yet more
accurate than previous measurements (”solution 2” of Nicholson et al. (1995);
de Pater et al. (2005)), $820.1118\leavevmode\nobreak\ \pm\leavevmode\nobreak\
0.0001\,$deg day-1. We used our new measurement to compare $n_{arcs}$ with the
mean motion of the 42:43 CIR, initially thought to confine dust within the
ring to form stable arcs (Goldreich et al., 1986; Porco, 1991).

Resonance type | drift in mean motion ($\Delta n$) | mismatch ($\Delta\,a$) |  $\Delta L$ ∗ |
---|---|---|---|---
| (deg/day) | (meters) | (deg) |
42:43 CIR with Galatea | $n_{CIR}-n_{arcs}\approx(5.95\pm 0.03)\,.\,10^{-3}$ | $a_{arcs}-a_{CIR}\approx 305\pm 2$ | $\sim(58.8\pm 0.3)$ |
42:43 CER with Galatea | $n_{CER}-n_{arcs}\approx(39.16\pm 0.03)\,.\,10^{-3}$ | $a_{arcs}-a_{CER}\approx 2004\,\pm\,2$ | $\sim(26.4\pm 0.3)$ |

Table 2: Investigating the 42:43 CIR and the 42:43 CER corotation models with
Galatea.

222∗ $\Delta\,L$ represents the difference for the arcs’ longitude (assuming a
trapping in the CIR or the CER) over $\sim$27.01364 years (between the 1989
Voyager and our VLT SPHERE-IRDIS 2016 data).

The 42:43 CIR with Galatea creates 86 equally spaced corotation sites around
Neptune, with a mean motion given by $n_{CIR}\leavevmode\nobreak\
=(42n_{G}+\dot{\Omega}_{G})/43$, where $\dot{\Omega}_{G}$ is Galatea’s nodal
precession rate. Using $\dot{\Omega}_{G}=-0.713675$ deg day-1 (Showalter et
al., 2019), we find $n_{CIR}=820.11773\pm 0.00003$ deg day-1. This value of
$n_{CIR}$ is similar to previous mean motion measurements (Nicholson et al.,
1995; Sicardy et al., 1999; Dumas et al., 1999, 2002; de Pater et al., 2005;
Renner et al., 2014), showing that the arcs are not at the location of the
42:43 CIR with Galatea. This mean motion drift translates into a mismatch in
semi-major axis $\Delta a=(305\pm 2)\,$m (cf. Table 2). This mismatch is
consistent with, yet more accurate than those reported in (Sicardy et al.,
1999; Renner et al., 2014); the half-width of the CIR being $(250\pm 100)\,$m
(Renner et al., 2014).

Furthermore, we investigate the 42:43 corotation eccentric resonance (CER)
with Galatea. This resonance is defined as follows
$n_{CER}\leavevmode\nobreak\ =\leavevmode\nobreak\
(42n_{G}+\dot{\omega}_{G})/43$, where $\dot{\omega}_{G}$ is Galatea’s apsidal
precession rate. With $\dot{\omega}_{G}=0.714282$ deg day-1 (Showalter et al.,
2019), we find $n_{CER}=820.15094\pm 0.00003$ deg day-1.

Neptune’s properties
---
$GM$ |  | 6 835 099.5 km3 s-2 |
$R$ |  | 25225 km |
$J_{2}$ |  | $3,408.43\times 10^{-6}$ |
$J_{4}$ |  | $-33.40\times 10^{-6}$ |
Galatea’s orbital elements and mass
$i$ (∘) |  | $0.0231\pm 0.0091$ |
$e$ |  | $0.00022\pm 0.00008$ |
$M_{G}$ |  |  $(2.12\pm 0.08)\,10^{18}\,$kg |
Table 3:  Parameters used to explore the validity of the 42:43 CER and CIR
models: Neptune’s physical properties from (Jacobson, 2009) as well as derived
orbital inclination and eccentricity of Galatea, derived by Showalter et al.
(2019). Galatea’s mass is that derived in (Porco, 1991).

For both the 42:43 CIR and CER, we give in table 2 the corresponding mismatch
values in semi-major axis, as well as the associated difference for the arcs’
longitude ($\Delta L$) over $\sim$27.01364 years (between the 1989 Voyager and
VLT SPHERE-IRDIS 2016 data). Note that these values are large and far from
what we observe.

Figure 4 illustrates a graphic comparison of the semi-major axis of the arcs,
as derived from the mean motion, as well the CER and CIR radii values derived
in this work following the same approach as Foryta & Sicardy (1996). We used
the parameters of Galatea and Neptune given in table 3. In particular, this
figure depicts clearly the widths ($\Delta\,a$) of each one of the CIR and CER
resonances, i.e. the interval in semi-major axis where particles would be
trapped by each one of these resonances.

Figure 4:  Graphical illustration of the arcs’ semi-major axis, as well as the
radii and widths of the CIR and CER motion resonances 42:43 with Galatea. The
orange continuous line depicts the geometrical orbital radius of the arcs,
with uncertainties (in black dotted lines) smaller than the thickness of the
orange line. The CIR and CER locations and widths are represented in blue and
purple, respectively. They represent the interval in semi-major axis where
particles would be trapped by these resonances. This figure clearly shows the
mismatch between the 42:43 CIR and the CER with Galatea; and the arc’s
position. The associated differences in longitude ($\Delta\,L$) in Table 2
support this conclusion.

From this analysis of the mismatch ($\Delta a$) and the resulting drifts in
longitude $\Delta L$, we conclude that neither the CIR nor the CER 42:43
(Namouni & Porco, 2002) corotation models can explain the confinement of the
arcs at their current locations. The valid model, so far, to explain the
azimuthal spacing of the arcs is the model based on small co-orbital
satellites (Renner et al., 2014).

Finally, we test here the three-body mean motion resonance hypothesis
(Showalter et al., 2017) of argument $(35\,n_{G}-\leavevmode\nobreak\
39\,n_{arcs}+\leavevmode\nobreak\ 4\,n_{L})$, which involves the satellites
Galatea and Larissa. We use the mean motions for Galatea and the arcs derived
in the work, and Larissa’s mean motion $n_{L}=\leavevmode\nobreak\
(649.054085\pm 0.000004)$ deg day-1 (Showalter et al., 2019); we obtain a
value $0.00382\,$deg day-1. Further work is needed to confirm wether or not
this three-body resonance is able to confine dust particles into corotation
sites, with spacings compatible with the observations.

##  4 Conclusions

We have analysed high-angular resolution near-infrared images of Neptune’s
ring arcs obtained in 2016 at the ESO VLT-UT3, with the adaptive-optics camera
SPHERE-IRDIS.

We detected the trailing arcs Fraternité and Egalité, as well as the satellite
Galatea for which we derived more accurate mean motion values. From the
accurate determination of the mean motions, we confirm the mismatch between
the arcs’ position and the location of the 42:43 CIR and the CER with Galatea;
thus showing that no 42:43 corotation model can explain the persistance of
Neptune’s incomplete ring arcs.

Moreover, our photometric analysis of the arcs confirms that the leading arcs
Courage and Liberté have faded away, while the trailing arcs Fraternité and
Égalité are persistent and stable since their discovery.

Regular imaging data, in particular with Galatea far from the arcs system, to
avoid light contamination, are needed to follow the global time evolution of
the system, and to propose a more global model for the arcs confinement that
takes into account the disappearance of the two leading arcs Liberté and
Courage.

##  5 Acknowledgment

This work has made use of the SPHERE Data Centre, jointly operated by
OSUG/IPAG (Grenoble), PYTHEAS/LAM/CeSAM (Marseille), OCA/Lagrange (Nice),
Observatoire de Paris/LESIA (Paris), and Observatoire de Lyon (OSUL/CRAL), and
supported by a grant from Labex OSUG@2020 (Investissements d’avenir – ANR10
LABX56).

This work was funded by the European Research Council under the European
Community’s H2020 (2014-2021/ERC Grant Agreement No. 669416).

We thank the reviewer for their thorough revision of our paper and
constructive comments that significantly improve the quality of this work.

## References

  * Beuzit et al. (2019) Beuzit, J. L., Vigan, A., Mouillet, D., et al. 2019, A&A, 631, A155
  * Covault et al. (1986) Covault, C. E., Glass, I. S., French, R. G., & Elliot, J. L. 1986, Icarus, 67, 126
  * de Pater et al. (2005) de Pater, I., Gibbard, S. G., Chiang, E., et al. 2005, Icarus, 174, 263
  * De Pater et al. (2018) De Pater, I., Renner, S., Showalter, M. R., & Sicardy, B. 2018, The Rings of Neptune, ed. M. S. Tiscareno & C. D. Murray, 112–124
  * Delorme et al. (2017) Delorme, P., Meunier, N., Albert, D., et al. 2017, in SF2A-2017: Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics, ed. C. Reylé, P. Di Matteo, F. Herpin, E. Lagadec, A. Lançon, Z. Meliani, & F. Royer, Di
  * Dohlen et al. (2008) Dohlen, K., Langlois, M., Saisse, M., et al. 2008, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7014, Ground-based and Airborne Instrumentation for Astronomy II, ed. I. S. McLean & M. M. Casali, 70143L
  * Dumas et al. (2002) Dumas, C., Terrile, R. J., Smith, B. A., & Schneider, G. 2002, AJ, 123, 1776
  * Dumas et al. (1999) Dumas, C., Terrile, R. J., Smith, B. A., Schneider, G., & Becklin, E. E. 1999, Nature, 400, 733
  * Foryta & Sicardy (1996) Foryta, D. W. & Sicardy, B. 1996, Icarus, 123, 129
  * Giuliatti Winter et al. (2020) Giuliatti Winter, S. M., Madeira, G., & Sfair, R. 2020, MNRAS, 496, 590
  * Goldreich et al. (1986) Goldreich, P., Tremaine, S., & Borderies, N. 1986, AJ, 92, 490
  * Hubbard et al. (1986) Hubbard, W. B., Brahic, A., Sicardy, B., et al. 1986, Nature, 319, 636
  * Jacobson (2009) Jacobson, R. A. 2009, AJ, 137, 4322
  * Langlois et al. (2010) Langlois, M., Vigan, A., Dohlen, K., et al. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7735, Ground-based and Airborne Instrumentation for Astronomy III, ed. I. S. McLean, S. K. Ramsay, & H. Takami, 773512
  * Maire et al. (2016) Maire, A.-L., Langlois, M., Dohlen, K., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9908, Ground-based and Airborne Instrumentation for Astronomy VI, ed. C. J. Evans, L. Simard, & H. Takami, 990834
  * Marsset et al. (2017) Marsset, M., Carry, B., Dumas, C., et al. 2017, A&A, 604, A64
  * Namouni & Porco (2002) Namouni, F. & Porco, C. 2002, Nature, 417, 45
  * Nicholson et al. (1990) Nicholson, P. D., Cooke, M. L., Matthews, K., Elias, J. H., & Gilmore, G. 1990, Icarus, 87, 1
  * Nicholson et al. (1995) Nicholson, P. D., Mosqueira, I., & Matthews, K. 1995, Icarus, 113, 295
  * Owen et al. (1991) Owen, W. M., Vaughan, R. M., & Synnott, S. P. 1991, AJ, 101, 1511
  * Porco (1991) Porco, C. C. 1991, Science, 253, 995
  * Porco et al. (1995) Porco, C. C., Nicholson, P. D., Cuzzi, J. N., Lissauer, J. J., & Esposito, L. W. 1995, in Neptune and Triton, 703–804
  * Renner & Sicardy (2004) Renner, S. & Sicardy, B. 2004, Celestial Mechanics and Dynamical Astronomy, 88, 397
  * Renner et al. (2014) Renner, S., Sicardy, B., Souami, D., Carry, B., & Dumas, C. 2014, A&A, 563, A133
  * Roques et al. (1984) Roques, F., Sicardy, B., Bouchet, P., et al. 1984, in Bulletin of the American Astronomical Society, Vol. 16, 1027–1028
  * Showalter et al. (2017) Showalter, M., Lissauer, J. J., de Pater, I., & French, R. S. 2017, in AAS/Division for Planetary Sciences Meeting Abstracts, Vol. 49, AAS/Division for Planetary Sciences Meeting Abstracts #49, 104.01
  * Showalter et al. (2019) Showalter, M. R., de Pater, I., Lissauer, J. J., & French, R. S. 2019, Nature, 566, 350
  * Sicardy et al. (1985) Sicardy, B., Hubbard, W. B., Brahic, A., & Roques, F. 1985, in Bulletin of the American Astronomical Society, Vol. 17, 923
  * Sicardy et al. (1999) Sicardy, B., Roddier, F., Roddier, C., et al. 1999, Nature, 400, 731
  * Smith et al. (1989) Smith, B. A., Soderblom, L. A., Banfield, D., et al. 1989, Science, 246, 1422
  * Viikinkoski et al. (2015) Viikinkoski, M., Kaasalainen, M., Ďurech, J., et al. 2015, A&A, 581, L3


# Exel-Pardo algebras of self-similar $k$-graphs

Hossein Larki  Department of Mathematics
Faculty of Mathematical Sciences and Computer
Shahid Chamran University of Ahvaz
P.O. Box: 83151-61357
Ahvaz
Iran<EMAIL_ADDRESS>
###### Abstract.

