# A study of the hydrostatic mass bias dependence and evolution within The
Three Hundred clusters

Giulia Gianfagna,1,2 Elena Rasia,3,4 Weiguang Cui,5,6 Marco De Petris,2
Gustavo Yepes,6,7 Ana Contreras-Santos,6 and Alexander Knebe,6,7,8
1INAF, Istituto di Astrofisica e Planetologia Spaziali, via Fosso del
Cavaliere 100, 00133 Rome, Italy
2Dipartimento di Fisica, Sapienza Universitá di Roma, Piazzale Aldo Moro,
5-00185 Roma, Italy
3IFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014
Trieste, Italy
4INAF Osservatorio Astronomico di Trieste, via Tiepolo 11, I-34131, Trieste,
Italy
5Institute for Astronomy, University of Edinburgh, Royal Observatory,
Edinburgh EH9 3HJ, UK
6Departamento deFísica Teórica, Módulo 15, Facultad de Ciencias, Universidad
Autónoma de Madrid, E-28049 Madrid, Spain
7Centro de Investigación Avanzada en Física Fundamental (CIAFF), Facultad de
Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain
8International Centre for Radio Astronomy Research, University of Western
Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia
E-mail:

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We use a set of about 300 simulated clusters from The Three Hundred Project to
calculate their hydrostatic masses and evaluate the associated bias by
comparing them with the true cluster mass. Over a redshift range from 0.07 to
1.3, we study the dependence of the hydrostatic bias on redshift,
concentration, mass growth, dynamical state, mass, and halo shapes. We find
almost no correlation between the bias and any of these parameters. However,
there is a clear evidence that the scatter of the mass-bias distribution is
larger for low-concentrated objects, high mass growth, and more generically
for disturbed systems. Moreover, we carefully study the evolution of the bias
of twelve clusters throughout a major-merger event. We find that the
hydrostatic-mass bias follows a particular evolution track along the merger
process: to an initial significant increase of the bias recorded at the begin
of merger, a constant plateaus follows until the end of merge, when there is a
dramatic decrease in the bias before the cluster finally become relaxed again.
This large variation of the bias is in agreement with the large scatter of the
hydrostatic bias for dynamical disturbed clusters. These objects should be
avoided in cosmological studies because their exact relaxation phase is
difficult to predict, hence their mass bias cannot be trivially accounted for.

###### keywords:

methods: numerical – galaxies: clusters: general – galaxies: clusters:
intracluster medium – large-scale structure of Universe.

††pubyear: 2021††pagerange: A study of the hydrostatic mass bias dependence
and evolution within The Three Hundred clusters–A study of the hydrostatic
mass bias dependence and evolution within The Three Hundred clusters

##  1 Introduction

Galaxy clusters are the most massive gravitationally-bound structures in the
Universe and, as such, their mass plays a key role in the estimation of the
cosmological parameters. The majority of their mass is composed by Dark Matter
(DM, almost 80%), the remaining part is composed by stars (galaxies) and hot
gas which is diffused between the galaxies and it is called Intra-Cluster
Medium, ICM (see Kravtsov & Borgani, 2012, for a review).

The mass of these objects in observations can be estimated in several ways:
from galaxy kinematic, applying the virial theorem to the distribution of
cluster galaxies (for example Li et al., 2021), or directly from the velocity
distribution of cluster galaxies (Zwicky, 1937; Pratt et al., 2019; Tian et
al., 2021; Herná ndez-Lang et al., 2022); from scaling laws, linking their
total mass to several observable quantities, like X-rays luminosity or SZ
(Sunyaev-Zeldovich) brightness (Nagarajan et al., 2018; Bulbul et al., 2019);
from weak lensing (Von der Linden et al., 2014; Hoekstra et al., 2015; Okabe &
Smith, 2016; Artis et al., 2022) and, recently, also combining the mentioned
methods and using machine learning (Ntampaka et al., 2015; De Andres et al.,
2022). Another frequently used approach is to derive the cluster mass under
the assumption of the hydrostatic equilibrium (HE). This method uses
temperature, density and pressure profiles of hot gas extracted from X-rays
alone or combined with SZ effect observations (see Pratt et al. 2019 for a
recent review). Three assumptions are the basis of the HE: the ICM must trace
the cluster potential well, it must be spherically symmetric, and its pressure
must be purely thermal. As shown in a recent review by Gianfagna et al.
(2021), the HE mass evaluated in simulated samples is on average
underestimating the true mass. In this paper we investigate more this subject
and in particular the dependence of the bias by several other parameters that
characterise the cluster dynamical state.

Numerical simulation studies find that the bias has a negligible dependence
with the redshift (Piffaretti & Valdarnini, 2008; Le Brun et al., 2017; Henson
et al., 2017; Gianfagna et al., 2021). On the contrary, observational data are
consistent with a decreasing mass bias along the redshift, see for instance
Salvati et al. (2019) and Wicker et al. (2022) on Planck clusters. This
dependence could be linked to a mass dependence of the bias, as clusters
observed at higher redshifts tend to have a higher mass. However, the mass
dependence is not detected in simulations, either using the spectroscopic-like
weighted temperature111Note that spectroscopic-like weighted temperature is
more similar to the X-ray temperature obtained from X-ray spectroscopic
analysis of observed data. (Mazzotta et al., 2004) to estimate the HE mass
(Piffaretti & Valdarnini, 2008; Le Brun et al., 2017; Henson et al., 2017;
Pearce et al., 2019; Ansarifard et al., 2020; Barnes et al., 2021) or the
mass-weighted temperature. Several authors also study the dependence of the
bias on the dynamical state, the general findings also pointed towards no
correlation, even if there is a hint that disturbed objects have a large
scatter with respect to the relaxed ones (Piffaretti & Valdarnini, 2008; Rasia
et al., 2012; Nelson et al., 2014; Henson et al., 2017; Ansarifard et al.,
2020). By comparing results in literature it appears that the hydrostatic mass
bias does not depend on various cosmological-simulation ingredients, such as
the baryon physics included in the simulations, the mass particles resolution
or the number of objects in the samples (Gianfagna et al., 2021). Due to this
consistency for simulated results, the discrepancy on the bias dependence on
redshift between simulation predictions and observation results is still
puzzling. This could be due to less numerous observational samples, due to a
different mass sampling at high redshift and due to the uncertainty in the
total physical mass estimation in observation.

The selection may also play a key role, for example the inclusion of disturbed
objects. Indeed, all of the HE assumptions are violated during major mergers
events which can induce strong dynamical perturbations, adding non-thermal
support to the equilibrium budget (Nelson et al., 2012; Rasia et al., 2012,
2013; Angelinelli et al., 2020; Sereno et al., 2021). Therefore, the
hydrostatic bias can significantly deviate from the expectation. In post-
merger situations, then the merging object might have already been destroyed
and thus it cannot be distinguished as a separate structure. Applying the
hydrostatic-equilibrium technique to these cases (assuming they are in a
relaxed state) will lead to a misinterpretation. Studying the extreme cases in
simulations, where the dynamical state is clearly identified, will allow us to
better understand the situations when the HE assumption is strongly violated
and how the HE mass bias is impacted by mergers.

The organisation of this paper is as follow: in Section 2 we introduce The
Three Hundred set of galaxy clusters, in Section 3 we give an overview on the
hydrostatic equilibrium method. In Section 4 the ICM radial profiles are
introduced, along with their analysis, and we introduce the parameters which
will be used to characterise the dynamical state, the accretion and the
geometry of clusters. In Section 5 we provide our the results, and we conclude
in Section 6.

##  2 The Three Hundred Project

This work is based on clusters simulated for The Three Hundred Project,
introduced in Cui et al. (2018). This project is based on the resimulations of
a set of 324 Lagrangian regions extracted from the MultiDark Planck (MDPL2)
dark-matter-only (DM) cosmological simulation (Klypin et al., 2016) with
different simulation codes. MDPL2 consists of a periodic cube of comoving
length $1h^{-1}$ Gpc and $3840^{3}$ N-body particles. The Lagrangian regions
are identified at $z=0$ around the most massive objects found in the parent
simulation with a comoving radius of $15h^{-1}$ Mpc. The particles in those
massive clusters and in their surrounding were traced back to $z=120$ to
generate the initial conditions for the zoomed-in re-simulations. The original
mass resolution was kept in the central region, and those particles were split
into dark matter and gas particles based on the cosmological baryon fraction
$\Omega_{b}$. The high-resolution dark-matter-particle mass is $1.87\times
10^{9}\rm M_{\odot}$, while the initial gas mass is $3.48\times 10^{8}\rm
M_{\odot}$. In the outer region, the dark matter particles were degraded with
multiple levels of mass refinements in several concentric shells. The 324
zoomed-in regions have been hydro-dynamically re-simulated with three
different baryon models: GADGET-X (Rasia et al., 2015), GADGET-MUSIC
(Sembolini et al., 2013) and GIZMO-Simba (Davé et al., 2019; Cui et al.,
2022). In this work we use the GADGET-X simulated clusters. This code includes
several radiative processes, like gas cooling, star formation, thermal
feedback from supernovae, chemical evolution and enrichment, super massive
black holes with AGN feedback (see Cui et al., 2018, for details).

The simulations assume a standard cosmological model according to the Planck
Collaboration et al. (2016) results: $h=0.6777$ for the reduced Hubble
parameter, $n=0.96$ for the primordial spectral index, $\sigma_{8}=0.8228$ for
the amplitude of the mass density fluctuations in a sphere of $8h^{-1}$ Mpc
comoving radius, $\Omega_{\Lambda}=0.692885$, $\Omega_{m}=0.307115$, and
$\Omega_{b}=0.048206$ respectively for the density parameters of dark energy,
dark matter, and baryonic matter.

###  2.1 Sample

For the analysis presented here, we select the most massive cluster for each
region that does not include any low-resolution particles within its virial
radius. The evolution of the bias is tested at 9 redshifts, covering the
redshift range between $z=0.07$ and $z=1.32$. The mean mass and the number of
objects considered at each redshift are reported in Table 1 for the three
studied overdensities: $\Delta=$ 2500, 500 and 200. These indicate the radius
$R_{\rm\Delta}$ of a sphere whose density is either 2500, 500 or 200 times the
critical density of the Universe at the consider cosmic time:

$\rho_{\rm crit}=(3/8\pi G)H_{0}^{2}[\Omega_{M}(1+z)^{3}+\Omega_{\Lambda}],$
(1)

where $H_{0}$ is the Hubble constant and G the gravitational constant.