We introduce the Exel-Pardo $*$-algebra $\mathrm{EP}_{R}(G,\Lambda)$
associated to a self-similar $k$-graph $(G,\Lambda,\varphi)$. We prove the
$\mathbb{Z}^{k}$-graded and Cuntz-Krieger uniqueness theorems for such
algebras and investigate their ideal structure. In particular, we modify the
graded uniqueness theorem for self-similar 1-graphs, and then apply it to
present $\mathrm{EP}_{R}(G,\Lambda)$ as a Steinberg algebra and to study the
ideal structure.

###### Key words and phrases:

self-similar $k$-graph, Exel-Pardo algebra, groupoid algebra

###### 2010 Mathematics Subject Classification:

16D70, 16W50

##  1\. introduction

To give a unified framework like graph $C^{*}$-algebras for the Katsura’s [10]
and Nekrashevyche’s algebras [16, 17], Exel and Pardo introduced self-similar
graphs and their $C^{*}$-algebras in [6]. They then associated an inverse
semigroup and groupoid model to this class of $C^{*}$-algebras and studied
structural features by underlying self-similar graphs. Note that although only
finite graphs are considered in [6], but many of arguments and results may be
easily generalized for countable row-finite graphs with no sources (see [7, 9]
for example). Inspired from [6], Li and Yang in [14, 15] introduced self-
similar action of a discrete countable group $G$ on a row-finite $k$-graph
$\Lambda$. They then associated a universal $C^{*}$-algebra
$\mathcal{O}_{G,\Lambda}$ to $(G,\Lambda)$ satisfying specific relations.

The algebraic analogues of Exel-Pardo $C^{*}$-algebras, denoted by
$\mathcal{O}_{(G,E)}^{\mathrm{alg}}$ in [5] and by $L_{R}(G,E)$ in [8], were
introduced and studied in [5, 8]. In particular, Hazrat et al. proved a
$\mathbb{Z}$-graded uniqueness theorem, and gave a model of Steinberg algebras
for $L_{R}(G,E)$ [8] among others. The initial idea of this paper is to modify
the $\mathbb{Z}$-graded uniqueness theorem [8, Theorem A] and then apply it to
give an easier proof for [8, Theorem B] (a groupoid model for $L_{R}(G,E)$).
However, we do these here in the self-similar $k$-graph setting, which is
interesting in its own right.

Let $R$ be a unital commutative $*$-ring. In this article, we associate a
universal $*$-algebras over $R$ to a self-similar $k$-graph $(G,\Lambda)$, and
then study it by properties of underlying $(G,\Lambda)$. In particular, we
investigate the ideal structure, where our results are new even in the 1-graph
case. Our algebras are the algebraic analogue of $\mathcal{O}_{G,\Lambda}$
[14, 15], and include many important known algebras such as the Exel-Pardo
$*$-algebras $L_{R}(G,E)$ [5, 8], algebraic Katsura algebras [8], Kumjian-Pask
algebras [1], and the quotient boundary algebras
$\mathcal{Q}_{R}^{\mathrm{alg}}(\Lambda\bowtie G)$ of a Zappa-
Sz$\acute{\mathrm{e}}$p product $\Lambda\bowtie G$ introduced in Section 3.

This article is organized as follows. In Section 2, we introduce a universal
$*$-algebra $\mathrm{EP}_{R}(G,\Lambda)$ of a self-similar $k$-graph
$(G,\Lambda)$ satisfying specific properties, which is called the _Exel-Pardo
algebras of $(G,\Lambda)$_. Our algebras are the higher rank generalization of
those in [8, Theorem 1.6]. In section 3, we give a specific example of Exel-
Pardo algebras using boundary quotient algebras of semigroups. Indeed, for a
single vertex self-similar $k$-graph $(G,\Lambda)$, the Zappa-Szép product
$\Lambda\bowtie G$ is a cancellative semigroup. We prove that the quotient
boundary algebra $\mathcal{Q}_{R}^{\mathrm{alg}}(\Lambda\bowtie G)$ (defined
in Definition 3.1) is isomorphic to $\mathrm{EP}_{R}(G,\Lambda)$. Section 4 is
devoted to prove a graded uniqueness theorem for the Exel-Pardo algebras. Note
that using the description in Proposition 2.7 below, there is a natural
$\mathbb{Z}^{k}$-grading on $\mathrm{EP}_{R}(G,\Lambda)$. Then, in Theorem
4.2, a $\mathbb{Z}^{k}$-graded uniqueness theorem is proved for
$\mathrm{EP}_{R}(G,\Lambda)$ which generalizes and modifies [8, Theorem A]. In
particular, we will see in Sections 5 and 6 that this modification makes it
more applicable.

In Sections 5 and 6, we assume that our self-similar $k$-graphs are pseudo
free. In Section 5, we prove that every Exel-Pardo algebra
$\mathrm{EP}_{R}(G,\Lambda)$ is isomorphic to the Steinberg algebra
$A_{R}(\mathcal{G}_{G,\Lambda})$, where $\mathcal{G}_{G,\Lambda}$ is the
groupoid introduced in [14]. We should note that the proof of this result is
completely different from that of [8, Theorem B]. Indeed, the main difference
between proof of Theorem 5.5 and that of [8, Theorem B] is due to showing the
injectivity of defined correspondence. In fact, in [8, Theorem B], the authors
try to define a representation for $\mathcal{S}_{(G,E)}$ in
$\mathrm{EP}_{R}(G,E)$ while we apply our graded uniqueness theorem, Theorem
4.2, here. This gives us an easier proof for Theorem 5.5, even in the 1-graph
case. Finally, in Section 6, we study the ideal structure of
$\mathrm{EP}_{R}(G,\Lambda)$. Using the Steinberg algebras, we can define a
conditional expectation $\mathcal{E}$ on $\mathrm{EP}_{R}(G,\Lambda)$ and then
characterize basic, $\mathbb{Z}^{k}$-graded, diagonal-invariant ideals of
$\mathrm{EP}_{R}(G,\Lambda)$ by $G$-saturated $G$-hereditary subsets of
$\Lambda^{0}$. These ideals are exactly basic,
$Q(\mathbb{N}^{k},G)\rtimes_{\mathcal{T}}\mathbb{Z}^{k}$-graded ideals of
$\mathrm{EP}_{R}(G,\Lambda)$.

###  1.1. Notation and terminology

Let $\mathbb{N}=\\{0,1,2,\ldots\\}$. For $k\geq 1$, we regard $\mathbb{N}^{k}$
as an additive semigroup with the generators $e_{1},\ldots,e_{k}$. We use
$\leq$ for the partial order on $\mathbb{N}^{k}$ given by $m\leq n$ if and
only if $m_{i}\leq n_{i}$ for $1\leq i\leq k$. We also write $m\vee n$ and
$m\wedge n$ for the coordinate-wise maximum and minimum, respectively.

A _$k$ -graph_ is a countable small category
$\Lambda=(\Lambda^{0},\Lambda,r,s)$ equipped with a _degree functor_
$d:\Lambda\rightarrow\mathbb{N}^{k}$ satisfying the unique factorization
property: for $\mu\in\Lambda$ and $m,n\in\mathbb{N}^{k}$ with $d(\mu)=m+n$,
then there exist unique $\alpha,\beta\in\Lambda$ such that $d(\alpha)=m$,
$d(\beta)=n$, and $\mu=\alpha\beta$. We usually denote $\mu(0,m):=\alpha$ and
$\mu(m,d(\mu)):=\beta$. We refer to $\Lambda^{0}$ as the vertex set and define
$\Lambda^{n}:=\\{\mu\in\Lambda:d(\mu)=n\\}$ for every $n\in\mathbb{N}^{k}$.
For $A,B\subseteq\Lambda$, define $AB=\\{\mu\nu:\mu\in A,\nu\in
B,~{}\mathrm{and}~{}s(\mu)=r(\nu)\\}$. Also, for $\mu,\nu\in\Lambda$, define
$\Lambda^{min}(\mu,\nu)=\\{(\alpha,\beta)\in\Lambda\times\Lambda:\mu\alpha=\nu\beta,d(\mu\alpha)=d(\mu)\vee
d(\nu)\\}$.

We say that $\Lambda$ is _row-finite_ if $v\Lambda^{n}$ is finite for all
$n\in\mathbb{N}^{k}$ and $v\in\Lambda^{0}$. A _source_ in $\Lambda$ is a
vertex $v\in\Lambda^{0}$ such that $v\Lambda^{e_{i}}=\emptyset$ for all $1\leq
i\leq k$.

Standing assumption. Throughout the article, we work only with row-finite
$k$-graphs without sources.

Let $\Omega_{k}:=\\{(m,n)\in\mathbb{N}^{k}\times\mathbb{N}^{k}:m\leq n\\}$. By
defining $(m,n).(n,l):=(m,l)$, $r(m,n):=(m,m)$, and $s(m,n):=(n,n)$, then
$\Omega_{k}$ is a row-finite $k$-graph without sources. A graph homomorphism
$x:\Omega_{k}\rightarrow\Lambda$ is called an _infinite path of $\Lambda$_
with the range $r(x)=x(0,0)$, and we write $\Lambda^{\infty}$ for the set of
all infinite paths of $\Lambda$.

##  2\. Exel-Pardo algebras of self-similar $k$-graphs

In this section, we associate a $*$-algebra to a self-similar $k$-graph as the
algebraic analogue of [15, Definition 3.9]. Let us first review some
definitions and notations.

Following [8], we consider $*$-algebras over $*$-rings. Let $R$ be a unital
commutative $*$-ring. Recall that a _$*$ -algebra over $R$_ is an algebra $A$
equipped with an involution such that $(a^{*})^{*}=a$, $(ab)^{*}=b^{*}a^{*}$,
and $(ra+b)^{*}=r^{*}a^{*}+b^{*}$ for all $a,b\in A$ and $r\in R$. Then $p\in
A$ is called a _projection_ if $p^{2}=p=p^{*}$, and $s\in A$ a _partial
isometry_ if $s=ss^{*}s$.

######  Definition 2.1 ([1, Definition 3.1]).

Let $\Lambda$ be a row-finite $k$-graph without sources. A _Kumjian-Pask
$\Lambda$-family_ is a family $\\{s_{\mu}:\mu\in\Lambda\\}$ of partial
isometries in a $*$-algebra $A$ such that

  1. (CK1)

$\\{s_{v}:v\in\Lambda^{0}\\}$ is a family of pairwise orthogonal projections;

  2. (CK2)

$s_{\mu\nu}=s_{\mu}s_{\nu}$ for all $\mu,\nu\in\Lambda$ with $s(\mu)=r(\nu)$;

  3. (CK3)

$s_{\mu}^{*}s_{\mu}=s_{s(\mu)}$ for all $\mu\in\Lambda$; and

  4. (CK4)

$s_{v}=\sum_{\mu\in v\Lambda^{n}}s_{\mu}s_{\mu}^{*}$ for all $v\in\Lambda^{0}$
and $n\in\mathbb{N}^{k}$.

###  2.1. Self-similar $k$-graphs and their algebras

Let $\Lambda$ be a row-finite $k$-graph without sources. An _automorphism of
$\Lambda$_ is a bijection $\psi:\Lambda\rightarrow\Lambda$ such that
$\psi(\Lambda^{n})\subseteq\Lambda^{n}$ for all $n\in\mathbb{N}^{k}$ with the
properties $s\circ\psi=\psi\circ s$ and $r\circ\psi=\psi\circ r$. We denote by
$\mathrm{Aut}(\Lambda)$ the group of automorphisms on $\Lambda$. Let $G$ be a
countable discrete group. An _action_ of $G$ on $\Lambda$ is a group
homomorphism $g\mapsto\psi_{g}$ from $G$ into $\mathrm{Aut}(\Lambda)$.

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

Let $\Lambda$ be a row-finite $k$-graph without sources and $G$ a discrete
group with identity $e_{G}$. We say that a triple $(G,\Lambda,\varphi)$ is a
_self-similar $k$-graph_ whenever the following properties hold:

  1. (1)

$G$ acts on $\Lambda$ by a group homomorphism $g\mapsto\psi_{g}$. We prefer to
write $g.\mu$ for $\psi_{g}(\mu)$ to ease the notation.

  2. (2)

$\varphi:G\times\Lambda\rightarrow G$ is a 1-cocycle for the action
$G\curvearrowright\Lambda$ such that for every $g\in G$, $\mu,\nu\in\Lambda$
and $v\in\Lambda^{0}$ we have

     * (a)

$\varphi(gh,\mu)=\varphi(g,h.\mu)\varphi(h,\mu)\hskip 14.22636pt$ (the
1-cocycle property),

     * (b)

$g.(\mu\nu)=(g.\mu)(\varphi(g,\mu).\nu)\hskip 14.22636pt$ (the self-similar
equation),

     * (c)

$\varphi(g,\mu\nu)=\varphi(\varphi(g,\mu),\nu)$, and

     * (d)

$\varphi(g,v)=g$.

For convenience, we usually write $(G,\Lambda)$ instead of
$(G,\Lambda,\varphi)$.

######  Remark 2.3.

In [14], the authors used the notation $g|_{\mu}$ for $\varphi(g,\mu)$.
However, we would prefer to follow [6, 7, 8] for writing $\varphi(g,\mu)$.

######  Remark 2.4.

If in equation (2)(a) of Definition 2.2, we set $g=h=e_{G}$, then we get
$\varphi(e_{G},\mu)=e_{G}$ for every $\mu\in\Lambda$. Moreover, [14, Lemma
3.5(ii)] shows that $\varphi(g,\mu).v=g.v$ for all $g\in G$,
$v\in\Lambda^{0}$, and $\mu\in\Lambda$.

Now we generalize the definition of Exel-Pardo $*$-algebras [8] to the
$k$-graph case.

######  Definition 2.5.

Let $(G,\Lambda)$ be a self-similar $k$-graph. An Exel-Pardo
$(G,\Lambda)$-family (or briefly $(G,\Lambda)$-family) is a set

$\\{s_{\mu}:\mu\in\Lambda\\}\cup\\{u_{v,g}:v\in\Lambda^{0},g\in G\\}$

in a $*$-algebra satisfying

  1. (1)

$\\{s_{\mu}:\mu\in\Lambda\\}$ is a Kumjian-Pask $\Lambda$-family,

  2. (2)

$u_{v,e_{G}}=s_{v}$ for all $v\in\Lambda^{0}$,

  3. (3)

$u_{v,g}^{*}=u_{g^{-1}.v,g^{-1}}$ for all $v\in\Lambda^{0}$ and $g\in G$,

  4. (4)

$u_{v,g}s_{\mu}=\delta_{v,g.r(\mu)}s_{g.\mu}u_{g.s(\mu),\varphi(g,\mu)}$ for
all $v\in\Lambda^{0}$, $\mu\in\Lambda$, and $g\in G$,

  5. (5)

$u_{v,g}u_{w,h}=\delta_{v,g.w}u_{v,gh}$ for all $v,w\in\Lambda^{0}$ and
$g,h\in G$.

Then the Exel-Pardo algebra $\mathrm{EP}_{R}(G,\Lambda)$ is the universal
$*$-algebra over $R$ generated by a $(G,\Lambda)$-family. Throughout the
paper, we will denote by $\\{s_{\mu},u_{v,g}\\}$ the $(G,\Lambda)$-family
generating $\mathrm{EP}_{R}(G,\Lambda)$.

###  2.2. The construction of $\mathrm{EP}_{R}(G,\Lambda)$

Let $(G,\Lambda)$ be a self-similar $k$-graph as in Definition 2.2. The
following is a standard construction of a universal algebra
$\mathrm{EP}_{R}(G,\Lambda)$ subject to desired relations. Consider the set of
formal symbols

$S=\left\\{S_{\mu},S_{\mu}^{*},U_{v,g},U_{v,g}^{*}:\mu\in\Lambda,v\in\Lambda^{0},g\in
G\right\\}.$

Let $X=w(S)$ be the collection of finite words in $S$. We equip the free
$R$-module

$\mathbb{F}_{R}(X):=\big{\\{}\sum_{i=1}^{l}r_{i}x_{i}:l\geq 1,r_{i}\in
R,x_{i}\in X\big{\\}}$

with the multiplication

$\big{(}\sum_{i=1}^{l}r_{i}x_{i}\big{)}\big{(}\sum_{j=1}^{l^{\prime}}s_{j}y_{j}\big{)}:=\sum_{i,j}r_{i}s_{j}x_{i}y_{j},$

and the involution

$\big{(}\sum r_{i}x_{i}\big{)}^{*}:=\big{(}\sum r_{i}^{*}x_{i}^{*}\big{)}$

where $x^{*}=s_{l}^{*}\ldots s_{1}^{*}$ for each $x=s_{1}\ldots s_{l}$. Then
$\mathbb{F}_{R}(X)$ is a $*$-algebra over $R$. If $I$ is the (two-sided and
self-adjoint) ideal of $\mathbb{F}_{R}(X)$ containing the routs of relations
(1)-(5) in Definition 2.5, then the quotient $\mathbb{F}_{R}(X)/I$ is the
Exel-Pardo algebra $\mathrm{EP}_{R}(G,\Lambda)$ with the desired universal
property. Let us define $s_{\mu}:=S_{\mu}+I$ and $u_{v,g}:=U_{v,g}+I$ for
every $\mu\in\Lambda$, $v\in\Lambda^{0}$, and $g\in G$. In case $(G,\Lambda)$
is pseudo free (Definition 5.1), Theorem 4.2 insures that all generators
$\\{s_{\mu},u_{v,g}\\}$ of $\mathrm{EP}_{R}(G,\Lambda)$ are nonzero.

Proposition 2.7 below describes the elements of $\mathrm{EP}_{R}(G,\Lambda)$.
First, see a simple lemma.

######  Lemma 2.6.

Let $(G,\Lambda)$ be a self-similar graph (as in Definition 2.2) and
$\\{S,U\\}$ a $(G,\Lambda)$-family. If $S_{\mu}U_{v,g}S_{\nu}^{*}\neq 0$ where
$\mu,\nu\in\Lambda$, $v\in\Lambda^{0}$ and $g\in G$, then $s(\mu)=v=g.s(\nu)$.

###### Proof.

If $a=S_{\mu}U_{v,g}S_{\nu}^{*}$ is nonzero, then by Definition 2.5 we can
write

$\displaystyle S_{\mu}U_{v,g}S_{\nu}^{*}$
$\displaystyle=S_{\mu}(S_{s(\mu)}U_{v,g})S_{\nu}^{*}$
$\displaystyle=S_{\mu}(U_{s(\mu),e_{G}}U_{v,g})S_{\nu}^{*}$
$\displaystyle=S_{\mu}(\delta_{s(\mu),e_{G}.v}U_{s(\mu),g})S_{\nu}^{*}.$

Now, hypothesis $a\neq 0$ forces $s(\mu)=v$. On the other hand, a similar
computation gives

$\displaystyle a$ $\displaystyle=S_{\mu}(U_{v,g}S_{s(\nu)})S_{\nu}^{*}$
$\displaystyle=S_{\mu}(U_{v,g}U_{s(\nu),e_{G}})S_{\nu}^{*}$
$\displaystyle=S_{\mu}(\delta_{v,g.s(\nu)}U_{v,g})S_{\nu}^{*},$

and thus $v=g.s(\nu)$. ∎

######  Proposition 2.7.

Let $(G,\Lambda)$ be a self-similar graph. Then

(2.1)
$\mathrm{EP}_{R}(G,\Lambda)=\mathrm{span}_{R}\\{s_{\mu}u_{s(\mu),g}s_{\nu}^{*}:g\in
G,~{}\mu,\nu\in\Lambda,~{}\mathrm{and}~{}s(\mu)=g.s(\nu)\\}.$

###### Proof.

Define $M:=\mathrm{span}_{R}\\{s_{\mu}u_{s(\mu),g}s_{\nu}^{*}:g\in
G,~{}\mu,\nu\in\Lambda\\}$. For every $g,h\in G$ and
$\mu,\nu,\alpha,\beta\in\Lambda$ with $\alpha=\nu\alpha^{\prime}$ for some
$\alpha^{\prime}\in\Lambda$, relations of Definition 2.5 imply that

$\displaystyle\big{(}s_{\mu}u_{s(\mu),g}s_{\nu}^{*}\big{)}\big{(}s_{\alpha}u_{s(\alpha),h}s_{\beta}^{*}\big{)}$
$\displaystyle=s_{\mu}u_{s(\mu),g}(s_{\nu}^{*}s_{\alpha})u_{s(\alpha),h}s_{\beta}^{*}$
$\displaystyle=s_{\mu}u_{s(\mu),g}(s_{\alpha^{\prime}})u_{s(\alpha),h}s_{\beta}^{*}$
$\displaystyle=s_{\mu}\big{(}\delta_{s(\mu),g.r(\alpha^{\prime})}s_{g.\alpha^{\prime}}u_{g.s(\alpha^{\prime}),\varphi(g,\alpha^{\prime})}u_{s(\alpha),h}s_{\beta}^{*}$
$\displaystyle=\delta_{s(\mu),g.s(\nu)}s_{\mu(g.\alpha^{\prime})}\big{(}\delta_{g.s(\alpha^{\prime}),\varphi(g,\alpha^{\prime}).s(\alpha)}u_{g.s(\alpha^{\prime}),\varphi(g,\alpha^{\prime})h}\big{)}s_{\beta}^{*}\hskip
14.22636pt(\mathrm{as}~{}~{}r(\alpha^{\prime})=s(\nu)).$

In the case $\nu=\alpha\nu^{\prime}$ for some $\nu^{\prime}\in\Lambda$, the
above multiplication may be computed similarly, and otherwise is zero. Hence,
$M$ is closed under multiplication. Also, we have

$\big{(}s_{\mu}u_{s(\mu),g}s_{\nu}^{*}\big{)}^{*}=s_{\nu}u_{g^{-1}.s(\mu),g^{-1}}s_{\mu}^{*},$

so $M^{*}\subseteq M$. Since

$s_{\mu}=s_{\mu}u_{s(\mu),e_{G}}s_{s(\mu)}^{*}\hskip
14.22636pt\mathrm{and}\hskip 14.22636ptu_{v,g}=s_{v}u_{v,g}s_{g.v}^{*}$

for all $g\in G$, $v\in\Lambda^{0}$, and $\mu\in\Lambda$, it follows that $M$
is a $*$-subalgebra of $\mathrm{EP}_{R}(G,\Lambda)$ containing the generators
of $\mathrm{EP}_{R}(G,\Lambda)$. In light of Lemma 2.6, this concludes the
identification (2.1). ∎

###  2.3. The unital case

In case $\Lambda$ is a $k$-graph with finite $\Lambda^{0}$, we may give a
better description for Definition 2.5. Note that this case covers all unital
Exel-Pardo algebras $\mathrm{EP}_{R}(G,\Lambda)$.

######  Lemma 2.8.

Let $(G,\Lambda)$ be a self-similar $k$-graph. Then
$\mathrm{EP}_{R}(G,\Lambda)$ is a unital algebra if and only if the vertex set
$\Lambda^{0}$ is finite.

###### Proof.

If $\Lambda^{0}=\\{v_{1},\ldots,v_{l}\\}$ is finite, then using identification
(2.1), $P=\sum_{i=1}^{l}s_{v_{i}}$ is the unit of
$\mathrm{EP}_{R}(G,\Lambda)$. Conversely, if $\Lambda^{0}$ is infinite,
arrange $\Lambda^{0}=\\{v_{1},v_{2},\ldots\\}$, and for each $l\geq 1$ define
$P_{l}=\sum_{i=1}^{l}s_{v_{i}}$. Then by (2.1), $\\{P_{l}\\}_{l=1}^{\infty}$
is an increasing (infinite) local units for $\mathrm{EP}_{R}(G,\Lambda)$
converging to an element $P$ in the multiplier algebra
$\mathcal{M}(\mathrm{EP}_{R}(G,\Lambda))$. Then $P$ is an identity for all
elements of $\mathrm{EP}_{R}(G,\Lambda)$ while
$P\notin\mathrm{EP}_{R}(G,\Lambda)$ by eq. 2.1. This forces that $P$ is the
identity of $\mathcal{M}(\mathrm{EP}_{R}(G,\Lambda))$ being outside of
$\mathrm{EP}_{R}(G,\Lambda)$, and hence $\mathrm{EP}_{R}(G,\Lambda)$ is not
unital. ∎

Note that if $\\{s,u\\}$ is a $(G,\Lambda)$-family in a $*$-algebra $A$, then
for each $g\in G$ we may define $u_{g}:=\sum_{v\in\Lambda^{0}}u_{v,g}$ as an
element of the multiplier algebra $\mathcal{M}(A)$ with the property
$s_{v}u_{g}=u_{v,g}$ for all $v\in\Lambda^{0}$ (relations (2) and (5) of
Definition 2.5 yield $s_{v}u_{w,g}=\delta_{v,w}u_{v,g}$). Thus relations (3)
and (5) of Definition 2.5 imply that $u:G\rightarrow\mathcal{M}(A)$, defined
by $g\mapsto u_{g}$, is a unitary $*$-representation of $G$ on
$\mathcal{M}(A)$. In particular, in case $\Lambda^{0}$ is finite, $u_{g}$’s
lie all in $A$, and we may describe Definition 2.5 as the following:

######  Proposition 2.9.

Let $(G,\Lambda)$ be a self-similar $k$-graph. Suppose also that $\Lambda^{0}$
is finite. Then $\mathrm{EP}_{R}(G,\Lambda)$ is the universal $*$-algebra
generated by families $\\{s_{\mu}:\mu\in\Lambda\\}$ of partial isometries and
$\\{u_{g}:g\in G\\}$ of unitaries satisfying

  1. (1)

$\\{s_{\mu}:\mu\in\Lambda\\}$ is a Kumjian-Pask $\Lambda$-family;

  2. (2)

$u:G\rightarrow\mathrm{EP}_{R}(G,\Lambda)$, by $g\mapsto u_{g}$, is a unitary
$*$-representation of $G$ on $\mathrm{EP}_{R}(G,\Lambda)$, in the sense that

     * (a)

$u_{g}u_{h}=u_{gh}$ for all $g,h\in G$, and

     * (b)

$u^{*}_{g}=u^{-1}_{g}=u_{g^{-1}}$ for all $g\in G$;

  3. (3)

$u_{g}s_{\mu}=s_{g.\mu}u_{\varphi(g,\mu)}$ for all $g\in G$ and
$\mu\in\Lambda$.

##  3\. An example: The Zappa-Szép product $\Lambda\bowtie G$ and its
$*$-algebra

Let $(G,\Lambda)$ be a self-similar $k$-graph such that $|\Lambda^{0}|=1$. The
$C^{*}$-algebra and quotient boundary $C^{*}$-algebra associated to the Zappa-
Szép product $\Lambda\bowtie G$ as a semigroup were studied in [3, 13]. In
this section, we first define $\mathcal{Q}_{R}^{\mathrm{alg}}(S)$ the
algebraic analogue of quotient boundary $C^{*}$-algebra $\mathcal{Q}(S)$ of a
cancellative semigroup $S$. Then we show that
$\mathcal{Q}_{R}^{\mathrm{alg}}(\Lambda\bowtie G)$ is isomorphic to the Exel-
Pardo algebra $\mathrm{EP}_{R}(G,\Lambda)$.