Table 1: The redshifts analysed in this work are written in the first column. The number of objects are in the second column. In the third, fourth and fifth columns are the mean masses at the three nominal overdensities 2500, 500, 200 with units of $10^{14}M_{\odot}$. $z$ | $N$ | $<M_{2500}>$ | $<M_{500}>$ | $<M_{200}>$
---|---|---|---|---
|  | $\rm[10^{14}M_{\odot}]$ | $\rm[10^{14}M_{\odot}]$ | $\rm[10^{14}M_{\odot}]$
1.32 | 277 | 0.47 | 1.36 | 1.99
1.22 | 281 | 0.55 | 1.58 | 2.31
0.99 | 304 | 0.78 | 2.26 | 3.29
0.78 | 305 | 1.07 | 3.04 | 4.45
0.59 | 305 | 1.48 | 4.05 | 5.91
0.46 | 300 | 1.73 | 4.89 | 7.22
0.33 | 297 | 2.09 | 5.73 | 8.46
0.22 | 298 | 2.43 | 6.73 | 9.97
0.07 | 290 | 2.98 | 8.19 | 12.20

##  3 The hydrostatic equilibrium mass

As said in the introduction, the hydrostatic equilibrium hypothesis is often
at the basis of the procedure that lead to estimate the mass of galaxy
clusters from X-ray observations (Kravtsov & Borgani, 2012; Ettori et al.,
2013). This assumption foresees that the gas thermal pressure, which naturally
leads to expansion of the gas, is balanced by the gravitational forces, which,
on the other side, causes the systems to collapse. The equilibrium is
expressed as the equality between the gradient of the thermal pressure and
that of the gravitational potential and it is supposed to hold at every
concentric distance from the cluster centre. Connecting the gravitational
potential to the mass, under the spherical assumption, leads to the following
formula for the total mass inside a sphere of radius $r$:

$M_{\rm HE,SZ}(<r)=-\frac{r^{2}}{G\rho_{\rm g}(r)}\frac{{\rm d}P_{\rm
th}(r)}{{\rm d}\ r}$ (2)

where $\rho_{\rm g}$ and $P_{\rm th}$ are the density and the thermal pressure
of the gas, respectively. We will refer to this mass as $M_{\rm HE,SZ}$, since
the radial pressure profile can be derived from Compton-$y$ parameter maps
provided by clusters SZ observations at millimetre wavelengths.

Assuming the equation of state of an ideal gas, the cluster mass can also be
estimated with gas density and temperature separately:

$M_{\rm HE,X}(<r)=-\frac{rk_{\rm B}T(r)}{G\mu m_{\rm p}}\left[\frac{{\rm
d}\ln\rho_{\rm g}(r)}{{\rm d}\ln r}+\frac{{\rm d}\ln T(r)}{{\rm d}\ln
r}\right].$ (3)

We refer to this HE mass formulation as $M_{\rm HE,X}$ because historically
this expression was used in the analysis of X-rays observations (Ansarifard et
al., 2020). The detailed computation of the HE mass in our simulated sample
will be introduced in next section.

This mass is compared with the true total mass of the systems, obtained by
summing over all dark matter, stars and gas particle masses inside a fixed
aperture radius. The mass bias $b_{\rm SZ}$ or $b_{\rm X}$is defined as

$b=\frac{M_{\rm true}-M_{\rm HE}}{M_{\rm true}}.$ (4)

This bias can be either positive, when the HE mass is underestimating the true
cluster mass, or negative, when there is an overestimation of the true mass. A
null bias results in a perfect estimate of the true mass through HE.

##  4 Methods

In this work, we aim to estimate the two hydrostatic masses, as presented in
Eqs. (2) and (3). The ICM temperature, gas density and pressure radial
profiles play a fundamental role in the calculation of these masses. In this
section, we describe how we compute them in The Three Hundred simulations
together with other relevant quantities which we use to study the correlations
with the bias. The quantities linked to the study of the mass bias during a
major merger event will be described in Section 5.2.

###  4.1 The ICM profiles

The cluster profiles are extracted in logarithmically equal-spaced radial
bins, centred at the maximum density (mostly coinciding with the minimum of
the potential well, see also Cui et al. 2016; Ansarifard et al. 2020). Only
gas particles with density below the star-formation density threshold and with
temperature above 0.3 keV are considered as ICM for calculation, this is to
insure that we are selecting the hot particles which can generate X-ray or SZ
signals, in order to match the observations.

Once the profiles are extracted, we fit them with analytic models which will
be described below. We then use the analytical best-fitting curves to estimate
the hydrostatic masses. Following this strategy will allow to avoid the small
fluctuations that are presented in the 3D numerical profiles which strongly
impact the calculation of the derivatives in the hydrostatic-equilibrium mass
equations (see Gianfagna et al., 2021). On average, this still allow to well
fit the large fluctuations (bumps or deeps in the profiles), caused by
substructures, which should instead taken into account. This is valid
especially for the temperature and density models, that have the largest
number of parameters. The pressure profile model could instead cancel some
large fluctuation, due to the smaller number of parameters, leading to a non-
reliable SZ bias, but the two biases are compatible between each other (see
Section 5) and also the scatter is not significantly different. The best-fit
procedure of each profile is performed in the radial range [0.2-3]$\rm
R_{500}$. This range was chosen to include the three studied overdensities:
$\Delta=$ 2500, 500 and 200. We study the goodness of the fits using the
$\chi^{2}$ test and found that all the fits can be classified as good fits.

####  4.1.1 Gas Density

The 3D gas density profiles are estimated as the total gas mass in a spherical
shell, divided by the shell volume: $\rho=\sum m_{\rm i}/V_{\rm shell}$. The
model chosen to fit this profile is the one proposed by Vikhlinin et al.
(2006):

$\rho_{g}(r)=\rho_{0}^{2}\frac{(r/r_{\rm d})^{-a}}{(1+(r/r_{\rm
d})^{2})^{3b-a/2}}\frac{1}{(1+(r/r_{\rm s})^{c})^{e/c}}+$ (5) $\hskip
28.45274pt+\frac{\rho_{02}^{2}}{(1+(r/r_{\rm d2})^{2})^{3b_{2}}},$

where we fix the parameter $c$ equal to 3 and we impose the following
limitation: $e<5$. The other 8 parameters are left free. Often in literature
this model is simplified by discarding the second beta model, here we keep it
to have a reliable fit also in the cluster region near $\rm R_{2500}$. For
reference, we report in Table 2 the medians of all best-fit parameters
computed from the sample at $z=0.07$.

Table 2:  The medians and $16^{\rm th}-84^{\rm th}$ percentiles of best-fit free parameters in Eq.5 for gas density profiles and the reduced $\chi^{2}$. The results only for all $z=0.07$ clusters are exhibited here. The $c$ parameter is fixed to 3. $\rho_{0}$[$10^{13}\rm M_{\odot}/Mpc^{3}$] |  $r_{d}$[Mpc] |  $r_{s}$[Mpc] | $a$
---|---|---|---
$3.5^{+1.5}_{-1.3}$ | $0.71^{+0.20}_{-0.17}$ | $0.73^{+0.65}_{-0.29}$ | $1.2^{+1.2}_{-0.9}$
$b$ | $e$ |  $\rho_{0,2}$[$10^{13}\rm M_{\odot}/Mpc^{3}$] |  $r_{d,2}$[Mpc]
$2.6^{+0.4}_{-0.7}$ | $2.5^{+0.4}_{-0.6}$ | $2.4^{+0.6}_{-0.8}$ | $0.95^{+0.36}_{-0.28}$
$b_{2}$ | $\chi^{2}$ |  |
$1.2^{+0.3}_{-0.2}$ | $1.1^{+0.8}_{-0.4}$ |  |

####  4.1.2 Gas Temperature

The mass-weighted temperature profile is estimated as a weighted average over
the gas particles in the same spherical shells used for the gas density:

$T=\frac{\sum_{i}(m_{i}T_{i})}{\sum_{i}m_{i}}.$ (6)

where $m_{i}$ and $T_{i}$ are the mass and temperature of $i^{th}$ gas
particle. Using the mass-weighted temperature is the optimal choice for
hydrostatic equilibrium studies which theoretically connect the temperature
with the gravitational mass (Biffi et al., 2014) and it is favorable with
respect to the spectroscopic-like temperature (Mazzotta et al., 2004). The
analytical model for the mass-weighted temperature used in this work was
introduced again by Vikhlinin et al. (2006):

$T(r)=T_{0}\ \frac{x+\tau}{x+1}\ \frac{(r/r_{\rm t})^{-\alpha}}{(1+(r/r_{\rm
t})^{\beta})^{\gamma/\beta}},$ (7) $x=(r/r_{\rm cool})^{\alpha_{\rm cool}}.$

All the 8 parameters are free to vary. Their medians for the $z=0.07$ clusters
are reported in Table 3.

Table 3: The medians and $16^{\rm th}$ and $84^{\rm th}$ percentiles of best-fit free parameters of Eq.7 and the reduced $\chi^{2}$ computed for all $z=0.07$ clusters. $T_{0}$[keV] | $\tau$ |  $r_{t}$[Mpc] | $\alpha$ | $\beta$ | $\gamma$
---|---|---|---|---|---
$15.3^{+56.4}_{-10.3}$ | $0.37^{+0.32}_{-0.33}$ | $2.3^{+22.6}_{-1.29}$ | $0.29^{+0.71}_{-0.29}$ | $4.6^{+7.4}_{-4.6}$ | $1.3^{+2.2}_{-1.3}$
$r_{cool}$[Mpc] | $\alpha_{cool}$ | $\chi^{2}$ |  |  |
$1.4^{+1.5}_{-0.7}$ | $5.4^{+4.6}_{-2.8}$ | $1.4^{+0.4}_{-0.1}$ |  |  |

####  4.1.3 Gas Pressure

The radial thermal pressure profile is measured from the gas density and
temperature of each gas particle. It is modelled by the generalised Navarro-
Frenk-White (gNFW) model (Nagai et al., 2007):

$P(r)=\frac{P_{0}}{x^{i}(1+x^{g})^{\frac{h-i}{g}}},$ (8)

where $x=r/r_{s}$ is a dimensionless radial distance normalised to the scale
radius, $r_{s}$. The parameters $h$ and $i$ are the slopes for outer and inner
region, respectively, and $g$ is the steepness of the transition between the
two regimes. This model has 4 free parameters, since $i=0.31$ is fixed (Arnaud
et al., 2010). The medians of the best-fit parameters for the $z=0.07$ sample
are listed in Table 4.