Let us recall some terminology from [12, 3]. Let $S$ be a left-cancelative
semigroup with an identity. Given $X\subseteq S$ and $s\in S$, define
$sX:=\\{sx:x\in X\\}$ and $s^{-1}X:=\\{r\in S:sr\in X\\}$. Also, the set of
_constructible right ideals in $S$_ is defined as

$\mathcal{J}(S):=\\{s^{-1}_{1}r_{1}\ldots s_{l}^{-1}r_{l}S:l\geq
1,~{}s_{i},r_{i}\in S\\}\cup\\{\emptyset\\}.$

Then, a _foundation set in $\mathcal{J}(S)$_ is a finite subset
$F\subseteq\mathcal{J}(S)$ such that for each $Y\in\mathcal{J}(S)$, there
exists $X\in F$ with $X\cap Y\neq\emptyset$.

The following is the algebraic analogue of [12, Definition 2.2].

######  Definition 3.1.

Let $S$ be a left-cancelative semigroup and $R$ be a unital commutative
$*$-ring. The _boundary quotient $*$-algebra of $S$_ is the universal unital
$*$-algebra $\mathcal{Q}_{R}^{\mathrm{alg}}(S)$ over $R$ generated by a set of
isometries $\\{t_{s}:s\in S\\}$ and a set of projections
$\\{q_{X}:X\in\mathcal{J}(S)\\}$ satisfying

  1. (1)

$t_{s}t_{r}=t_{sr}$,

  2. (2)

$t_{s}q_{X}t_{s}^{*}=q_{sX}$,

  3. (3)

$q_{S}=1$ and $q_{\emptyset}=0$,

  4. (4)

$q_{X}q_{Y}=q_{X\cap Y}$, and moreover

  5. (5)

$\prod_{X\in F}(1-q_{X})=0$

for all $s,r\in S$, $X,Y\in\mathcal{J}(S)$, and foundation sets
$F\subseteq\mathcal{J}(S)$.

Let $\Lambda$ be a $k$-graph such that $\Lambda^{0}=\\{v\\}$. Then $\mu\nu$ is
composable for all $\mu,\nu\in\Lambda$, and hence $\Lambda$ may be considered
as a semigroup with the identity $v$. Also, the unique factorization property
follows that $\Lambda$ is cancelative.

######  Definition 3.2.

Let $(G,\Lambda)$ is a single-vertex self-similar $k$-graph. If we considered
$\Lambda$ as a semigroup, according to [3, Definition 3.1], the _Zappa-Szép
product $\Lambda\bowtie G$_ is the semigroup $\Lambda\times G$ with the
multiplication

$(\mu,g)(\nu,h):=(\mu(g.\nu),\varphi(g,\nu).h)\hskip
14.22636pt(\mu,\nu\in\Lambda~{}\mathrm{and}~{}g,h\in G).$

######  Remark 3.3.

If $\Lambda$ is a single-vertex $k$-graph, then [14, Lemma 3.2 (iv)] follows
that

$\mathcal{J}(\Lambda)=\big{\\{}\bigcup_{i=1}^{l}\mu_{i}\Lambda:l\geq
1,\mu_{i}\in\Lambda,d(\mu_{1})=\cdots=d(\mu_{l})\big{\\}}.$

In order to prove Theorem 3.6, the following lemmas are useful.

######  Lemma 3.4.

Let $(G,\Lambda)$ be a self-similar $k$-graph with $\Lambda^{0}=\\{v\\}$.
Suppose that for each $\mu\in\Lambda$, the map $g\mapsto\varphi(g,\mu)$ is
surjective. Then

  1. (1)

$\mathcal{J}(\Lambda\bowtie G)=\mathcal{J}(\Lambda)\times\\{G\\}$, where
$\emptyset\times G:=\emptyset$.

  2. (2)

A finite subset $F\subseteq\mathcal{J}(\Lambda)$ is a foundation set if and
only if $F^{\prime}=F\times\\{G\\}$ is a foundation set in
$\mathcal{J}(\Lambda\bowtie G)$.

###### Proof.

Statement (1) is just [14, Lemma 2.13]. For (2), suppose that
$F\subseteq\mathcal{J}(\Lambda)$ is a foundation set, and let $Y\times
G\in\mathcal{J}(\Lambda\bowtie G)$. Then there exists $X\in F$ such that
$X\cap Y\neq\emptyset$. Thus $(X\times G)\cap(Y\times G)\neq\emptyset$,
concluding that $F\times\\{G\\}$ is a foundation set in
$\mathcal{J}(\Lambda\bowtie G)$. The converse may be shown analogously. ∎

In the following, for $\mu\in\Lambda$ and $E\subseteq\Lambda$ we define

$\mathrm{Ext}(\mu;E):=\\{\alpha:(\alpha,\beta)\in\Lambda^{\mathrm{min}}(\mu,\nu)\mathrm{~{}for~{}some~{}}\nu\in
E\\}.$

######  Lemma 3.5.

Let $(G,\Lambda)$ be a self-similar $k$-graph with $\Lambda^{0}=\\{v\\}$. For
every $X=\cup_{i=1}^{l}\mu_{i}\Lambda$ and
$Y=\cup_{j=1}^{l^{\prime}}\nu_{j}\Lambda$ in $\mathcal{J}(\Lambda)$, we have

$X\cap Y=\cup\\{\mu_{i}\alpha\Lambda:1\leq i\leq
l,\alpha\in\mathrm{Ext}(\mu_{i};\\{\nu_{i},\ldots,\nu_{l^{\prime}}\\})\\}.$

###### Proof.

For any $\lambda\in X\cap Y$, there are $\alpha,\beta\in\Lambda$, $1\leq i\leq
l$, and $1\leq j\leq l^{\prime}$ such that
$\lambda=\mu_{i}\alpha=\nu_{j}\beta$. Define

$\alpha^{\prime}:=\alpha(0,d(\mu_{i})\vee
d(\nu_{j})-d(\mu_{i}))~{}~{}~{}\mathrm{and}~{}~{}~{}\beta^{\prime}:=\beta(0,d(\mu_{i})\vee
d(\nu_{j})-d(\nu_{j})).$

Then the factorizations property implies that
$\lambda=\mu_{i}\alpha^{\prime}\lambda^{\prime}=\nu_{j}\beta^{\prime}\lambda^{\prime}$
where $d(\mu_{i}\alpha^{\prime})=d(\nu_{j}\beta^{\prime})=d(\mu_{i})\vee
d(\nu_{j})$ and $\lambda^{\prime}=\lambda(d(\mu_{i})\vee
d(\nu_{j}),d(\lambda))$. This follows that
$\lambda\in\mu_{i}\alpha^{\prime}\Lambda$ with
$\alpha^{\prime}\in\mathrm{Ext}(\mu_{i};\\{\nu_{j}\\})$ as desired. The
reverse containment is trivial. ∎

The following result is inspired from [13, Theorem 3.3].

######  Theorem 3.6.

Let $(G,\Lambda)$ be a self-similar $k$-graph with $\Lambda^{0}=\\{v\\}$ and
let $\\{s_{\mu},u_{g}\\}$ be the $(G,\Lambda)$-family generating
$\mathrm{EP}_{R}(G,\Lambda)$ as in Proposition 2.9. Suppose that for every
$\mu\in\Lambda$ the map $g\mapsto\varphi(g,\mu)$ is surjective. If the family
$\\{t_{(\mu,g)},q_{X}:(\mu,g)\in\Lambda\bowtie
G,X\in\mathcal{J}(\Lambda\bowtie G)\\}$ generates
$\mathcal{Q}_{R}^{\mathrm{alg}}(\Lambda\bowtie G)$, then there exists an
$R$-algebra $*$-isomorphism
$\pi:\mathrm{EP}_{R}(G,\Lambda)\rightarrow\mathcal{Q}_{R}^{\mathrm{alg}}(\Lambda\bowtie
G)$ such that $\pi(s_{\mu})=t_{(\mu,e_{G})}$ and $\pi(u_{g})=q_{(v,g)}$ for
all $\mu\in\Lambda$ and $g\in G$.

###### Proof.

For every $\mu\in\Lambda$ and $g\in G$, define

$S_{\mu}:=t_{(\mu,e_{G})}~{}~{}\mathrm{and}~{}~{}U_{g}:=t_{(v,g)}.$

We will show that $\\{S,U\\}$ is a $(G,\Lambda)$-family in
$\mathcal{Q}_{R}^{\mathrm{alg}}(\Lambda\bowtie G)$, which is described in
Proposition 2.9. First, for each $g\in G$ we have

$U_{g}U_{g}^{*}=t_{(v,g)}~{}q_{\Lambda\bowtie
G}~{}t_{(v,g)}^{*}=q_{(v,g)\Lambda\bowtie G}=q_{\Lambda\bowtie
G}=1_{\mathcal{Q}_{R}^{\mathrm{alg}}(\Lambda\bowtie G)}.$

So, $g\mapsto U_{g}$ is a unitary $*$-representation of $G$ into
$\mathcal{Q}_{R}^{\mathrm{alg}}(\Lambda\bowtie G)$. Moreover, (KP1)-(KP3) can
be easily checked, so we verify (KP4) for $\\{s_{\mu}:\mu\in\Lambda\\}$. Fix
some $n\in\mathbb{N}^{k}$. Then $\\{\mu\Lambda:\mu\in\Lambda^{n}\\}$ is a
foundation set in $\mathcal{J}(\Lambda)$, so is $F=\\{\mu\Lambda\times
G:\mu\in\Lambda^{n}\\}$ in $\mathcal{J}(\Lambda\bowtie G)$ by Lemma 3.4. Hence
we have

$\displaystyle 1-\sum_{\mu\in\Lambda^{n}}S_{\mu}S_{\mu}^{*}$
$\displaystyle=\prod_{\mu\in\Lambda^{n}}(1-S_{\mu}S_{\mu}^{*})\hskip
45.5244pt(\mathrm{because~{}}S_{\mu}S_{\mu}^{*}\mathrm{s~{}are~{}pairwise~{}orthogonal})$
$\displaystyle=\prod_{\mu\in\Lambda^{n}}(1-t_{(\mu,e_{G})}~{}q_{\Lambda\bowtie
G}~{}t_{(\mu,e_{G})}^{*})$
$\displaystyle=\prod_{\mu\in\Lambda^{n}}(1-q_{\mu\Lambda\times G})\hskip
28.45274pt(\mathrm{by~{}eq.~{}(2)~{}of~{}Definition~{}\ref{defn3.1}})$
$\displaystyle=\prod_{X\in F}(1-q_{X})=0\hskip
28.45274pt(\mathrm{by~{}eq.~{}(5)~{}of~{}Definition~{}\ref{defn3.1}}).$

Because $S_{v}=U_{e_{G}}=1$, (KP4) is verified, and therefore
$\\{S_{\mu}:\mu\in\Lambda\\}$ is a Kumjian-Pask $\Lambda$-family. Since for
each $\mu\in\Lambda$ and $g\in G$,

$U_{g}S_{\mu}=t_{(v,g)}t_{(\mu,e_{G})}=t_{(v,g)(\mu,e_{G})}=t_{(g.\mu,\varphi(g,\mu))}=S_{g.\mu}U_{\varphi(g,\mu)},$

we have shown that $\\{S,U\\}$ is a $(G,\Lambda)$-family in
$\mathcal{Q}_{R}^{\mathrm{alg}}(\Lambda\bowtie G)$. Now the universality
deduces that the desired $*$-homomorphism
$\pi:\mathrm{EP}_{R}(G,\Lambda)\rightarrow\mathcal{Q}_{R}^{\mathrm{alg}}(\Lambda\bowtie
G)$ exists.

It is easy to see that $\pi$ is surjective. Indeed, given any
$(\mu,g)\in\Lambda\bowtie G$, eq. (1) of Definition 3.1 gives

$S_{\mu}U_{g}=t_{(\mu,e_{G})}t_{(v,g)}=t_{(\mu,e_{G})(v,g)}=t_{(\mu,g)}.$

Also, for every $X=(\cup_{i=1}^{l}\mu_{i}\Lambda)\times
G\in\mathcal{J}(\Lambda\bowtie G)$ with $d(\mu_{1})=\cdots=d(\mu_{l})$, since

$(\mu_{i}\Lambda\times G)\cap(\mu_{j}\Lambda\times
G)=(\mu_{i}\Lambda\cap\mu_{j}\Lambda)\times G=\emptyset\times G=\emptyset$

for $i\neq j$, relation (4) of Definition 3.1 implies that
$\\{q_{\mu_{i}\Lambda\times G}\\}_{i=1}^{l}$ are mutually orthogonal
projections. Thus

$q_{X}=\sum_{i=1}^{l}q_{\mu_{i}\Lambda\times
G}=\sum_{i=1}^{l}t_{(\mu_{i},e_{G})}~{}q_{\Lambda\bowtie
G}~{}t_{(\mu_{i},e_{G})}^{*}=\sum_{i=1}^{l}S_{\mu_{i}}S_{\mu_{i}}^{*},$

concluding that $\pi$ is surjective.

To prove the injectivity, it suffices to find a homomorphism
$\rho:\mathcal{Q}_{R}^{\mathrm{alg}}(\Lambda\bowtie
G)\rightarrow\mathrm{EP}_{R}(G,\Lambda)$ such that
$\rho\circ\pi=\mathrm{id}_{\mathrm{EP}_{R}(G,\Lambda)}$. To do this, for any
$(\mu,g)\in\Lambda\bowtie G$ and $X=(\cup_{i=1}^{l}\mu_{i}\Lambda)\times
G\in\mathcal{J}(\Lambda\bowtie G)$, define

$T_{(\mu,g)}:=s_{\mu}u_{g}\hskip 14.22636pt\mathrm{and}\hskip
14.22636ptQ_{X}:=\sum_{i=1}^{l}s_{\mu_{i}}s_{\mu_{i}}^{*}.$

We will show that the family $\\{T,Q\\}$ satisfies the properties of
Definitions 3.1. Relations (1)-(3) easily hold by the $(G,\Lambda)$-relations
for $\\{s,u\\}$. Also, for every $X=(\cup_{i=1}^{l}\mu_{i}\Lambda)\times G$
and $Y=(\cup_{j=1}^{l^{\prime}}\nu_{j}\Lambda)\times G$ in
$\mathcal{J}(\Lambda\bowtie G)$, we have

$\displaystyle Q_{X}Q_{Y}$
$\displaystyle=\big{(}\sum_{i=1}^{l}s_{\mu_{i}}s_{\mu_{i}}^{*}\big{)}\big{(}\sum_{j=1}^{l^{\prime}}s_{\nu_{j}}s_{\nu_{j}}^{*}\big{)}$
$\displaystyle=\sum_{i,j}s_{\mu_{i}}(s_{\mu_{i}}^{*}s_{\nu_{j}})s_{\nu_{j}}^{*}$
$\displaystyle=\sum_{i,j}s_{\mu_{i}}\big{(}\sum_{(\alpha,\beta)\in\Lambda^{\mathrm{min}}(\mu_{i},\nu_{j})}s_{\alpha}s_{\beta}^{*}\big{)}s_{\nu_{j}}^{*}\hskip
14.22636pt(\mathrm{by~{}[1,Lemma~{}3.3]})$
$\displaystyle=\sum_{i,j}\sum_{\begin{subarray}{c}\mu_{i}\alpha=\nu_{j}\beta\\\
d(\mu_{i}\alpha)=d(\mu_{i})\vee
d(\nu_{j})\end{subarray}}s_{\mu_{i}\alpha}s_{\nu_{j}\beta}^{*}$
$\displaystyle=Q_{X\cap Y}\hskip
42.67912pt(\mathrm{by~{}Lemma}~{}\ref{lem3.5}).$

It remains to verify eq. (5) of Definition 3.1. For, let $F=\\{X_{i}\times
G:=\cup_{j=1}^{t_{i}}\mu_{ij}\Lambda\times G:1\leq i\leq l\\}$ be a foundation
set in $\mathcal{J}(\Lambda\bowtie G)$. Then
$F^{\prime}=\\{X_{i}=\cup_{j=1}^{t_{i}}\mu_{ij}\Lambda\\}_{i=1}^{l}$ is a
foundation set in $\mathcal{J}(\Lambda)$ by Lemma 3.4(2). Define
$n:=\bigvee_{i,j}d(\mu_{ij})$. We claim that the set
$M=\\{\mu_{ij}\alpha:\alpha\in\Lambda^{n-d(\mu_{ij})},1\leq i\leq l,1\leq
j\leq t_{i}\\}$ coincides with $\Lambda^{n}$. Indeed, if on the contrary there
exists some $\lambda\in\Lambda^{n}\setminus M$, then
$\Lambda^{\mathrm{min}}(\lambda,\mu_{ij})=\emptyset$, and hence
$\lambda\Lambda\cap\mu_{ij}\Lambda=\emptyset$ for all $i$ and $j$. This yields
that $\lambda\Lambda\cap X_{i}=\emptyset$ for every $X_{i}\in F^{\prime}$,
contradicting that $F^{\prime}$ is a foundation set in $\mathcal{J}(\Lambda)$.

Now one may compute

$\displaystyle\prod_{X_{i}\times G\in F}(1-Q_{X_{i}\times G})$
$\displaystyle=\prod_{i=1}^{l}(1-\sum_{j=1}^{t_{i}}s_{\mu_{ij}}s_{\mu_{ij}}^{*})$
$\displaystyle=\prod_{i=1}^{l}\big{(}1-\sum_{j=1}^{t_{i}}s_{\mu_{ij}}(\sum_{\alpha\in\Lambda^{n-d(\mu_{ij})}}s_{\alpha}s_{\alpha}^{*})s_{\mu_{ij}}^{*}\big{)}$
$\displaystyle=\prod_{i=1}^{l}\big{(}1-\sum_{j=1}^{t_{i}}\sum_{\alpha\in\Lambda^{n-d(\mu_{ij})}}s_{\mu_{ij}\alpha}s_{\mu_{ij}\alpha}^{*}\big{)}\hskip
28.45274pt(\star).$

Note that projections $s_{\mu_{ij}\alpha}s_{\mu_{ij}\alpha}^{*}$ are pairwise
orthogonal because $d(\mu_{ij}\alpha)=n$ for all $i,j$ (see [1, Remark
3.2(c)]). Hence, using the above claim, expression $(\star)$ equals to

$(\star)=1-\sum_{\lambda\in\Lambda^{n}}s_{\lambda}s_{\lambda}^{*}=0\hskip
28.45274pt(\mathrm{by~{}(KP4)}).$

Therefore, the family $\\{T,Q\\}$ satisfies the relations of Definition 3.1,
and by the universality there exists an algebra $*$-homomorphism
$\rho:\mathcal{Q}_{R}^{\mathrm{alg}}(\Lambda\bowtie
G)\rightarrow\mathrm{EP}_{R}(G,\Lambda)$ such that
$\rho(t_{(\mu,g)})=T_{(\mu,g)}$ and $\rho(q_{X})=Q_{X}$ for
$(\mu,g)\in\Lambda\bowtie G$ and $X\in\mathcal{J}(\Lambda\bowtie G)$. It is
clear that $\rho\circ\pi=\mathrm{id}_{\mathrm{EP}_{R}(G,\Lambda)}$ because it
fixes the generators of $\mathrm{EP}_{R}(G,\Lambda)$. This follows that $\pi$
is injective, completing the proof. ∎

##  4\. A graded Uniqueness Theorem

In this section, we prove a graded uniqueness theorem for
$\mathrm{EP}_{R}(G,\Lambda)$. Our result here generalizes and modifies [8,
Theorem A] for self-similar $k$-graphs. This modification, in particular,
helps us to prove Theorems 5.5 and 6.8.

Let us first recall some definitions. Let $\Gamma$ be a group and $A$ be an
algebra over a ring $R$. $A$ is called _$\Gamma$ -graded_ (or briefly,
_graded_ whenever the group is clear) if there is a family of $R$-submodules
$\\{A_{\gamma}:\gamma\in\Gamma\\}$ of $A$ such that
$A=\bigoplus_{\gamma\in\Gamma}A_{\gamma}$ and
$A_{\gamma}A_{\gamma^{\prime}}\subseteq A_{\gamma\gamma^{\prime}}$ for all
$\gamma,\gamma^{\prime}\in\Gamma$. Then each set $A_{\gamma}$ is said
_$\gamma$ -homogeneous component of $A$_. In this case, we say an ideal $I$ of
$A$ to be _$\Gamma$ -graded_ if $I=\bigoplus_{\gamma\in\Gamma}(I\cap
A_{\gamma})$. Note that an ideal $I$ of $A$ is $\Gamma$-graded if and only if
it is generated be a subset of $\bigcup_{\gamma\in\Gamma}A_{\gamma}$ the
homogenous elements of $A$.

Furthermore, if $A$ and $B$ are two $\Gamma$-graded algebra over $R$, a
homomorphism $\phi:A\rightarrow B$ is said to be a _graded homomorphism_ if
$\phi(A_{\gamma})\subseteq B_{\gamma}$ for all $\gamma\in\Gamma$. Recall that
the kernel of a graded homomorphism is always a graded ideal. Also, if $I$ is
a graded ideal of $A$, then there is a natural $\Gamma$-grading
$(A_{\gamma}+I)_{\gamma\in\Gamma}$ on the quotient algebra $A/I$, and thus the
quotient map $A\rightarrow A/I$ is a graded homomorphism.

######  Lemma 4.1.

Let $(G,\Lambda)$ be a self-similar $k$-graph. If for every
$n\in\mathbb{Z}^{k}$, define

$\mathrm{EP}_{R}(G,\Lambda)_{n}:=\mathrm{span}_{R}\big{\\{}s_{\mu}u_{s(\mu),g}s_{\nu}^{*}:g\in
G,~{}\mu,\nu\in\Lambda,~{}\mathrm{and}~{}d(\mu)-d(\nu)=n\big{\\}},$

then $(\mathrm{EP}_{R}(G,\Lambda)_{n})_{n\in\mathbb{Z}^{k}}$ is a
$\mathbb{Z}^{k}$-grading on $\mathrm{EP}_{R}(G,\Lambda)$.

###### Proof.

Consider the free $*$-algebra $\mathbb{F}_{R}(X)$ and its ideal $I$ as in
Subsection 2.2. Definitions

$\theta(S_{\mu}):=d(\mu),~{}\theta(S_{\mu}^{*}):=-d(\mu),~{}\mathrm{and}~{}\theta(U_{v,g}):=0$

for all $g\in G$, $v\in\Lambda^{0}$, and $\mu\in\Lambda$, induce a
$\mathbb{Z}^{k}$-grading on $\mathbb{F}_{R}(X)$. Also, since the generators of
$I$ are homogenous, $I$ is a graded ideal. Therefore,
$\mathrm{EP}_{R}(G,\Lambda)\cong\mathbb{F}_{R}(X)/I$ is a
$\mathbb{Z}^{k}$-graded algebra, and Proposition 2.7 concludes the result. ∎

######  Theorem 4.2 (Graded Uniqueness).

Let $(G,\Lambda)$ be a self-similar $k$-graph. Let
$\phi:\mathrm{EP}_{R}(G,\Lambda)\rightarrow B$ be a $\mathbb{Z}^{k}$-graded
$R$-algebra $*$-homomorphism into a $\mathbb{Z}^{k}$-graded $*$-algebra $B$.
If $\phi(a)\neq 0$ for every nonzero element of the form
$a=\sum_{i=1}^{l}r_{i}u_{v,g_{i}}$ with $v\in\Lambda^{0}$ and
$g_{i}^{-1}.v=g_{j}^{-1}.v$ for $1\leq i,j\leq l$, then $\phi$ is injective.

###### Proof.

For convenience we write $A=\mathrm{EP}_{R}(G,\Lambda)$. Since
$A=\bigoplus_{n\in\mathbb{Z}^{k}}A_{n}$ and $\phi$ preserves the grading, it
suffices to show that $\phi$ is injective on each $A_{n}$. So, fix some $b\in
A_{n}$, and assume $\phi(b)=0$. By equation (2.1), we can write

(4.1) $b=\sum_{i=1}^{l}r_{i}s_{\mu_{i}}u_{w_{i},g_{i}}s_{\nu_{i}}^{*}$

where $w_{i}=s(\mu_{i})=g_{i}.s(\nu_{i})$ and $d(\mu_{i})-d(\nu_{i})=n$ for
$1\leq i\leq l$. Define $n^{\prime}=\vee_{1\leq i\leq l}d(\mu_{i})$. Then, for
each $i\in\\{1\ldots l\\}$, (KP4) says that

$s_{w_{i}}=\sum_{\lambda\in
w_{i}\Lambda^{n^{\prime}-d(\mu_{i})}}s_{\lambda}s_{\lambda}^{*},$

and we can write

$\displaystyle s_{\mu_{i}}u_{w_{i},g_{i}}s_{\nu_{i}}^{*}$
$\displaystyle=s_{\mu_{i}}(s_{s(\mu_{i})})u_{w_{i},g_{i}}s_{\nu_{i}}^{*}$
$\displaystyle=\sum
s_{\mu_{i}}(s_{\lambda}s_{\lambda}^{*})u_{w_{i},g_{i}}s_{\nu_{i}}^{*}$
$\displaystyle=\sum
s_{\mu_{i}\lambda}\big{(}s_{\nu_{i}}u_{w_{i},g_{i}}^{*}s_{\lambda}\big{)}^{*}$
$\displaystyle=\sum
s_{\mu_{i}\lambda}\big{(}s_{\nu_{i}}u_{g_{i}^{-1}.w_{i},g_{i}^{-1}}s_{\lambda}\big{)}^{*}$
$\displaystyle=\sum
s_{\mu_{i}\lambda}\big{(}s_{\nu_{i}}s_{g_{i}^{-1}.\lambda}u_{g_{i}^{-1}.s(\lambda),\varphi(g_{i}^{-1},\lambda)}\big{)}^{*}$
$\displaystyle=\sum
s_{\mu_{i}\lambda}u_{\varphi(g_{i}^{-1},\lambda)^{-1}g_{i}^{-1}.s(\lambda),\varphi(g_{i}^{-1},\lambda)^{-1}}s_{\nu_{i}(g_{i}^{-1}.\lambda)}^{*}$

where the above summations are on $\lambda\in
w_{i}\Lambda^{n^{\prime}-d(\mu_{i})}$. So, in each term of (4.1), we may
assume $d(\mu_{i})=n^{\prime}$ and $d(\nu_{i})=n^{\prime}-n$. Now, for given
any $1\leq j\leq l$, (KP3) implies that

$s_{\mu_{j}}^{*}bs_{\nu_{j}}=s_{\mu_{j}}^{*}\big{(}\sum_{i=1}^{l}r_{i}s_{\mu_{i}}u_{w_{i},g_{i}}s_{\nu_{i}}^{*}\big{)}s_{\nu_{j}}=\sum_{i\in[j]}r_{i}u_{w_{i},g_{i}},$

where $[j]:=\\{1\leq i\leq l:(\mu_{i},\nu_{i})=(\mu_{j},\nu_{j})\\}$. Thus

$\phi\big{(}\sum_{i\in[j]}r_{i}u_{w_{i},g_{i}}\big{)}=\phi(s_{\mu_{j}}^{*})\phi(b)\phi(s_{\nu_{j}})=0$

and hypothesis forces $\sum_{i\in[j]}r_{i}u_{w_{i},g_{i}}=0$. Therefore,

$\sum_{i\in[j]}r_{i}s_{\mu_{i}}u_{w_{i},g_{i}}s_{\nu_{i}}^{*}=s_{\mu_{j}}\big{(}\sum_{i\in[j]}r_{i}u_{w_{i},g_{i}}\big{)}s_{\nu_{j}}^{*}=0.$

Since the index set $\\{1,\ldots,l\\}$ is disjoint unions of $[j]$’s, we
obtain $b=0$. This follows that $\phi$ is injective. ∎

##  5\. $\mathrm{EP}_{R}(G,\Lambda)$ as a Steinberg algebra

Inspired from [6], Li and Yang in [14, 15] gave a groupoid model for the
$C^{*}$-algebra $\mathcal{O}_{G,\Lambda}$. In this section, after recalling
the groupoid $\mathcal{G}_{G,\Lambda}$ from [14], we prove, provided
$(G,\Lambda)$ being pseudo free, that $\mathrm{EP}_{R}(G,\Lambda)$ is
isomorphic to the Steinberg algebra $A_{R}(\mathcal{G}_{G,\Lambda})$. Although
this result is the $k$-graph generalization of [8, Theorem B], but our proof
here relies on the graded uniqueness theorem, Theorem 4.2, and is completely
deferent from that of [8, Theorem B]. This gives us a much easier and shorter
proof.