Table 4: The medians and $16^{\rm th}$ and $84^{\rm th}$ percentiles of best-fit free parameters of Eq.8 and the reduced $\chi^{2}$ computed for all $z=0.07$ clusters. The $i$ parameter is fixed equal to $0.31$. $P_{0}[10^{-2}\rm keV/cm^{3}]$ |  $r_{s}$ [Mpc] | $g$ | $h$ | $\chi^{2}$
---|---|---|---|---
$2.6^{+5.5}_{-1.5}$ | $3.1^{+6.9}_{-2.5}$ | $1.1^{+1.3}_{-0.4}$ | $8.2^{+6.8}_{-4.0}$ | $1.1^{+0.8}_{-0.3}$

###  4.2 The NFW concentration

Similarly to the thermodynamical profiles, we compute the total density
profiles which include the contribution from all particles in the simulations
(dark matter, gas, stars). The total mass profile is fitted with a NFW model
(Navarro et al., 1997):

$M(<r)=M_{0}\left[\log(1+x)-\frac{x}{1+x}\right].$ (9)

As before the radial coordinate is related to the scale radius, $x=r/r_{s}$.
This parameter, together with the normalisation, is derived from a best-
fitting procedure applied in the radial range between 100 kpc and $R_{200}$,
which roughly reproduce a typical radial range used in weak-lensing analysis.
The concentration within R500 is defined from the scale radius:
$c_{500}=R_{500}/r_{s}$.

###  4.3 Relative mass growth

We straightforwardly define the cluster mass growth as the relative mass
difference between an initial, $t_{1}$, and a final time, $t_{2}$:

$\frac{\Delta M/M}{{\rm
d}t}=\frac{(M(t_{2})-M(t_{1}))/M(t_{1})}{[t_{2}-t_{1}]}.$ (10)

Specifically, we estimated the mass growth from the merger tree between
$z_{2}=0.33$ and $z_{1}=0.46$, so that $t_{2}-t_{1}$ corresponds to about 1
Gyr (precisely it is equal to $1.046$ Gyr). This interval corresponds to 5
simulation snapshots.

###  4.4 Dynamical state parameter

The dynamical state of the clusters has been inferred by the combination of
two indicators computed in 3D (Neto et al., 2007; Cui et al., 2017; Cialone et
al., 2018; De Luca et al., 2021):

  * •

$f_{s}=M_{\rm sub}/M_{\Delta}$, which is the fraction of cluster mass included
in substructures inside a fixed overdensity;

  * •

$\Delta_{\rm r}=|\textbf{r}_{\delta}-\textbf{r}_{\rm cm}|/R_{\Delta}$, which
is the offset between the positions of the maximum density peak and the centre
of mass of the cluster within $R_{\Delta}$ and normalised to that aperture
radius.

When correlating the dynamical-state and the HE mass bias, we evaluate all
quantities of interest at the same overdensity value. In order to have a
relaxed cluster, both indicators should be smaller than 0.1 (Cialone et al.,
2018) while they should be both greater than 0.1 to have a disturbed one. The
other cases are classified as intermediate or hybrid. The percentage of
relaxed clusters is almost 50% at low redshifts, and decreases to 30% at high
redshifts, while the percentage of disturbed clusters is 20% at low redshifts
and increases to 30% at high redshifts (De Luca et al., 2021). As previously
said, in our study we consider one combined parameter as in Haggar et al.
(2020) but derived exclusively from the two mentioned indicators as in De Luca
et al. (2021):

$\chi_{\rm DS}=\sqrt{\frac{2}{\left(\frac{\Delta_{\rm
r}}{0.1}\right)^{2}+\left(\frac{f_{s}}{0.1}\right)^{2}}}.$ (11)

Using the $\chi_{\rm DS}$ parameter is a continuous way to classify the state
of the cluster where the relaxed ones satisfy $\chi_{\rm DS}>1$. We use this
parameter to study the dependence of the dynamical state on the HE bias (see
Section 5.1.4). An extensive study of the relaxation state of The Three
Hundred clusters including its dependence from redshift is presented in De
Luca et al. (2021).

###  4.5 Triaxiality

In order to test how much the lack of spherical symmetry is affecting the HE
mass estimate, we calculate, using all particles within $R_{500}$, the three
axes of the inertia tensor as in (Vega-Ferrero et al., 2017). We then use the
ratio $c/a$ and the triaxiality of the halos:

$t=\frac{1-(b/a)^{2}}{1-(c/a)^{2}},$ (12)

where $a,b$ and $c$ indicate the major, intermediate, and minor axes,
respectively. Depending on the value of the parameter $t$, the halo is
considered as oblate if $0<t<1/3$; triaxial if $1/3<t<2/3$ and prolate if
$2/3<t<1$. More simply, when $c/a$ is equal to $1$ the cluster is spherical.

##  5 Results

In this section we present the results of our work. First, in Section 5.1, we
study the HE mass bias dependencies on the redshift, mass, and on all other
parameters: the concentration, the mass growth, the relaxation parameter and
the triaxiality. We performed all these analyses at the three overdensities
($\Delta=2500,500,200$) and at all considered redshifts. However, most of the
figures related to this part of the work (with the exception of the first) are
only presented with $\Delta=500$ and at one specific redshift because we do
not find a strong dependence of the results on either of these two quantities.

Secondly, in Section 5.2, we analyse how the HE mass bias varies throughout
strong merger events. In this case, we only analyse the behaviour of the bias
at $\rm R_{200}$, since we take as reference the mergers analysis in
Contreras-Santos et al. (2022), which is done considering the entire cluster
volume and thus at overdensity 200.

###  5.1 Bias correlations

####  5.1.1 Redshift dependence

The variation of $b_{SZ}$ and $b_{X}$ along the redshift range is presented in
Fig. 1 at the 3 overdensities and for the different cluster dynamical states
defined with the dynamical-state parameter, $\chi_{\rm DS}$. For all
overdensities, the un-relaxed clusters (purple shaded region) have the largest
scatter. Interestingly, this large scatter is mostly caused by the presence of
low-value bias. We will present the reason in Section 5.2. No dependence of
the bias on the redshift up to $z=1.25$ is detected in agreement with Le Brun
et al. 2017; Henson et al. 2017; Salvati et al. 2018; Koukoufilippas et al.
2020. Furthermore, we notice that the median $b_{2500}$ is very similar to the
median $b_{500}$ and that both are close to 0.1 (10% of bias) and as such
systematically lower than $b_{200}$ (close to $\sim 0.2$). This is in
agreement with Gianfagna et al. (2021), which showed a declining $b$ from
outer to inner radius. This is occurring despite the differences both in the
code (GADGET2 versus GADGET-X) and in the baryon models. Note, however, that
the SZ and X median $b_{500}$ in Gianfagna et al. (2021) is slightly larger
than this work. The biases dispersion at $R_{2500}$ seems to be slightly
larger with respect to the other overdensities, this is probably due to the
larger deviations in the cluster core properties, which could marginally
affect the profile at $R_{2500}$. In The Three Hundred, the simulation produce
a more diverse variety of simulated cores with respect to the GADGET2 code,
and in some situations, the simulated clusters are extremely peaked at the
centre (see Campitiello et al., 2022).

Figure 1: The redshift evolution of the biases, $b_{\rm SZ}$ (left panels) and
$b_{\rm X}$ (right panels). The median values of the bias for all clusters,
relaxed and un-relaxed clusters are represented with dark cyan crosses, green
diamonds and purple stars respectively. The shaded regions represent the
$16^{th}$ and $84^{th}$ percentiles. The bias estimated at $\rm R_{200}$, $\rm
R_{500}$ and $\rm R_{2500}$ are represented in the top, middle and bottom
panels respectively. The dashed lines show the 0 and 0.2 bias for reference.

####  5.1.2 Concentration

The concentration parameter $c_{500}$ of a halo is representative of the
halo’s central density. In presence of a disturbed system the concentration is
typically lower since the X-ray peak might have been destroyed by a merger
event which also could have brought more mass in the external regions. For
this it might be interesting to see if there is any correlation between the HE
mass bias and the NFW concentration parameter.

The bias as a function of the concentration is represented in Fig. 2 where the
relaxed, disturbed and hybrid clusters are introduced with different symbols
and colours. Hybrid clusters are these with either $f_{s}<0.1$ or
$\Delta_{r}<0.1$. The two quantities do not show any dependence, as clear from
the trend of the median value shown with a black line. We notice however that
the scatter in the bias substantially decreases going towards larger
concentration values. This is quantified by the half difference between the
$16^{\rm th}$ and the $84^{\rm th}$ percentiles, $d2=0.5\times(p_{84}-p_{16})$
reported in Table 5 together with the bias percentiles. The value of $d2$
decreases from low (third row) to high concentrated clusters (first row) by
70-80%. This trend – reduced scatter with increasing concentration – is
expected because most of the clusters with higher concentration are relaxed.

Figure 2: The bias dependence on the concentration at $\rm R_{500}$ and at $z=0.07$, $b_{\rm SZ}$ is represented in the left panel, $b_{\rm X}$ in the right one. The black line represents the median bias value, with the shaded regions as 16th and 84th percentiles. The relaxed clusters are shown in green diamonds, the disturbed in purple stars and the hybrids in blue dots. Table 5: Table of the biases for the 50 clusters with the highest (first row) and lowest (third row) NFW concentration and the remaining clusters (second row) for the $z=0.59$ 0 sample. We report the median value ($m$), the $16^{\rm th}$ and $84^{\rm th}$ percentiles ($p_{16}$ and $p_{84}$), and their half difference ($d_{2}$) at $R_{500}$.  $c_{500}$ | $b_{SZ}$ |  |  |  | $b_{X}$ |  |  |
---|---|---|---|---|---|---|---|---
| $m$ | $p_{16}$ | $p_{84}$ | $d_{2}$ | m | $p_{16}$ | $p_{84}$ | $d_{2}$
high | 0.11 | 0.07 | 0.19 | 0.06 | 0.12 | 0.05 | 0.24 | 0.10
med | 0.11 | 0.00 | 0.18 | 0.09 | 0.13 | 0.00 | 0.24 | 0.12
low | 0.06 | -0.03 | 0.22 | 0.12 | 0.11 | -0.07 | 0.24 | 0.15

####  5.1.3 Relative Mass growth

The dependence of the bias at $z=0.333$ on the mass growth recorded in the
last Gyr (and specifically since $z=0.46$) is shown in the top panel of Fig.
3, where both quantities are estimated at $\rm R_{500}$. The relative median
values for high, low, and intermediate accreting clusters are listed in Table
6. The correlation is again very weak, the Pearson correlation coefficient is
-0.20 for $b_{\rm SZ}$ and -0.36 for $b_{\rm X}$. However, the systems with
the largest mass growth have also the largest dispersion. This is expected
because the systems with the largest mass growth are also the most disturbed
(purple stars in Fig. 3). This relation will be discussed more afterwards,
however, we point out that clusters with large mass accretion will also have
abundant non-thermal gas motions and, thus, a large non-thermal pressure
component. The same trend is seen for the quantities estimated at $\rm
R_{200}$, with a Pearson correlation coefficient of 0.16 for $b_{\rm SZ}$ and
0.21 for $b_{\rm X}$.