We first review some terminology about groupoids; for more details see [18]. A
_groupoid_ is a small category $\mathcal{G}$ with inverses. For each
$\alpha\in\mathcal{G}$, we may define the range $r(\alpha):=\alpha\alpha^{-1}$
and the source $s(\alpha):=\alpha^{-1}\alpha$ satisfying
$r(\alpha)\alpha=\alpha=\alpha s(\alpha)$. This follows that for every
$\alpha,\beta\in\mathcal{G}$, the composition $\alpha\beta$ is well-defined if
and only if $s(\alpha)=r(\beta)$. The _unit space of $\mathcal{G}$_ is
$\mathcal{G}^{(0)}:=\\{\alpha^{-1}\alpha:\alpha\in\mathcal{G}\\}$. Throughout
the paper we work with _topological groupoids_ , which are ones equipped with
a topology such that the maps $r$ and $s$ are continuous. Then a _bisecion_ is
a subset $B\subseteq\mathcal{G}$ such that both restrictions $r|_{B}$ and
$s|_{B}$ are homeomorphisms. In case $\mathcal{G}$ has a basis of compact open
bisections, $\mathcal{G}$ is called an _ample groupoid_.

Let $(G,\Lambda)$ be a self-similar $k$-graph. Recall here the groupoid
$\mathcal{G}_{G,\Lambda}$ introduced in [14]. Let $C(\mathbb{N}^{k},G)$ be the
group of all map form $\mathbb{N}^{k}$ to $G$ with the pointwise
multiplication. For $f,g\in C(\mathbb{N}^{k},G)$, define the equivalent
relation $f\sim g$ in case there exists $n_{0}\in\mathbb{N}^{k}$ such that
$f(n)=g(n)$ for all $n\geq n_{0}$. Write
$Q(\mathbb{N}^{k},G):=C(\mathbb{N}^{k},G)/\sim$. Also, for each
$z\in\mathbb{Z}^{k}$, let $\mathcal{T}_{z}:C(\mathbb{N}^{k},G)\rightarrow
C(\mathbb{N}^{k},G)$ be the automorphism defined by

$\mathcal{T}_{z}(f)(n)=\left\\{\begin{array}[]{ll}f(n-z)&n-z\geq 0\\\
e_{G}&\mathrm{oth.}\end{array}\right.\hskip 14.22636pt(f\in
C(\mathbb{N}^{k},G),~{}n\in\mathbb{N}^{k})$

Then $\mathcal{T}_{z}$ induces an automorphism, denoted again by
$\mathcal{T}_{z}$, on $Q(\mathbb{N}^{k},G)$, which is
$\mathcal{T}_{z}([f])=[\mathcal{T}_{z}(f)]$. So,
$\mathcal{T}:\mathbb{Z}^{k}\rightarrow\mathrm{Aut}Q(\mathbb{N}^{k},G)$ is a
homomorphism and we consider the semidirect product group
$Q(\mathbb{N}^{k},G)\rtimes_{\mathcal{T}}\mathbb{Z}^{k}$.

Note that for every $g\in G$ and $x\in\Lambda^{\infty}$, one may define
$\varphi(g,x)\in C(\mathbb{N}^{k},G)$ by

$\varphi(g,x)(n):=\varphi(g,x(0,n))\hskip 14.22636pt(n\in\mathbb{N}^{k}).$

Moreover, [14, Lemma 3.7] say that there exists a unique action
$G\curvearrowright\Lambda^{\infty}$ by defining

$(g.x)(m,n):=\varphi(g,x(0,m)).x(m,n)\hskip 19.91692pt((m,n)\in\Omega_{k})$

for every $g\in G$ and $x\in\Lambda^{\infty}$.

######  Definition 5.1.

A self-similar $k$-graph $(G,\Lambda)$ is said to be _pseudo free_ if for any
$g\in G$ and $\mu\in\Lambda$, $g.\mu=\mu$ and $\varphi(g,\mu)=e_{G}$ imply
$g=e_{G}$.

According to [14, Lemma 5.6], in case $(G,\Lambda)$ is pseudo free, for every
$g,h\in G$ and $\mu\in\Lambda$ we have

$g.\mu=h.\mu~{}~{}\mathrm{and}~{}~{}\varphi(g,\mu)=\varphi(h,\mu)~{}~{}\Longrightarrow~{}~{}g=h.$

######  Definition 5.2.

Associated to $(G,\Lambda)$ we define the subgroupoid

$\mathcal{G}_{G,\Lambda}:=\left\\{\big{(}\mu(g.x);\mathcal{T}_{d(\mu)}([\varphi(g,x)]),d(\mu)-d(\nu);\nu
x\big{)}:g\in G,~{}\mu,\nu\in\Lambda,~{}s(\mu)=g.s(\nu)\right\\}$

of
$\Lambda^{\infty}\times\big{(}Q(\mathbb{N}^{k},G)\rtimes_{\mathcal{T}}\mathbb{Z}^{k}\big{)}\times\Lambda^{\infty}$
with the range and source maps

$r(x;[f],n-m;y)=x\hskip 14.22636pt\mathrm{and}\hskip
14.22636pts(x;[f],n-m;y)=y.$

If we set

$Z(\mu,g,\nu):=\left\\{\big{(}\mu(g.x);\mathcal{T}_{d(\mu)}([\varphi(g,x)]),d(\mu)-d(\nu);\nu
x\big{)}:x\in s(\nu)\Lambda^{\infty}\right\\},$

then the basis

$\mathcal{B}_{G,\Lambda}:=\left\\{Z(\mu,g,\nu):\mu,\nu\in\Lambda,g\in
G,s(\mu)g.s(\nu)\right\\}$

induces a topology on $\mathcal{G}_{G,\Lambda}$. In case $(G,\Lambda)$ is
pseudo free, [15, Proposition 3.11] shows that $\mathcal{G}_{G,\Lambda}$ is a
Hausdorff groupoid with compact open base $\mathcal{B}_{G,\Lambda}$.

######  Definition 5.3.

Let $(G,\Lambda)$ be a pseudo free self-similar $k$-graph and $R$ a unital
commutative $*$-ring. Then _the Steinberg algebra associated to $(G,\Lambda)$_
is the $R$-algebra

$A_{R}(\mathcal{G}_{G,\Lambda}):=\mathrm{span}_{R}\\{1_{B}:B~{}\mathrm{is~{}a~{}compact~{}open~{}bisection}\\}$

endowed with the pointwise addition, the multiplication
$fg(\gamma):=\sum_{\alpha\beta=\gamma}f(\alpha)g(\beta)$, and the involution
$f^{*}(\gamma):=f(\gamma^{-1})^{*}$ for all
$\gamma\in\mathcal{G}_{G,\Lambda}$.

To prove Theorem 5.5 below, we need the following lemma.

######  Lemma 5.4.

Let $(G,\Lambda)$ be a pseudo free self-similar $k$-graph. Let
$v,w\in\Lambda^{0}$ and $g,h\in G$ with $g.v=h.v=w$. Then $Z(v,g,w)\cap
Z(v,h,w)=\emptyset$ whenever $g\neq h$.

###### Proof.

Suppose that $(g.x;[\varphi(g,x)],0;x)=(h.y,[\varphi(h,y)],0;y)\in
Z(v,g,w)\cap Z(v,h,w)$ where $x,y\in Z(w)$. Then $y=x$, $g.x=h.x$ and
$[\varphi(g,x)]=[\varphi(h,x)]$. Since $(G,\Lambda)$ is pseudo free, [14,
Corollary 5.6] implies that $g=h$. ∎

######  Theorem 5.5.

Let $(G,\Lambda)$ be a pseudo free self-similar $k$-graph. Then there is a
(unique) $*$-algebra isomorphism $\phi:\mathrm{EP}_{R}(G,\Lambda)\rightarrow
A_{R}(\mathcal{G}_{G,\Lambda})$ such that

$\phi(s_{\mu})=1_{Z(\mu,e_{G},s(\mu))}\hskip 14.22636pt\mathrm{and}\hskip
14.22636pt\phi(u_{v,g})=1_{Z(v,g,g^{-1}.v)}$

for every $\mu\in\Lambda$, $v\in\Lambda^{0}$, and $g\in G$. In particular,
elements $rs_{\mu}$ and $ru_{v,g}$ with $r\in R\setminus\\{0\\}$ are all
nonzero.

###### Proof.

For each $v\in\Lambda^{0}$, $\mu\in\Lambda$, and $g\in G$ define

$S_{\mu}:=1_{Z(\mu,e_{G},s(\mu))}\hskip 14.22636pt\mathrm{and}\hskip
14.22636ptU_{v,g}:=1_{Z(v,g,g^{-1}.v)}.$

Since $S_{\mu}^{*}=1_{Z(\mu,e_{G},s(\mu))^{-1}}=1_{Z(s(\mu),e_{G},\mu)}$ and
$U_{v,g}^{*}=1_{Z(v,g,g^{-1}.v)^{-1}}=1_{Z(g^{-1}.v,g^{-1},v)}$, a long but
straightforward computation shows that $\\{S_{\mu},U_{v,g}\\}$ is a
$(G,\Lambda)$-family in $A_{R}(\mathcal{G}_{G,\Lambda})$. Then, by the
universal property, such $*$-homomorphism $\phi$ exists.

[15, Proposition 3.11] says that $\mathcal{G}_{G,\Lambda}$ is ample with
compact open base $\mathcal{B}_{G,\Lambda}$. Since each element $Z(\mu,g,\nu)$
of $\mathcal{B}_{G,\Lambda}$ can be written as

$Z(\mu,g,\nu)=Z(\mu,e_{G},s(\mu))Z(s(\mu),g,s(\nu))Z(\nu,e_{G},s(\nu))^{-1},$

$\phi$ is surjective.

We will show the injectivity of $\phi$ by applying the graded uniqueness
theorem. Note that the continuous 1-cocycle
$c:\mathcal{G}_{G,\Lambda}\rightarrow\mathbb{Z}^{k}$, defined by
$c(\mu(g.x);[f],d(\mu)-d(\nu);\nu x):=d(\mu)-d(\nu)$, induces a
$\mathbb{Z}^{k}$-grading on $A_{R}(\mathcal{G}_{G,\Lambda})$. Also, $\phi$
preserves the $\mathbb{Z}^{k}$-grading because it does on the generators. Now,
to apply Theorem 4.2, we assume $\phi(a)=0$ for an element of the form
$a=\sum_{i=1}^{l}r_{i}u_{v,g_{i}}$ with $g_{i}^{-1}.v=g_{j}^{-1}.v$ for $1\leq
i,j\leq l$. We may also assume $g_{i}$’s are distinct (otherwise, combine the
terms with same $g_{i}$’s). We then have

$\phi(a)=\sum_{i=1}^{l}r_{i}1_{Z(v,g_{i},g_{i}^{-1}.v)}=0.$

Lemma 5.4 says that bisections $Z(v,g_{i},g_{i}^{-1}.v)$ are pairwise
disjoint. Hence, for each $i$, if $\alpha\in Z(v,g_{i},g_{i}^{-1}.v)$, then we
have $r_{i}=\phi(a)(\alpha)=0$. Therefore $a=0$, and Theorem 4.2 concludes
that $\phi$ is injective. We are done. ∎

Combining [19, Theorem 6.7], [14, Theorem 5.9], and Theorem 5.5 gives the
following. (Although in [14] it is supposed $|\Lambda^{0}|<\infty$, but [14,
Theorem 5.9] holds also for $\Lambda$ with infinite vertices.)