In the bottom panel of Fig. 3 the difference between the bias at $z_{2}=0.33$
and at $z_{1}=0.46$ is shown ($\Delta b=b(z_{2})-b(z_{1})$). The variation in
the bias for disturbed clusters (at large accretion rates) is negative,
implying that the HE mass bias overall decreases. This phenomenon which might
seem surprising will be better studied in Section 5.2, but we can already see
from the top panel the origin of this trend: after a major merger (with a mass
variation of at least 50% $\Delta M/M>0.5$) the mass computed by Eqs. 2 or 3
can be significantly larger than the true mass with a large scatter as well
(see also Nelson et al., 2012).

Table 6: Table of the biases values for the 50 clusters with the largest mass growth (first row), the 50 with the smallest (last row) and the remaining clusters (second row) for the $z=0.33$ sample. We report the median value ($m$), the $16^{\rm th}$ and $84^{\rm th}$ percentiles ($p_{16}$ and $p_{84}$), and their half difference ($d_{2}$) at R500.  Mass | $b_{SZ}$ |  |  |  | $b_{X}$ |  |  |
---|---|---|---|---|---|---|---|---
growth | m | $p_{16}$ | $p_{84}$ | d2 | m | $p_{16}$ | $p_{84}$ | d2
high | 0.09 | -0.05 | 0.17 | 0.11 | 0.09 | -0.12 | 0.20 | 0.16
med | 0.10 | -0.02 | 0.17 | 0.08 | 0.12 | 0.00 | 0.22 | 0.11
low | 0.14 | 0.06 | 0.22 | 0.08 | 0.24 | 0.10 | 0.35 | 0.13

Overall Fig.3 shows that the mass growth is a parameter that is capable to
distinguish between relaxed and disturbed clusters, as seen also in Fig. 4.
We, indeed, find a correlation betwee the mass growth parameter and $\chi_{\rm
DS}$ equal to -0.46 both at $\rm R_{500}$ (shown in the Figure) and at $\rm
R_{200}$. The good correlation is actually expected since one of the
indicators defining $\chi_{\rm DS}$ is the mass in substructures. That said
there is a certain level of contamination with the presence of disturbed
objects with reduced mass growth or viceversa relaxed objects with significant
mass growth. Notice, however, that both these peculiar cases have a HE bias
value which is quite consistent with the average expectation of the samples,
while extreme bias values such as those with $b<-0.2$ are only present for the
largest mass growth.

Figure 3: Top panel: the bias dependence on the mass growth $\frac{\Delta
M/M}{dt}$ is represented. $\frac{\Delta M/M}{dt}$ is estimated from $z=0.46$
to $z=0.33$, which corresponds to 1 Gyr, while the biases are estimated at
$z=0.33$. Bottom panel: the bias variation from $z=0.46$ to $z=0.33$ as a
function of $\frac{\Delta M/M}{dt}$. The relaxed and disturbed clusters are
represented with green diamonds and purple stars respectively, while the
hybrids with blue dots. The cluster dynamical states shown in this plot are
estimated at $z=0.33$.  Figure 4: The relaxation parameter at $z=0.33$ is
represented as a function of the mass growth from $z=0.46$ to $z=0.33$, all
estimated at $\rm R_{500}$. The dot-dashed line represents the threshold under
which a cluster is considered disturbed.

####  5.1.4 Relaxation parameter

The bias as a function of the relaxation parameter is represented in Fig. 5,
where, per definition, the coloured points of relaxed and disturbed clusters
are completely separated. Weak or no correlations is found, indeed the Pearson
correlation coefficients with the highest values are only equal to 0.13 for
$b_{\rm SZ}$ and 0.23 for $b_{\rm X}$ at $z=1.32$; and 0.01 for $b_{\rm SZ}$
and 0.01 for $b_{\rm X}$ at $z=0.07$. Also in this case, the disturbed
clusters show a wider dispersion with respect to the relaxed ones, see Table
7, where the difference between the bias percentiles grows by almost a factor
of 2 from the lowest to the largest $\chi_{\rm DS}$ clusters. This is in
agreement with Piffaretti & Valdarnini (2008); Rasia et al. (2012); Nelson et
al. (2014); Henson et al. (2017); Ansarifard et al. (2020).

Figure 5: The biases are represented as a function of the relaxation parameter $\chi_{\rm DS}$. The hybrids, relaxed and disturbed clusters are represented with blue dots, green diamonds and purple stars respectively. In the top panel we show the quantities at $z=0.07$ and in the bottom one at $z=1.3$. The lines are the results of a simple linear fitting result. The black line and the shaded region represent the median bias and the percentiles (16th and 84th). Table 7: Table of the bias values for the 50 highest (first row) and 50 lowest (third row) $\chi_{\rm DS}$ clusters and for the remaining clusters (second row) for the $z=0.59$ sample. We report the median value ($m$), the $16^{\rm th}$ and $84^{\rm th}$ percentiles ($p_{16}$ and $p_{84}$), and their half difference ($d_{2}$) at R500. $\chi_{\rm DS}$ | $b_{SZ}$ |  |  |  | $b_{X}$ |  |  |
---|---|---|---|---|---|---|---|---
| m | $p_{16}$ | $p_{84}$ | d2 | m | $p_{16}$ | $p_{84}$ | d2
high | 0.10 | 0.05 | 0.16 | 0.06 | 0.12 | 0.05 | 0.23 | 0.09
med | 0.11 | -0.02 | 0.19 | 0.09 | 0.14 | 0.01 | 0.25 | 0.12
low | 0.11 | -0.04 | 0.21 | 0.13 | 0.07 | -0.12 | 0.20 | 0.16

####  5.1.5 Total mass

We continue to investigate the correlation between the HE bias and the cluster
mass at $\rm R_{500}$ in Fig.6. Since we detect no variation on cosmic time
(see Fig. 1), in this plot, we simultaneously show the biases at four
redshifts (1.32, 0.78, 0.33, 0.07) to increase the halo mass range. We find no
dependence of the bias on the total mass of the clusters in agreement with
Piffaretti & Valdarnini (2008); Le Brun et al. (2017); Henson et al. (2017);
Pearce et al. (2019); Ansarifard et al. (2020); Barnes et al. (2021).

Figure 6: The biases are represented as a function of the cluster total mass,
both at $\rm R_{500}$. The redshifts 1.32, 0.78, 0.33, 0.07 are represented in
red, blue, magenta and green, respectively. The black line and the shaded
region represent the binned median and the 16th-84th percentiles.

####  5.1.6 Triaxiality

We study the dependence of the bias at $R_{500}$ on the sphericity of the
halos, one of the HE main assumptions, quantified with the parameter $t$ and
on $c/a$, the ratio of the minor and major axis. All the particles within
$R_{500}$ are used to estimate the halo shape (Vega-Ferrero et al., 2017).

The relation between the biases and $c/a$ is represented in Fig. 7, the
results for $t$ are not shown, being very similar. From that Figure we notice
that the overall correlation and the amplitude of the scatter seem to be
uncorrelated with the axial ratio. Although, if we restrict to the most
aspherical cases (third line in Table 8) we see that the both biases increases
by almost 50% with respect to the objects with the highest $c/a$ values. Note
that the halo shape estimated based on total mass is similar to the one based
on hot gas (see Velliscig et al., 2015, for details). Therefore, we believe a
similar result will be in place when the halo shape is estimated only from the
gas particles.

Figure 7: The biases at $\rm R_{500}$ are represented as a function of the axis ratio $c/a$. The hybrids, relaxed and disturbed are represented with blue dots, green diamonds and purple stars respectively. Both $b_{SZ}$ and $b_{X}$ are shown for $z=0.33$. The median value of the bias is represented with the black line, the 16th and 84th percentiles with the shaded region.  Table 8: Table of the biases values for the 50 clusters with the highest (first row) and lowest (last row) axial ratio and for the remaining clusters (second row) for the $z=0.46$ sample. We report the median value ($m$), the $16^{\rm th}$ and $84^{\rm th}$ percentiles ($p_{16}$ and $p_{84}$), and their half difference ($d_{2}$) at R500.  $c/a$ | $b_{SZ}$ |  |  |  | $b_{X}$ |  |  |
---|---|---|---|---|---|---|---|---
| m | $p_{16}$ | $p_{84}$ | $d_{2}$ | m | $p_{16}$ | $p_{84}$ | $d_{2}$
high | 0.09 | 0.01 | 0.18 | 0.07 | 0.11 | -0.00 | 0.18 | 0.09
med | 0.09 | 0.00 | 0.17 | 0.08 | 0.14 | -0.02 | 0.23 | 0.13
low | 0.12 | 0.02 | 0.24 | 0.11 | 0.20 | 0.07 | 0.32 | 0.12

###  5.2 HE mass bias and the merger history of the clusters

Unlike observations, hydrodynamical simulations allow for tracking the whole
history of a cluster, and thus to follow a merging event during all its
phases. Specifically in this subsection, we study major merger events that are
identified when a halo experiences a very rapid mass increase, primarily
caused by the accretion of a single massive object (Contreras-Santos et al.,
2022). Our main goal is to understand the evolution of the bias throughout
these violent merging processes.

To define and compute the relevant times to the merger event, we closely
follow the definitions in Contreras-Santos et al. (2022) to which we refer for
a more detailed explanations. In summary, instead of using absolute time as
the variable in the merger history, which can vary with redshift, we normalise
it to the halo dynamical time:

$t_{dyn}=\sqrt{\frac{3}{4\pi}\frac{1}{200G\rho_{\rm crit}}}.$ (13)

Since the critical density depends only on the cosmology and on the redshift,
see Eq. (1), the dynamical time does not depend on the cluster features, but
evolves only with redshift. As described before, we only consider the major
merger events defined as an event that produces a mass increase of 100% within
half of the dynamical time

$\frac{\Delta M}{M}=\frac{M_{f}-M_{i}}{M_{i}}\geq 1,$ (14)

where $M_{i}$ is the initial mass and $M_{f}$ is the final mass. Note that the
masses in this case refer to the overdensity of 200. We use this larger radius
for better capturing the evolution process inside the virial region. The
merger event can easily be characterised by 4 particular redshifts (see Figure
1 of Contreras-Santos et al. 2022):

  * •

$z_{\rm before}$: the last time before the merger begins. It is characterised
by a relaxation parameter with high values (typically larger than 1). Soon
after the two systems start to influence each other and the relaxation
parameter decreases;

  * •

$z_{\rm start}$: the merging halo enters in the main cluster $R_{200}$ and as
a consequence the mass of the latter starts to grow;

  * •

$z_{\rm end}$: the end point of the merger identified as the moment when the
mass growth stops and the $\chi_{\rm DS}$ starts to grow again implying that
the relaxation process is beginning.

  * •

$z_{\rm after}$: the end of the whole merger phase when the cluster approaches
a relaxed state by showing $\chi_{DS}$ closer or above $1$.

In this work, we analyse how the bias evolves with the major merger event and
how much time is needed for the bias to return to the average value in the
relaxed population. In our sample, we select 12 clusters which experience a
major merger, from $z_{\rm before}$ to $z_{\rm after}$, in the investigated
redshift range [0.07,1.32]. In the top panel of Fig. 8 the two stacked biases,
presented by mean bias values with $16^{th}-84^{th}$ percentiles as error
bars, are shown as a function of $\Delta t/t_{\rm dyn}$, which is defined as

$\frac{\Delta t}{t_{\rm dyn}}=\frac{t_{0+\textit{i}}-t_{0}}{t_{\rm dyn}}$ (15)

where $t_{0}$ corresponds to the analysed redshift right before $z_{\rm
before}$, and $t_{0+i}$ corresponds to the analysed redshifts following
$t_{0}$. The vertical grey shaded areas represent the $\pm 1\sigma$ regions of
the averaged time before, after, start and end of the merger. The yellow
shaded region shows the $\pm 1\sigma$ region of the SZ bias for the relaxed
clusters at $\rm R_{200}$, averaged over all the redshifts.

At $\Delta t/t_{\rm dyn}=0$ both biases are in agreement with the typical
relaxed clusters values. Right at the beginning of the merger phase ($z_{\rm
before}$), the biases increase to $\sim 0.25-0.3$ due to the incoming
substructure, that causes the true mass of the cluster to increase faster than
the HE mass, leading to a slight increase of the bias. The latter stays almost
constant until the end of the accretion of the secondary object. In this
situation the HE mass is always underestimating the true mass of the object. A
small increase of the bias is detected at around $z_{\rm end}$ followed by a
quick decrease that lasts until $z_{\rm after}$, where a minimum is reached.
Right at this time, around the end of the merger phase, the HE bias can even
reach negative values. The in-fall of the substructure generates shocks
propagating outward. These shocks cause a steep increase of the derivatives of
the pressure, gas density and temperature profiles (see the bottom panel
described later), leading to an increase of the HE mass, and a negative bias.
This trend is in agreement with Bennett & Sijacki (2022) and Nelson et al.
(2012), who conducted a similar study on simulated clusters. When the cluster
approaches a relaxation phase at $\sim z_{\rm after}$, the bias returns to a
value that is closer to the mean $b$ of relaxed clusters. This trend is again
in agreement with what seen in the FABLE simulated clusters (Bennett &
Sijacki, 2022).

In the middle panel of Fig. 8 we show the evolution of the relaxation
parameter during the merger process. The clusters are classified as relaxed at
$t_{0}$ by definition, and then as the secondary cluster approaches (before
$z_{\rm start}$) the $\chi_{\rm DS}$ drops. The relaxation parameter shows
some fluctuation between $z_{\rm start}$ and $z_{\rm end}$ to finally grow
again after the secondary objects is completely incorporated into the main
cluster. At the end of the merger phase ($z_{\rm after}$), per definition
$\chi_{\rm DS}$ is approaching 1. Notice that even after 4 dynamical times
from the beginning of the merger the main cluster does not reach the original
relaxation state, consistently with Contreras-Santos et al. (2022).

In the bottom panel, we shown the relative evolution of all quantities
entering the HE mass equations (both thermodynamic quantities and their
derivatives). To be consistent with the analysis shown in the upper panels,
all the values are computed at $R_{200}$. The pressure derivative is
represented with blue triangles, the temperature and its derivative are
represented as olive green dots and triangles, the gas density and its
derivative are represented with orange dots and triangles. The temperature
derivative does not show any particular trend and fluctuates around the
original values with large error bars. The temperature values instead grows
immediately as the secondary object approaches and keeps growing until the end
of the process, this is expected as it is a consequence of the shock heating
produced by the merger.

The density value does not show any particular change until the secondary
structure merge into the main halo after $z_{\rm start}$. The slope of the gas
density at $R_{200}$ also increases for the presence of the massive companion.
In both cases, the variation continues up to $z_{\rm end}$ before reaching a
plateau. Finally, the pressure derivative is on the mid way between the
temperature and the gas density derivatives. The increase in the derivative of
the gas density, pressure and temperature due to the shocks generated in the
merger or to the sloshing of the subclusters moving around the cluster core
drive the HE masses of Eqs. 2 and 3 to be closer or even above the true mass.

At $\rm R_{500}$ these considerations are still valid, the behaviour of all
the quantities is the same. To conclude, the bias shows a stable evolution
along the whole major merger event, albeit slightly difference between
$b_{SZ}$ and $b_{X}$. However, the dramatic change of the bias value,
especially in the period between $z_{\rm end}$ to $z_{\rm after}$ when the
cluster is identified as un-relaxed, is the origin of the large scatter of the
bias for the disturbed systems shown in all previous figures. For this reason,
it is not suggested to estimate the HE masses for these dynamical un-relaxed
clusters, unless we can correctly identify their states in the merging events
which can be very difficult in observation.

Figure 8: Top panel: the stacked biases at $\rm R_{200}$ (means and standard
deviations) are represented as a function of the difference in times from
$t_{0}$, divided by the dynamical time. The SZ bias is represented in green,
while the X one in orange. The yellow shaded region represents the biases
range of the relaxed clusters. The null bias is represented with a dashed
line. Central panel: the stacked relaxation parameter estimated inside $\rm
R_{200}$, the means and the standard deviations are represented as a function
of $\Delta t/t_{dyn}$. The dashed line represents the threshold between
relaxed and disturbed clusters. Bottom panel: each thermodynamic quantity
which takes part in the estimation of the HE mass (and bias) divided by their
value before the merger (indicated with a subscript 0) is shown as a function
of $\Delta t/t_{dyn}$. The dashed line shows the value of 1 (the quantity does
not change with respect to the age prior to the merger). The temperature and
its derivative are represented in olive green dots and triangles respectively.
Instead, the density and its derivative are represented as orange dots and
triangles, and the pressure derivative is represented with blue triangles. In
the plot the means and the standard deviations are represented. In all the
panels, the vertical grey shaded areas represent the $\pm 1\sigma$ regions of
the averaged time before, after, start and end of the merger.

##  6 Conclusions

Galaxy cluster mass plays a major role in cluster cosmology. Several different
methods are used to estimate it. In this paper we focus on the hydrostatic
equilibrium approximation, which makes use of the temperature, pressure and
density radial profiles of the ICM. These 3D profiles are computed from The
Three Hundred simulations, which include a set of almost 300 simulated galaxy
clusters that we studied at 10 different redshifts. From the profiles, we
recover the masses under the HE approximation and we estimate the bias against
the true cluster total mass. In this work the focus is specifically on the
connection between the hydrostatic mass bias and different cluster properties,
such as dynamical state, cluster total mass, mass-growth rate, NFW
concentration and axial ratio. Moreover, we follow the bias during 12 major-
merger events to understand the bias evolution along these events. The main
findings are as follows:

  * •

We do not find correlation between the biases, estimated at different radii,
and the redshift, in agreement with other simulations.

  * •

The bias and its scatter are influenced by the radii within which the bias is
estimated, i.e. $R_{2500},R_{500}$ and $R_{200}$ in this work. The largest
bias is at $R_{200}$ (almost 20%), while at $R_{500}$ and $R_{2500}$, the
median value is around 10%.

  * •

There is no correlation between the hydrostatic mass bias and the dynamical
state of the clusters as measured with different indicators: the dynamical-
state parameter $\chi_{\rm DS}$, the NFW concentration, the relative mass
growth, and the triaxiality. However, whenever one of these parameters is
typically associated to a disturbed objects (e.g. low $\chi_{\rm DS}$, low
concentration, high mass growth) we detect a bias scatter that can be almost
twice as that of the relaxed systems. The scatter has, instead, little
dependence on mass, redshift, and sphericity.

  * •

A moderate correlation is found between the relaxation parameter and the mass
growth. This is expected because the larger is the mass growth, the more
likely the cluster is disturbed.

  * •

By stacking clusters while experiencing major merger events we find that the
main object becomes more disturbed along the merger with a slightly higher
$b$, when the secondary structure collides. After $z_{\rm end}$ – the halo
mass growth stops and $b$ drops significantly because of the shocks
propagating outward generated by the in-falling substructure. These shocks
cause a steep increase of the derivatives of the pressure, gas density and
temperature profiles, even leading to an overestimation of the HE mass. At the
end of the merger phase when clusters get closer to a relaxed dynamical state
with $\chi\sim 1$, the bias assumes the typical values of relaxed clusters.

Although no correlation is found between the HE bias and the dynamical state
of the cluster at a given redshift, we find that selecting a sample with the
typical characteristics of regular objects, such as high NFW concentration,
low mass accretion rate, high $\chi_{\rm DS}$, and high $c/a$ ratio, leads to
a reduced cluster-to-cluster scatter. We also found a correlation between
various merger phases and the HE bias which can be close to 0 when the cluster
is at the end of merger process. This, however, occurs when the cluster is not
yet relaxed confirming that including yet-no relaxed clusters could largely
increase the bias scatter.