######  Corollary 5.6.

Let $(G,\Lambda)$ be a pseudo free self-similar $k$-graph over an amenable
group $G$. Then the complex algebra $\mathrm{EP}_{\mathbb{C}}(G,\Lambda)$ is a
dense subalgebra of $\mathcal{O}_{G,\Lambda}$ introduced in [15].

In the following, we see that the Kumjian-Pask algebra
$\mathrm{KP}_{R}(\Lambda)$ from [1] can be embedded in
$\mathrm{EP}_{R}(G,\Lambda)$.

######  Corollary 5.7.

Let $(G,\Lambda)$ be a pseudo free self-similar $k$-graph. Let the Kumjisn-
Pask algebra $\mathrm{KP}_{R}(\Lambda)$ be generated by a Kumjian-Pask
$\Lambda$-family $\\{t_{\mu}:\mu\in\Lambda\\}$. Then the map $t_{\mu}\mapsto
s_{\mu}$ embeds $\mathrm{KP}_{R}(\Lambda)$ into $\mathrm{EP}_{R}(G,\Lambda)$
as a $*$-subalgebra.

###### Proof.

We know that $\mathrm{KP}_{R}(\Lambda)$ is $\mathbb{Z}^{k}$-graded by
homogenous components

$\mathrm{KP}_{R}(\Lambda)_{n}:=\mathrm{span}_{R}\left\\{t_{\mu}t_{\nu}^{*}:\mu,\nu\in\Lambda,d(\mu)-d(\nu)=n\right\\}.$

for all $n\in\mathbb{Z}^{k}$. Then, the universal property of Kumjian-Pask
algebras gives a graded $*$-algebra homomorphism
$\phi:\mathrm{KP}_{R}(\Lambda)\rightarrow\mathrm{EP}_{R}(G,\Lambda)$ such that
$\phi(t_{\mu}):=s_{\mu}$ and $\phi(t_{\mu}^{*}):=s_{\mu}^{*}$ for every
$\mu\in\Lambda$. Moreover, Theorem 5.5 shows that $\phi(rt_{\mu})=rs_{\mu}\neq
0$ for all $r\in R\setminus\\{0\\}$ and $\mu\in\Lambda$. Therefore, the graded
uniqueness theorem for Kumjian-Pask algebras [1, Theorem 4.1] concludes that
$\phi$ is injective. ∎

######  Definition 5.8.

Let $\mathcal{G}$ be a topological groupoid. We say that $\mathcal{G}$ is
_topological principal_ if the set of units with trivial isotropy group, that
is $\\{u\in\mathcal{G}^{(0)}:s^{-1}(u)\cap r^{-1}(u)=\\{u\\}\\}$, is dense in
$\mathcal{G}^{(0)}$.

The analogue of the topological principal property for self-similar $k$-graphs
is $G$-aperiodicity (see [14, Proposition 6.5]).

######  Definition 5.9.

Let $(G,\Lambda)$ be a self-similar $k$-graph. Then $\Lambda$ is said to be
_$G$ -aperiodic_ if for every $v\in\Lambda^{0}$, there exists $x\in
v\Lambda^{\infty}$ with the property that

$x(p,\infty)=g.x(q,\infty)~{}~{}\Longrightarrow~{}~{}g=e_{G}~{}~{}\mathrm{and}~{}~{}p=q\hskip
14.22636pt(\forall g\in G,~{}\forall p,q\in\mathbb{N}^{k}).$

######  Theorem 5.10 (Cuntz-Krieger uniqueness).

Let $(G,\Lambda)$ be a pseudo free self-similar $k$-graph. Let $(G,\Lambda)$
be also $G$-aperiodic. Suppose that
$\phi:\mathrm{EP}_{R}(G,\Lambda)\rightarrow A$ is a $*$-algebra homomorphism
from $\mathrm{EP}_{R}(G,\Lambda)$ into a $*$-algebra $A$ such that
$\phi(rs_{v})\neq 0$ for all $0\neq r\in R$ and $v\in\Lambda^{0}$. Then $\phi$
is injective.

###### Proof.

First note that $\mathcal{G}_{G,\Lambda}$ is a Hausdorff ample groupoid by
[15, Proposition 3.11], that $\mathcal{B}_{G,\Lambda}$ is a basis for
$\mathcal{G}_{G,\Lambda}$ consisting compact open bisections. Also, [15, Lemma
3.12] says that $\mathcal{G}_{G,\Lambda}$ is topologically principal. So, we
may apply [4, Theorem 3.2].

Denote by $\psi:\mathrm{EP}_{R}(G,\Lambda)\rightarrow
A_{R}(\mathcal{G}_{G,\Lambda})$ the isomorphism of Theorem 5.5. If on the
contrary $\phi$ is not injective, then so is not
$\widetilde{\phi}:=\phi\circ\psi^{-1}:A_{R}(\mathcal{G}_{G,\Lambda})\rightarrow
A$. Thus, by [4, Theorem 3.2], there exists a compact open subset
$K\subseteq\mathcal{G}_{G,\Lambda}^{(0)}$ and $r\neq 0$ such that
$\widetilde{\phi}(r1_{K})=0$. Since $K$ is open, there is a unit
$U=Z(\mu,e_{G},\mu)\in\mathcal{B}_{G,\Lambda}$ such that $U\subseteq K$. So we
get

$\phi(rs_{\mu}s_{\mu}^{*})=\widetilde{\phi}(r1_{U})=\widetilde{\phi}(r1_{U\cap
K})=\widetilde{\phi}(r1_{K})\widetilde{\phi}(1_{U})=0,$

and hence

$\phi(rs_{s(\mu)})=\phi(s_{\mu}^{*})\phi(rs_{\mu}s_{\mu}^{*})\phi(s_{\mu})=0.$

This contradicts hypothesis, and therefore $\phi$ is injective. ∎

##  6\. Ideal structure

By an _ideal_ we mean a two-sided and self-adjoint one. In this section, we
characterize basic, $\mathbb{Z}^{k}$-graded and diagonal-invariant ideals of
$\mathrm{EP}_{R}(G,\Lambda)$, which are exactly all basic
$Q(\mathbb{N}^{k},G)\rtimes_{\mathcal{T}}\mathbb{Z}^{k}$-graded ones.

Let $(G,\Lambda)$ be a pseudo free self-similar $k$-graph. Since
$\mathcal{G}_{G,\Lambda}$ is an ample groupoid [14, Theorem 5.8],
$\mathcal{G}_{G,\Lambda}^{(0)}$ is both open and closed, and for every $f\in
A_{R}(\mathcal{G}_{G,\Lambda})$ the restricted function
$f|_{\mathcal{G}_{G,\Lambda}^{(0)}}=f\chi_{\mathcal{G}_{G,\Lambda}^{(0)}}$
lies again in $A_{R}(\mathcal{G}_{G,\Lambda})$. Then
$A_{R}(\mathcal{G}_{G,\Lambda}^{(0)})$ is a $*$-subalgebra of
$A_{R}(\mathcal{G}_{G,\Lambda})$ and there is a conditional expectation
$\mathcal{E}:A_{R}(\mathcal{G}_{G,\Lambda})\rightarrow
A_{R}(\mathcal{G}_{G,\Lambda}^{(0)})$ defined by
$\mathcal{E}(f)=f|_{\mathcal{G}_{G,\Lambda}^{(0)}}$ for $f\in
A_{R}(\mathcal{G}_{G,\Lambda})$. Let
$D:=\mathrm{span}_{R}\\{s_{\mu}s_{\mu}^{*}:\mu\in\Lambda\\}$ be the diagonal
of $\mathrm{EP}_{R}(G,\Lambda)$. In light of Theorem 5.5, it is easy to check
that the expectation is $\mathcal{E}:\mathrm{EP}_{R}(G,\Lambda)\rightarrow D$
defined by

$\mathcal{E}\big{(}s_{\mu}u_{s(\mu),g}s_{\nu}^{*}\big{)}=\delta_{\mu,\nu}\delta_{g,e_{G}}s_{\mu}s_{\mu}^{*}\hskip
14.22636pt(\mu,\nu\in\Lambda,~{}s(\mu)=g.s(\nu)).$

######  Definition 6.1.

An ideal $I$ of $\mathrm{EP}_{R}(G,\Lambda)$ is called diagonal-invariant
whenever $\mathcal{E}(I)\subseteq I$. Also, $I$ is said to be _basic_ if
$rs_{v}\in I$ implies $s_{v}\in I$ for all $v\in\Lambda^{0}$ and $r\in
R\setminus\\{0\\}$.

######  Definition 6.2.

Let $(G,\Lambda)$ be a self-similar $k$-graph. A subset
$H\subseteq\Lambda^{0}$ is called

  1. (1)

_$G$ -hereditary_ if $r(\mu)\in H~{}~{}\Longrightarrow~{}~{}g.s(\mu)\in H$ for
all $g\in G$ and $\mu\in\Lambda$;

  2. (2)

_$G$ -saturated_ if $v\in\Lambda^{0}$ and $s(v\Lambda^{n})\subseteq
H~{}~{}\mathrm{for~{}some~{}}n\in\mathbb{N}^{k}~{}~{}\Longrightarrow~{}~{}v\in
H$.

In the following, given any $H\subseteq\Lambda^{0}$, we denote by $I_{H}$ the
ideal of $\mathrm{EP}_{R}(G,\Lambda)$ generated by $\\{s_{v}:v\in H\\}$. Also,
for each ideal $I$ of $\mathrm{EP}_{R}(G,\Lambda)$, we define
$H_{I}:=\\{v\in\Lambda^{0}:s_{v}\in I\\}$.

To prove Theorem 6.8 we need some Lemmas.

######  Lemma 6.3.

If $I$ is an ideal of $\mathrm{EP}_{R}(G,\Lambda)$, then
$H_{I}:=\\{v\in\Lambda^{0}:s_{v}\in I\\}$ is a $G$-saturated $G$-hereditary
subset of $\Lambda^{0}$.

###### Proof.

The proof is straightforward. ∎

######  Lemma 6.4.

Let $H$ be a $G$-saturated $G$-hereditary subset of $\Lambda^{0}$ and $I_{H}$
the ideal of $\mathrm{EP}_{R}(G,\Lambda)$ generated by $\\{s_{v}:v\in H\\}$.
Then we have

(6.1) $I_{H}=\mathrm{span}_{R}\left\\{s_{\mu}u_{s(\mu),g}s_{\nu}^{*}:g\in
G,~{}s(\mu)=g.s(\nu)\in H\right\\},$

and $I_{H}$ is a $\mathbb{Z}^{k}$-graded diagonal-invariant ideal.

###### Proof.

Denote by $J$ the right-hand side of (6.1). The fact

$s_{\mu}u_{s(\mu),g}s_{\nu}^{*}=s_{\mu}(s_{s(\mu)})u_{s(\mu),g}s_{\nu}^{*},$

yields $J\subseteq I_{H}$. Also, using the description of
$\mathrm{EP}_{R}(G,\Lambda)$ in Proposition 2.7, it is straightforward to
check that $J$ is an ideal of $\mathrm{EP}_{R}(G,\Lambda)$. So, by
$s_{v}=s_{v}u_{v,e_{G}}s_{v}^{*}$, $J$ contains all generators of $I_{H}$, and
we have proved (6.1).

Now, (6.1) says that $I_{H}$ is spanned by homogenous elements, hence it is a
graded ideal. Moreover, let
$a=\sum_{i=1}^{l}s_{\mu_{i}}u_{s(\mu_{i}),g_{i}}s_{\nu_{i}}^{*}\in I_{H}$ such
that $s(\mu_{i})=g.s(\nu_{i})\in H$. Then, in particular, each term of $a$
with $g_{i}=e_{G}$ belongs to $I_{H}$. Therefore, $\mathcal{E}(a)\in I_{H}$,
and $I_{H}$ is diagonal-invariant. ∎

Let $H$ be a $G$-saturated $G$-hereditary subset of $\Lambda^{0}$ and consider
the $k$-subgraph $\Lambda\setminus\Lambda H$. Then the restricted action
$G\curvearrowright\Lambda\setminus\Lambda H$ is well-defined, and hence
$(G,\Lambda\setminus\Lambda H,\varphi|_{G\times\Lambda\setminus\Lambda H})$ is
also a self-similar $k$-graph. So we have:

######  Lemma 6.5.

Let $(G,\Lambda)$ be a pseudo free self-similar $k$-graph. If $H$ is a
$G$-saturated $G$-hereditary subset of $\Lambda^{0}$ then
$(G,\Lambda\setminus\Lambda H)$ is a pseudo free self-similar $k$-graph.

###### Proof.

The proof is straightforward. ∎

######  Lemma 6.6.

Let $H$ be a $G$-saturated $G$-hereditary subset of $\Lambda^{0}$. For
$v\in\Lambda^{0}$ and $0\neq r\in R$, then $v\in H$ if and only if $rs_{v}\in
I_{H}$. In particular, $I_{H}$ is a basic ideal.