Our results are in agreement with other simulations, where a large bias is
found mostly generated by deviations from spherical symmetry (which is at the
base of HE), temperature dishomogeneities, but also the presence of
substructures or gas motions, generating non-thermal pressure component. In
the first part of the paper, we have proved that the HE bias is not dependent
on the latter, while in the last part of the paper we explore how the bias
changes during a merger. Here we show that the presence of a substructure, and
consequently a merger, can indeed deeply affect the cluster dynamical state
and the HE mass bias, leading even to an overestimation of the true mass.
During the merger events, see Fig. 8, the bias is almost always positive, i.e.
the HE mass is underestimating the true cluster mass. However, even with the
highest bias value during the merger events, we can only come close to the
bias suggested by Planck. Therefore, we need to look for other reasons for
explaining this bias tension. The next-generation satellites, like XRISM
(X-Ray Imaging and Spectroscopy Mission), hopefully Athena and the proposed
probe LEM (Line Emission Mapper), but also eROSITA-data analysis, are going to
give precise indication on the gas velocities and dispersion and, consequently
on cluster dynamical state.

## Acknowledgements

The authors would like to thank the Referee for the constructive and helpful
comments. The simulations used in this work have been performed in the
MareNostrum Supercomputer at the Barcelona Supercomputing Center, thanks to
CPU time granted by the Red Española de Supercomputación. WC is supported by
the STFC AGP Grant ST/V000594/1 and the Atracción de Talento Contract no.
2020-T1/TIC-19882 granted by the Comunidad de Madrid in Spain. He further
acknowledges the science research grants from the China Manned Space Project
with NO. CMS-CSST-2021-A01 and CMS-CSST-2021-B01. GY acknowledges financial
support from the MICIU/FEDER (Spain) under project grant PGC2018-094975-C21.
MDP acknowledges support from Sapienza Università di Roma thanks to Progetti
di Ricerca Medi 2019, RM11916B7540DD8D. AK is supported by the Ministerio de
Ciencia, Innovación y Universidades (MICIU/FEDER) under research grant
PGC2018-094975-C21 and further thanks Piero Umiliani for ‘svezia, inferno e
paradiso’.

## Data Availability

The data underlying this article were produced as part of The Three Hundred
Project (Cui et al., 2018). They will be shared on request to The Three
Hundred Collaboration, at https://www.the300-project.org.

## References

  * Angelinelli et al. (2020) Angelinelli M., Vazza F., Giocoli C., Ettori S., Jones T. W., Brunetti G., Brüggen M., Eckert D., 2020, MNRAS, 495, 864–885
  * Ansarifard et al. (2020) Ansarifard S., et al., 2020, A&A, 634, A113
  * Arnaud et al. (2010) Arnaud M., Pratt G. W., Piffaretti R., Böhringer H., Croston J. H., Pointecouteau E., 2010, A&A, 517, 1
  * Artis et al. (2022) Artis E., Melin J.-B., Bartlett J., Murray C., 2022, EPJ Web of Conferences, 257, 00004
  * Barnes et al. (2021) Barnes D. J., Vogelsberger M., Pearce F. A., Pop A.-R., Kannan R., Cao K., Kay S. T., Hernquist L., 2021, MNRAS, 506
  * Bennett & Sijacki (2022) Bennett J. S., Sijacki D., 2022, MNRAS, 514, 313
  * Biffi et al. (2014) Biffi V., Sembolini F., De Petris M., Valdarnini R., Yepes G., Gottlöber S., 2014, MNRAS, 439, 588
  * Bulbul et al. (2019) Bulbul E., et al., 2019, ApJ, 871, 50
  * Campitiello et al. (2022) Campitiello M. G., et al., 2022, A&A, 665, A117
  * Cialone et al. (2018) Cialone G., De Petris M., Sembolini F., Yepes G., Baldi A. S., Rasia E., 2018, MNRAS, 477, 139
  * Contreras-Santos et al. (2022) Contreras-Santos A., et al., 2022, MNRAS, 511, 2897
  * Cui et al. (2016) Cui W., et al., 2016, MNRAS, 456, 2566
  * Cui et al. (2017) Cui W., Power C., Borgani S., Knebe A., Lewis G. F., Murante G., Poole G. B., 2017, MNRAS, 464, 2502
  * Cui et al. (2018) Cui W., et al., 2018, MNRAS, 480, 2898
  * Cui et al. (2022) Cui W., et al., 2022, MNRAS, 514, 977
  * Davé et al. (2019) Davé R., Anglés-Alcázar D., Narayanan D., Li Q., Rafieferantsoa M. H., Appleby S., 2019, MNRAS, 486, 2827
  * De Andres et al. (2022) De Andres D., et al., 2022, EPJ Web of Conferences, 257, 00013
  * De Luca et al. (2021) De Luca F., De Petris M., Yepes G., Cui W., Knebe A., Rasia E., 2021, MNRAS, 504, 5383
  * Ettori et al. (2013) Ettori S., Donnarumma A., Pointecouteau E., Reiprich H. T., Giodini S., Lovisari L., Schmidt R. W., 2013, Space Sci Rev, 177, 119–154
  * Gianfagna et al. (2021) Gianfagna G., et al., 2021, MNRAS, 502, 5115
  * Haggar et al. (2020) Haggar R., Gray M. E., Pearce F. R., Knebe A., Cui W., Mostoghiu R., Yepes G., 2020, MNRAS, 492, 6074
  * Henson et al. (2017) Henson M. A., Barnes D. J., Kay S. T., McCarthy I. G., Schaye J., 2017, MNRAS, 465, 3361
  * Herná ndez-Lang et al. (2022) Herná ndez-Lang D., et al., 2022, MNRAS, 517, 4355
  * Hoekstra et al. (2015) Hoekstra H., Herbonnet R., Muzzin A., Babul A., Mahdavi A., Viola M., Cacciato M., 2015, MNRAS, 449, 685
  * Klypin et al. (2016) Klypin A., Yepes G., Gottlöber S., Prada F., Heß S., 2016, MNRAS, 457, 4340
  * Koukoufilippas et al. (2020) Koukoufilippas N., Alonso D., Bilicki M., Peacock J. A., 2020, MNRAS, 491, 5464
  * Kravtsov & Borgani (2012) Kravtsov A. V., Borgani S., 2012, ARA&A, 50, 353
  * Le Brun et al. (2017) Le Brun A. M. C., McCarthy I. G., Schaye J., Ponman T. J., 2017, MNRAS, 466, 4442
  * Li et al. (2021) Li Q., Han J., Wang W., Cui W., Li Z., Yang X., 2021, MNRAS, 505, 3907
  * Mazzotta et al. (2004) Mazzotta P., Rasia E., Moscardini L., Tormen G., 2004, MNRAS, 354, 10
  * Nagai et al. (2007) Nagai D., Kravtsov A. V., Vikhlinin A., 2007, ApJ, 668, 1
  * Nagarajan et al. (2018) Nagarajan A., et al., 2018, MNRAS, 488, 1728
  * Navarro et al. (1997) Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ, 490, 493
  * Nelson et al. (2012) Nelson K., Rudd D. H., Shaw L., Nagai D., 2012, ApJ, 751, 121
  * Nelson et al. (2014) Nelson K., Lau E. T., Nagai D., Rudd D. H., Yu L., 2014, ApJ, 782, 107
  * Neto et al. (2007) Neto A. F., et al., 2007, MNRAS, 381, 1450
  * Ntampaka et al. (2015) Ntampaka M., Trac H., Sutherland D. J., Battaglia N., Póczos B., Schneider J., 2015, ApJ, 803, 50
  * Okabe & Smith (2016) Okabe N., Smith G. P., 2016, MNRAS, 461, 3794
  * Pearce et al. (2019) Pearce F. A., Kay S. T., Barnes D. J., Bower R. G., Schaller M., 2019, MNRAS, 491, 1622
  * Piffaretti & Valdarnini (2008) Piffaretti R., Valdarnini R., 2008, A&A, 491, 71
  * Planck Collaboration et al. (2016) Planck Collaboration et al., 2016, A&A, 594, A13
  * Pratt et al. (2019) Pratt G. W., Arnaud M., Biviano A., Eckert D., Ettori S., Nagai D., Okabe N., Reiprich T. H., 2019, Space Sci Rev, 215
  * Rasia et al. (2012) Rasia E., et al., 2012, New Journal of Physics, 14, 055018
  * Rasia et al. (2013) Rasia E., Borgani S., Ettori S., Mazzotta P., Meneghetti M., 2013, ApJ, 776, 39
  * Rasia et al. (2015) Rasia E., et al., 2015, ApJ, 813, L17
  * Salvati et al. (2018) Salvati L., Douspis M., Aghanim N., 2018, A&A, 614, A13
  * Salvati et al. (2019) Salvati L., Douspis M., Ritz A., Aghanim N., Babul A., 2019, A&A, 626, A27
  * Sembolini et al. (2013) Sembolini F., Yepes G., De Petris M., Gottlöber S., Lamagna L., Comis B., 2013, MNRAS, 429, 323
  * Sereno et al. (2021) Sereno M., Lovisari L., Cui W., Schellenberger G., 2021, MNRAS, 507, 5214–5223
  * Tian et al. (2021) Tian Y., Cheng H., McGaugh S. S., Ko C.-M., Hsu Y.-H., 2021, ApJ Letters, 917, L24
  * Vega-Ferrero et al. (2017) Vega-Ferrero J., Yepes G., Gottlöber S., 2017, MNRAS, 467, 3226
  * Velliscig et al. (2015) Velliscig M., et al., 2015, MNRAS, 453, 721
  * Vikhlinin et al. (2006) Vikhlinin A., Kravtsov A., Forman W., Jones C., Markevitch M., Murray S. S., Speybroeck L. V., 2006, ApJ, 640, 691
  * Von der Linden et al. (2014) Von der Linden A., et al., 2014, MNRAS, 443, 1973–1978
  * Wicker et al. (2022) Wicker R., Douspis M., Salvati L., Aghanim N., 2022, EPJ Web of Conferences, 257, 00046
  * Zwicky (1937) Zwicky F., 1937, ApJ, 86, 217


# A criterion for hypersymmetry on discrete groupoids

F. Flores  F. Flores 111 2010 Mathematics Subject Classification: Primary
43A20, Secondary 47L65, 47L30.
Key Words: Groupoid, symmetric Banach algebra, Fell Bundle, C* algebra.

###### Abstract

Given a Fell bundle $\mathscr{C}\overset{q}{\to}\Xi$ over the discrete
groupoid $\Xi$, we study the symmetry of the associated Hahn algebra
$\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$ in terms of the isotropy subgroups
of $\Xi$. We prove that $\Xi$ is symmetric (resp. hypersymmetric) if and only
if all of the isotropy subgroups are symmetric (resp. hypersymmetric). We also
characterize hypersymmetry using Fell bundles with constant fibers, showing
that for discrete groupoids, ’hypersymmetry’ equals ’rigid symmetry’.

##  1 Introduction

This article treats the symmetry of certain Banach ∗-algebras connected with
Fell bundles over discrete groupoids. The study of symmetry for groupoid
algebras started not long ago, with Austad and Ortega [1], and found a
continuation in [2], where Jauré, Măntoiu and myself first treated the problem
of symmetry for Fell bundles over discrete groupoids.

######  Definition 1.1.

A Banach ∗-algebra $\mathfrak{B}$ is called symmetric if the spectrum of
$b^{*}b$ is positive for every $b\in\mathfrak{B}$ (this happens if and only if
the spectrum of any self-adjoint element is real).

In this regard, the main result of the paper is the following.

######  Theorem 1.2.

Let $\Xi$ be a discrete groupoid. Then:

  1. 1.

The algebra $\ell^{\infty,1}(\Xi)$ is symmetric if and only if every isotropy
group is symmetric.

  2. 2.

The algebra $\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$ is symmetric for every
Fell bundle $\mathscr{C}$ over $\Xi$ if and only if every isotropy group is
hypersymmetric.

This result is interesting as it reduces the question of the (hyper)symmetry
of a given groupoid to the (hyper)symmetry of its isotropy subgroups and the
study of group $\ell^{1}$-algebras is far more developed. For example, see [1,
2, 4, 10, 13, 9, 5, 8] and the reference therein.

The article is divided in 3 parts: Section 2 deals with preliminares; there we
introduce Fell bundles, the Hahn algebra of a Fell bundle and define what
(hyper)symmetry for a groupoid means. In Section 3 we introduce a
characterization for hypersymmetry using only Fell bundles with constant
fibers. Its analogous to the characterization made for groups by Jauré and
Măntoiu in [4]. Finally, Section 4 deals with the proof of Theorem 1.2. This
is achieved by writing a general discrete groupoid as a disjoint union of
transitive groupoids and proving that transitive groupoids are isomorphic to
transitive transformation groups. Using the available theory for groups yields
the desired result.

##  2 Preliminaries

Let $\Xi$ be a groupoid, with unit space $\mathcal{U}:=\Xi^{(0)}$ , source map
${\rm d}$ and range map ${\rm r}$ . The ${\rm d}$ and ${\rm r}$-fibers are
$\Xi_{u}=\\{\xi\in\Xi\mid{\rm d}(\xi)=u\\}$ and $\Xi^{u}=\\{\xi\in\Xi\mid{\rm
r}(\xi)=u\\}$. The set of composable pairs is

$\Xi^{(2)}\\!:=\\{(x,y)\\!\mid\\!{\rm r}(y)={\rm d}(x)\\}\,.$

The isotropy group associated with the unit $u\in\mathcal{U}$ is
$\Xi_{u}^{u}=\Xi_{u}\cap\Xi^{u}$, it is a subgroupoid of $\Xi$, which happens
to be a group. We endow the groupoid $\Xi$ with the discrete topology. Let us
introduce an important class of groupoids which will be useful.

######  Definition 2.1.

A groupoid $\Xi$ is called transitive if for any pair $u,v\in\mathcal{U}$,
there exists $x\in\Xi$ such that ${\rm r}(x)=u$ and ${\rm d}(x)=v$.

The concept of a transitivity is borrowed from the theory of dynamical
systems. And it comes from a natural but hidden action of the groupoid on its
unit space. So the groupoid is transitive if and only if this action is
transitive, cf [3, Example 2.2, Corollary 7.8].

In this article we are going to work with Fell bundles
$\mathscr{C}\overset{q}{\to}\Xi$ over the groupoid ${\Xi}$ (see [6, 11]). A
Fell bundle is composed of fibers and it satisfies that each fiber
$\mathfrak{C}_{x}\\!:=q^{-1}(\\{x\\})$ is a Banach space with norm
$\parallel\\!\cdot\\!\parallel_{\mathfrak{C}_{x}}$ , the topology of
$\mathscr{C}$ coincides with the norm topology on each fiber, there are
antilinear continuous involutions

$\mathfrak{C}_{x}\ni\\!a\to a^{\bullet}\\!\in\mathfrak{C}_{x^{-1}}$

and for all $(x,y)\in\Xi^{(2)}$ there are continuous multiplications

$\mathfrak{C}_{x}\\!\times\\!\mathfrak{C}_{y}\ni(a,b)\to a\bullet
b\in\mathfrak{C}_{xy}$

satisfying the following axioms valid for
$a\in\mathfrak{C}_{x}\,,b\in\mathfrak{C}_{y}\,$ and $(x,y)\in\Xi^{(2)}$ :

  * •

$\parallel\\!ab\\!\parallel_{\mathfrak{C}_{xy}}\,\leq\,\parallel\\!a\\!\parallel_{\mathfrak{C}_{x}}\parallel\\!b\\!\parallel_{\mathfrak{C}_{y}}$
,

  * •

$(ab)^{\bullet}=b^{\bullet}a^{\bullet}$,

  * •

$\parallel\\!a^{\bullet}a\\!\parallel_{\mathfrak{C}_{{\rm
d}(x)}}=\,\parallel\\!a\\!\parallel_{\mathfrak{C}_{x}}^{2}$ ,

  * •

$a^{\bullet}a$ is positive in $\mathfrak{C}_{{\rm d}(x)}$ .

From these axioms it follows that $\mathfrak{C}_{x}$ is a $C^{*}$-algebra for
every unit $x\in\mathcal{U}$ . Sometimes we simply write
$\mathscr{C}=\bigsqcup_{x\in\Xi}\mathfrak{C}_{x}$ for the Fell bundle.

Our object of study is the Hahn algebra
$\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$ adapted to Fell bundles [11],
which in our case it is formed by the sections $\Phi:\Xi\to\mathfrak{C}$ (thus
satisfying $\Phi(x)\in\mathfrak{C}_{x}$ for every $x\in\Xi$) that can be
obtained as a limit of finitely-supported sections in the Hahn-type norm

$\lVert\Phi\rVert_{\infty,1}\,:=\max\Big{\\{}\sup_{u\in\mathcal{U}}\sum_{{\rm
r}(x)=u}\\!\parallel\\!\Phi(x)\\!\parallel_{\mathfrak{C}_{x}}\,,\,\sup_{u\in\mathcal{U}}\sum_{{\rm
d}(x)=u}\\!\parallel\\!\Phi(x)\\!\parallel_{\mathfrak{C}_{x}}\\!\Big{\\}}.$
(2.1)

It is a Banach ∗-algebra under the multiplication

$(\Phi*\Psi)(x):=\sum_{yz=x}\Phi(y)\bullet\Psi\big{(}z)$ (2.2)

and the involution

$\Phi^{*}(x):=\Phi(x^{-1})^{\bullet}.$ (2.3)

######  Remark 2.2.

Let us point out some of the nature of the functions in
$\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$. If
$\Phi_{n}\in\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$ is a sequence of
sections with finite support and $\Phi_{n}\to\Phi$, then the convergence is
uniform. Indeed, let $x\in\Xi$ and observe that

$\displaystyle\lVert\Phi_{n}(x)-\Phi(x)\rVert_{\mathfrak{C}_{x}}$
$\displaystyle\leq\sum_{y\in\Xi^{r(x)}}\lVert\Phi_{n}(y)-\Phi(y)\rVert_{\mathfrak{C}_{y}}$
$\displaystyle\leq\sup_{u\in\mathcal{U}}\sum_{y\in\Xi^{u}}\lVert\Phi_{n}(y)-\Phi(y)\rVert_{\mathfrak{C}_{y}}$
$\displaystyle\leq\lVert\Phi_{n}-\Phi\rVert_{\ell^{\infty,1}(\Xi\,\mid\,\mathscr{C})}.$

So
$\lim_{n}\sup_{x\in\Xi}\lVert\Phi_{n}(x)-\Phi(x)\rVert_{\mathfrak{C}_{x}}=0$.
This implies that the function $\Phi$ has countable support and vanishes at
$\infty$.

We denote by $C^{*}(\Xi\,|\,\mathscr{C})$ the enveloping $C^{*}$-algebra of
the Hahn algebra $\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$. It is a known
fact that $\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$ is a dense ∗-subalgebra
of $C^{*}(\Xi\\!\mid\\!\mathscr{C})$.

######  Definition 2.3.

  1. (i)

The discrete groupoid $\Xi$ is called symmetric if the convolution Banach
∗-algebra $\ell^{\infty,1}(\Xi)$ is symmetric.

  2. (ii)

The discrete groupoid $\Xi$ is called hypersymmetric if given any Fell bundle
$\mathscr{C}\\!=\bigsqcup_{x\in\Xi}\mathfrak{C}_{x}$ , the Banach ∗-algebra
$\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$ is symmetric.

######  Example 2.4.

If $\Pi\subset X\\!\times\\!X$ is an equivalence relation on $X$, one can make
$\Xi=\Pi$ a discrete groupoid by defining the operations

$\begin{split}{\rm d}(x,y)=(y,y)\,,\quad&{\rm
r}(x,y)=(x,x)\,,\quad(x,y)(y,z)=(x,z)\,,\quad(x,y)^{-1}\\!=(y,x)\,.\end{split}$

The unit space is $\,\mathcal{U}={\sf Diag}(X)$ and it gets canonically
identified with $X$, via the homeomorphism $(x,x)\mapsto x$ . In this case all
of the isotropy groups correspond to the trivial group
$\Pi_{u}^{u}=\\{(u,u)\\}$, so Theorem 4.9 will guarantee that $\Pi$ is
hypersymmetric. A particular examples is the so called pair groupoid,
$\Pi=X\times X$.

##  3 A characterization of hypersymmetry for discrete groupoids

In [2] we introduced some special Fell bundles arising from Hilbert bundles to
characterize the hypersymmetry of a discrete groupoid. However, in this paper
we will improve this characterization: One can actually verify hypersymmetry
by looking at much simpler algebras, associated to Fell bundles with constant
fibers. Let us make precise the Fell bundles of interest.