###### Proof.

We follow the proof of [1, Lemma 5.4] with some modifications. If $v\in H$,
then $s_{v}\in I_{H}$ and $rs_{v}\in I_{H}$. To see the converse, suppose
$v_{0}\in\Lambda^{0}\setminus H$. Let $\\{t_{\mu},w_{v,g}\\}$ be the
generators of $\mathrm{EP}_{R}(G,\Lambda\setminus\Lambda H)$. If we define

$S_{\mu}:=\left\\{\begin{array}[]{ll}t_{\mu}&s(\mu)\notin H\\\
0&\mathrm{oth.}\end{array}\right.\hskip 14.22636pt\mathrm{and}\hskip
14.22636ptU_{v,g}:=\left\\{\begin{array}[]{ll}w_{v,g}&v\notin H\\\
0&\mathrm{oth.}\end{array}\right.,$

then $\\{S_{\mu},U_{v,g}\\}$ is a $(G,\Lambda)$-family in
$\mathrm{EP}_{R}(G,\Lambda\setminus\Lambda H)$, and by the universality, there
is a $*$-homomorphism
$\psi:\mathrm{EP}_{R}(G,\Lambda)\rightarrow\mathrm{EP}_{R}(G,\Lambda\setminus\Lambda
H)$ such that $\psi(s_{\mu})=S_{\mu}$ and $\psi(u_{v,g})=U_{v,g}$ for all
$\mu\in\Lambda$, $v\in\Lambda^{0}$ and $g\in G$. Since $\psi(s_{v})=0$ for
$v\in H$, we have $I_{H}\subseteq\ker\psi$. On the other hand, Theorem 5.5
implies that all $\psi(rs_{v})=rt_{v}$ are nonzero for
$v\in\Lambda^{0}\setminus H$ and $r\in R\setminus\\{0\\}$. In particular,
$rs_{v_{0}}\notin\ker\psi\supseteq I_{H}$, as desired. ∎

######  Proposition 6.7.

Let $H$ be a $G$-saturated $G$-hereditary subset of $\Lambda^{0}$. Let
$\\{t_{\mu},w_{v,g}\\}$ be the $(G,\Lambda\setminus\Lambda H)$-family
generating $\mathrm{EP}_{R}(G,\Lambda\setminus\Lambda H)$. Then the map
$\psi:\mathrm{EP}_{R}(G,\Lambda\setminus\Lambda
H)\rightarrow\mathrm{EP}_{R}(G,\Lambda)/I_{H}$ defined by

$\psi(t_{\mu}w_{s(\mu),g}t_{\nu}^{*}):=s_{\mu}u_{s(\mu),g}s_{\nu}^{*}+I_{H}\hskip
14.22636pt(\mu,\nu\in\Lambda\setminus\Lambda H,~{}g\in G)$

is an ($R$-algebra) $*$-isomorphism.

###### Proof.

If we set $T_{\mu}:=s_{\mu}+I_{H}$ and $W_{v,g}:=u_{v,g}+I_{H}$ for every
$v\in\Lambda^{0},~{}\mu\in\Lambda$, and $g\in G$, then $\\{T_{\mu},W_{v,g}\\}$
is a $(G,\Lambda\setminus\Lambda H)$-family in
$\mathrm{EP}_{R}(G,\Lambda)/I_{H}$ (the relations of Definition 2.5 for
$\\{T_{\mu},W_{v,g}\\}$ immediately follow from those for
$\\{s_{\mu},u_{v,g}\\}$). So, the universality of
$\mathrm{EP}_{R}(G,\Lambda\setminus\Lambda H)$ gives such $*$-homomorphism
$\psi$. Note that $s_{\mu}\in I_{H}$ for each $\mu\in\Lambda H$ by (6.1). This
follows the surjectivity of $\psi$.

To prove the injectivity, we apply the graded uniqueness theorem, Theorem 4.2.
First, since $I_{H}$ is a $\mathbb{Z}^{k}$-graded ideal,
$\mathrm{EP}_{R}(G,\Lambda)/I_{H}$ has a natural $\mathbb{Z}^{k}$-grading and
$\psi$ is a graded homomorphism. Thus, we fix an element in
$\mathrm{EP}_{R}(G,\Lambda\setminus\Lambda H)$ of the form
$a=\sum_{i=1}^{l}r_{i}w_{v,g_{i}}$ such that $v\in\Lambda^{0}\setminus H$ and
$g_{i}^{-1}.v=g_{j}^{-1}.v$ for all $1\leq i,j\leq l$. Without loss of
generality, we may also suppose that $g_{i}$’s are distinct. If $\psi(a)=0$,
then $\psi(a)=\sum_{i=1}^{l}r_{i}u_{v,g_{i}}+I_{H}=I_{H}$ and
$\sum_{i=1}^{l}r_{i}u_{v,g_{i}}\in I_{H}$. Thus, for each $1\leq j\leq l$, we
have

$\big{(}\sum_{i=1}^{l}r_{i}u_{v,g_{i}}\big{)}u_{g_{j}^{-1}.v,g_{j}^{-1}}=\sum_{i=1}^{l}r_{i}u_{v,g_{i}g_{j}^{-1}}\in
I_{H}\hskip
14.22636pt(\mathrm{by~{}eq.~{}(5)~{}in~{}Definition~{}\ref{defn2.4}})$

and since $I_{H}$ is diagonal-invariant,

$r_{j}s_{v}=r_{j}u_{v,e_{G}}=\mathcal{E}(\sum_{i=1}^{l}r_{i}u_{v,g_{i}g_{j}^{-1}})\in
I_{H}.$

As $v\notin H$, Lemma 6.6 forces $r_{j}=0$ for each $1\leq j\leq l$, hence
$a=0$. Now Theorem 4.2 concludes that $\psi$ is an isomorphism. ∎

######  Theorem 6.8.

Let $(G,\Lambda)$ be pseudo-free self-similar $k$-graph. Then $H\mapsto I_{H}$
is a one-to-one correspondence between $G$-saturated $G$-hereditary subsets of
$\Lambda^{0}$ and basic, $\mathbb{Z}^{k}$-graded and diagonal-invariant ideals
of $\mathrm{EP}_{R}(G,\Lambda)$ with the inverse $I\mapsto H_{I}$.

###### Proof.

The injectivity of $H\mapsto I_{H}$ follows from Lemma 6.6. Indeed, if
$I_{H}=I_{K}$ for $G$-saturated $G$-hereditary subsets
$H,K\subseteq\Lambda^{0}$, then Lemma 6.6 yields that
$H=H_{I_{H}}=H_{I_{K}}=K$.

To see the surjectivity, we take a basic, $\mathbb{Z}^{k}$-graded and
diagonal-invariant ideal $I$ of $\mathrm{EP}_{R}(G,\Lambda)$, and then prove
$I=I_{H_{I}}$. Write $J:=I_{H_{I}}$ for convenience. By Proposition 6.7 we may
consider $\mathrm{EP}_{R}(G,\Lambda\setminus\Lambda
H_{I})\cong\mathrm{EP}_{R}(G,\Lambda)/J$ as a $*$-$R$-algebra. Let
$\\{s_{\mu},u_{v,g}\\}$ and $\\{t_{\mu},w_{v,g}\\}$ be the generators of
$\mathrm{EP}_{R}(G,\Lambda)$ and $\mathrm{EP}_{R}(G,\Lambda)/J$, respectively.
Since $J\subseteq I$, we may define the quotient map
$q:\mathrm{EP}_{R}(G,\Lambda)/J\rightarrow\mathrm{EP}_{R}(G,\Lambda)/I$ such
that

$q(t_{\mu})=s_{\mu}+I\hskip 14.22636pt\mathrm{and}\hskip
14.22636ptq(w_{v,g})=u_{v,g}+I$

for all $\mu\in\Lambda$, $v\in\Lambda^{0}$ and $g\in G$. Notice that $q$
preserves the grading because $I$ is a $\mathbb{Z}^{k}$-graded ideal. So, we
can apply Theorem 4.2 to show that $q$ is an isomorphism. To do this, fix an
element of the form $a=\sum_{i=1}^{l}r_{i}w_{v,g_{i}}$ with
$v\in\Lambda^{0}\setminus H_{I}$ and same $g_{i}^{-1}.v$’s such that $q(a)=0$.
Then $\sum_{i=1}^{l}r_{i}u_{v,g_{i}}\in I$. As before, we assume also that
$g_{i}$’s are distinct. Thus, for each $1\leq j\leq l$, we have

$b_{j}:=\big{(}\sum_{i=1}^{l}r_{i}u_{v,g_{i}}\big{)}\big{(}u_{g_{j}^{-1}.v,g_{j}^{-1}}\big{)}=\sum_{i=1}^{l}r_{i}u_{v,g_{i}g_{j}^{-1}}\in
I.$

Since $I$ is diagonal invariant and basic, in case $r_{j}\neq 0$, we have
$r_{j}s_{v}=r_{j}u_{v,e_{G}}=\mathcal{E}(b_{j})\in I$, thus $s_{v}\in I$ and
$v\in H_{I}$, which contradicts the choice of $v$. This forces $r_{j}$’s to be
all zero, hence $a=0$. Now Theorem 4.2 implies that $q$ is injective,
concluding $I=J=I_{H_{I}}$ as desired. ∎

In the end, we remark the following about
$Q(\mathbb{N}^{k},G)\rtimes_{\mathcal{T}}\mathbb{Z}^{k}$-graded ideals of
$\mathrm{EP}_{R}(G,\Lambda)$.

######  Remark 6.9.

Let $(G,\Lambda)$ be a pseudo free self-similar $k$-graph and
$\mathcal{G}_{G,\Lambda}$ be the associated groupoid. Denote
$\Gamma:=Q(\mathbb{N}^{k},G)\rtimes_{\mathcal{T}}\mathbb{Z}^{k}$ the group
introduced in Section 5. If we define
$c:\mathcal{G}_{G,\Lambda}\rightarrow\Gamma$ by $c(x;\gamma;y):=\gamma$, then
$c$ is a cocycle on $\mathcal{G}_{G,\Lambda}$ because
$c(\alpha\beta)=c(\alpha)*_{\Gamma}c(\beta)$ for all
$\alpha,\beta\in\mathcal{G}_{G,\Lambda}$ with $s(\alpha)=r(\beta)$. Hence, it
induces a $\Gamma$-grading on
$A_{R}(\mathcal{G}_{G,\Lambda})=\mathrm{EP}_{R}(G,\Lambda)$ with the
homogenous components

$A_{\gamma}:=\mathrm{span}_{R}\\{1_{V}:V\subseteq
c^{-1}(\gamma)\mathrm{~{}is~{}a~{}compact~{}open~{}bisection~{}}\\}$

(see [5, Propositin 5.1] for example). By a similar argument as in [5, §6.5]
and combining Theorem 5.5 and [5, Theorem 5.3], we may obtain that the ideals
of the form $I_{H}$, described in Theorem 6.8 above, are precisely basic,
$\Gamma$-graded ideals of $\mathrm{EP}_{R}(G,\Lambda)$.

## References

  * [1] G. Aranda Pino, J. Clark, A. an Huef, and I. Raeburn, _Kumjian-Pask algebras of higher rank graphs_ , Trans. Amer. Math. Soc. 365 (2013), 3613-3641.
  * [2] E. Bédos, S. Kaliszewski, and J. Quigg, _On Exel-Pardo algebras_ , J. Operator Theory 78(2) (2017), 309-345.
  * [3] N. Brownlowe, J. Ramagge, D. Roberston, and M.F. Whittaker, _Zappa-Szép products of semigroups and their $C^{*}$-algebras_. J. Funct. Anal. 266 (2014), 3937-3967.
  * [4] L.O. Clark and C. Edie-Michell, _Uniqueness theorems for Steinberg algebras_ , Algebra Represent. Theory 18 (2015), 907-916.
  * [5] L.O. Clark, R. Exel, and E. Pardo, _A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras_ , Forum Math. 30(3) (2018), 533-552.
  * [6] R. Exel, E. Pardo, _Self-similar graphs, a unified treatment of Katsura and Nekrashevych $C^{*}$-algebras_, Adv. Math. 306 (2017), 1046-1129.
  * [7] R. Exel, E. Pardo, and C. Starling, _$C^{*}$ -algebras of self-similar graphs over arbitrary graphs_, preprint, arXiv:1807.01686 (2018).
  * [8] R. Hazrat, D. Pask, A. Sierakowski, and A. Sims, _An algebraic analogue of Exel-Pardo $C^{*}$-algebras_, Algebra Represent. Theory 24(4) (2021), 877-909.
  * [9] H. Larki, _A dichotomy for simple self-similar graph $C^{*}$-algebras_, J. Math. Anal. Appl. 494(2) (2021), 124622.
  * [10] T. Katsura, _A construction of actions on Kirchberg algebras which induce given actions on their $K$-groups_, J. Reine Angew. Math. 617 (2008), 27-65.
  * [11] A. Kumjian and D. Pask, _Higher rank graph $C^{*}$-algebras_, New York J. Math. 6 (2000), 1-20.
  * [12] X. Li, _semigroup $C^{*}$-algebras and amenability of semigroups_, J. Funct. Anal. 262 (2012), 4302-4340.
  * [13] H. Li and D. Yang, _Boundary quotient $C^{*}$-algebras of products of odometers_, Canad. J. Math. 71(1) (2019), 183-212.
  * [14] H. Li and D. Yang, _Self-similar $k$-graph $C^{*}$-algebras_, Int. Math. Res. Not. 15 (2021), 11270-11305.
  * [15] H. Li and D. Yang, _The ideal structures of self-similar $k$-graph $C^{*}$-algebras_, Ergodic Theory Dynam. Systems 41(8) (2021), 2480-2507.
  * [16] V. Nekrashevych, _Cuntz-Pimsner algebras of group actions_ , J. Operator Theory 52 (2004), 223-249.
  * [17] V. Nekrashevych, Self-Similar Groups, Mathematical Surveys and Monographs, vol. 117, Amer. Math. Soc., Providence RI 2005.
  * [18] J. Renault, A groupoid approach to $C^{\ast}$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980.
  * [19] B. Steinberg. _A groupoid approach to discrete inverse semigroup algebras_. Adv. Math. 223(2) (2010), 689-727.