######  Definition 3.1.

By a (left) groupoid action of a discrete groupoid $\Xi$ on the $C^{*}$-bundle
$\mathscr{A}\\!:=\bigsqcup_{u\in\mathcal{U}}\mathfrak{A}_{u}\overset{p}{\to}\mathcal{U}$
over its unit space we understand a continuous map

$\mathscr{A}\rtimes\Xi:=\\{(\alpha,x)\in\mathscr{A}\\!\times\\!\Xi\\!\mid\\!p(\alpha)={\rm
d}(x)\\}\ni(\alpha,x)\to\mathcal{T}(\alpha,x)\equiv\mathcal{T}_{x}(\alpha)\in\mathscr{A},$

satifying the axioms

  1. (a)

$p\big{[}\mathcal{T}_{x}(\alpha)\big{]}={\rm r}(x)$ ,
$\forall\,x\in\Xi\,,\,\alpha\in\mathfrak{A}_{{\rm d}(x)}$ ,

  2. (b)

each $\mathcal{T}_{x}$ is a ∗-isomorphism $:\mathfrak{A}_{{\rm
d}(x)}\\!\to\mathfrak{A}_{{\rm r}(x)}$ ,

  3. (c)

$\mathcal{T}_{u}={\rm id}_{\mathfrak{A}_{u}}$ , $\forall\,u\in\mathcal{U}$ ,

  4. (d)

if $(x,y)\in\Xi^{(2)}$ and $(\alpha,y)\in\mathscr{A}\rtimes\Xi$ , then
$\big{(}\mathcal{T}_{y}(\alpha),x\big{)}\in\mathscr{A}\rtimes\Xi$ and
$\mathcal{T}_{xy}(\alpha)=\mathcal{T}_{x}\big{[}\mathcal{T}_{y}(\alpha)\big{]}$
.

######  Definition 3.2.

Let $\mathcal{T}$ be a groupoid action of $\Xi$ on the $C^{*}$-bundle
$\mathscr{A}\\!:=\bigsqcup_{u\in\mathcal{U}}\mathfrak{A}_{u}\overset{p}{\to}\mathcal{U}$.
We define its associated Fell bundle as follows. The underlying space is
$\mathscr{C}_{\mathcal{T}}:=\mathscr{A}\rtimes\Xi$ with the topology inherited
from the product topology and the obvious projection $q$. We endow it with the
operations

$(\alpha,x)\bullet(\beta,y):=(\alpha\mathcal{T}_{x}(\beta),xy)\,,\textup{when
}(x,y)\in\Xi^{(2)}$

and

$(\alpha,x)^{\bullet}:=(\mathcal{T}_{x^{-1}}(\alpha^{*}),x^{-1})$

to get a Fell bundle over $\Xi$ . A section is now a map
$\Phi:\Xi\to\mathscr{C}_{\mathcal{T}}$ such that

$\Phi(x)\equiv\big{(}\varphi(x),x\big{)}\in\mathfrak{C}_{x}=\mathfrak{A}_{{{\rm
r}}(x)}\\!\times\\!\\{x\\}\,,\quad\forall\,x\in\Xi\,.$

So we may identify every section
$\Phi\in\ell^{\infty,1}(\Xi\,|\,\mathscr{C}_{\mathcal{T}})$ with a function
$\Phi:\Xi\to\mathscr{A}$ (note the abuse of notation), such that
$\Phi(x)\in\mathfrak{A}_{{{\rm r}}(x)}$.

We will also denote the algebra
$\ell^{\infty,1}(\Xi\,|\,\mathscr{C}_{\mathcal{T}})$ by
$\ell^{\infty,1}_{\mathcal{T}}(\Xi,\mathscr{A})$ to recall its particular
nature. If the action $\mathcal{T}$ is trivial, meaning that
$\mathfrak{A}_{u}\equiv\mathfrak{A}$ for all $u\in\mathcal{U}$ and
$\mathcal{T}_{x}\equiv{\rm id}_{\mathfrak{A}}$ for all $x\in\Xi$, then we
denote the resulting algebra simply by $\ell^{\infty,1}(\Xi,\mathfrak{A})$.

######  Remark 3.3.

Let $\Phi,\Psi\in\ell^{\infty,1}_{\mathcal{T}}(\Xi,\mathscr{A})$. In this
case, one may write the algebraic laws as

$\big{[}\Phi*\Psi\big{]}(x)=\\!\sum_{y\in\Xi^{{\rm
r}(x)}}\\!\Phi(y)\mathcal{T}_{y}\big{[}\Psi(y^{-1}x)\big{]}\,,$ (3.1)
$\Phi^{*}(x)=\mathcal{T}_{x}\big{[}\Phi(x^{-1})\big{]}^{*}.$ (3.2)

And the Hahn-type norm as

$\parallel\\!\Phi\\!\parallel_{\ell^{\infty,1}_{\mathcal{T}}(\Xi,\mathscr{A})}=\max\Big{\\{}\sup_{u\in\mathcal{U}}\sum_{{\rm
r}(x)=u}\\!\parallel\\!\Phi(x)\\!\parallel_{\mathfrak{A}_{{\rm
r}(x)}}\,,\,\sup_{u\in\mathcal{U}}\sum_{{\rm
d}(x)=u}\\!\parallel\\!\Phi(x)\\!\parallel_{\mathfrak{A}_{{\rm
r}(x)}}\\!\Big{\\}}\,.$ (3.3)

######  Lemma 3.4.

Let $\mathscr{C}\\!=\bigsqcup_{x\in\Xi}\mathfrak{C}_{x}$ be a Fell bundle over
the discrete groupoid $\Xi$. Then there exists and an isometric ∗-monomorphism

$\varphi:\ell^{\infty,1}(\Xi\,|\,\mathscr{C})\to\ell^{\infty,1}\big{(}\Xi,C^{*}(\Xi\,|\,\mathscr{C})\big{)}.$
(3.4)

###### Proof.

Set $\mathfrak{A}:=C^{*}(\Xi\,|\,\mathscr{C})$ and, for every $x\in\Xi$ we
embed $\mathfrak{C}_{x}$ into
$\ell^{\infty,1}(\Xi\,|\,\mathscr{C})\subset\mathfrak{A}$, by setting for each
$a\in\mathfrak{C}_{x}$

$(\theta_{x}a)(y):=a\ \ {\rm if}\ \
y=x\,,\quad(\theta_{x}a)(y):=0_{\mathfrak{C}_{x}}\ \ {\rm if}\ \ y\neq x\,.$

It is not hard to prove that if $(x,y)\in\Xi^{(2)}$, then

$\theta_{x}a*\theta_{y}b=\theta_{xy}(a\bullet b)\quad\textup{ and
}\quad(\theta_{x}a)^{*}=\theta_{x^{-1}}a^{\bullet}$

hold. On the other hand, one also has
$\lVert\theta_{x}a\rVert_{\mathfrak{A}}=\lVert a\rVert_{\mathfrak{C}_{x}}$: If
$x\in\mathcal{U}$, this equality holds because
$\theta_{x}:\mathfrak{C}_{x}\to\mathfrak{A}$ is a ∗-monomorphism of
$C^{*}$-algebras, hence isometric. If $x\in\Xi$ is not an unit, then one may
apply the -now standard- trick

$\lVert\theta_{x}a\rVert^{2}_{\mathfrak{A}}=\lVert(\theta_{x}a)^{*}*(\theta_{x}a)\rVert_{\mathfrak{A}}=\lVert\theta_{x^{-1}x}(a^{\bullet}\bullet
a)\rVert_{\mathfrak{A}}=\lVert a^{\bullet}\bullet
a\rVert_{\mathfrak{C}_{x^{-1}x}}=\lVert a\rVert^{2}_{\mathfrak{C}_{x}}$

to conclude that $\theta_{x}$ preserves the mentioned norms. This allows us to
successfully define

$\varphi(\Phi)(x)=\theta_{x}\Phi(x),\textup{ for
}\Phi\in\ell^{\infty,1}(\Xi\,|\,\mathscr{C})$

and have an isometry:

$\displaystyle\lVert\varphi(\Phi)\rVert_{\ell^{\infty,1}(\Xi,{\mathfrak{A}})}$
$\displaystyle=\max\Big{\\{}\sup_{u\in\mathcal{U}}\sum_{{\rm
r}(x)=u}\\!\parallel\\!\theta_{x}\Phi(x)\\!\parallel_{\mathfrak{A}}\,,\,\sup_{u\in\mathcal{U}}\sum_{{\rm
d}(x)=u}\\!\parallel\\!\theta_{x}\Phi(x)\\!\parallel_{\mathfrak{A}}\\!\Big{\\}}$
$\displaystyle=\max\Big{\\{}\sup_{u\in\mathcal{U}}\sum_{{\rm
r}(x)=u}\\!\parallel\\!\Phi(x)\\!\parallel_{\mathfrak{C}_{x}}\,,\,\sup_{u\in\mathcal{U}}\sum_{{\rm
d}(x)=u}\\!\parallel\\!\Phi(x)\\!\parallel_{\mathfrak{C}_{x}}\\!\Big{\\}}$
$\displaystyle=\lVert\Phi\rVert_{\ell^{\infty,1}(\Xi\,|\,\mathscr{C})}.$

Now we check that $\varphi$ is an ∗-homomorphism,

$\displaystyle\big{[}\varphi(\Phi)*\varphi(\Psi)\big{]}(x)$
$\displaystyle=\sum_{yz=x}\theta_{y}\Phi(y)*\theta_{z}\Psi(z)$
$\displaystyle=\theta_{x}\sum_{yz=x}\Phi(y)\bullet\Psi(z)$
$\displaystyle=\varphi(\Phi*\Psi)(x)$

and

$\varphi(\Phi^{*})(x)=\theta_{x}\Phi^{*}(x)=\theta_{x}\Phi(x^{-1})^{\bullet}=\big{[}\theta_{x^{-1}}\Phi(x^{-1})\big{]}^{*}=\varphi(\Phi)(x^{-1})^{*}=\varphi(\Phi)^{*}(x).$

This finishes the proof.∎

######  Remark 3.5.

Observe that
$\ell^{\infty,1}(\Xi,\mathfrak{A})\cong\ell^{\infty,1}(\Xi)\,\hat{\otimes}\,\mathfrak{A}$,
where $\hat{\otimes}$ denotes the projective tensor product. Indeed, given
$(\varphi,a)\in\ell^{\infty,1}(\Xi)\times\mathfrak{A}$, define the function
$\varphi\otimes a$ by

$\big{[}\varphi\otimes a\big{]}(x):=a\varphi(x),\quad\forall x\in\Xi.$

The map $(\varphi,a)\mapsto\varphi\otimes a$ defined in
$\ell^{\infty,1}(\Xi)\times\mathfrak{A}\to\ell^{\infty,1}(\Xi,\mathfrak{A})$
is bilinear, has norm $1$ (it satisfies $\lVert\varphi\otimes
a\rVert_{\ell^{\infty,1}(\Xi,\mathfrak{A})}\leq\lVert
a\rVert_{\mathfrak{A}}\lVert\varphi\rVert_{\ell^{\infty,1}(\Xi)}$) and it
extends to ∗-isomorphism.

$\iota:\ell^{\infty,1}(\Xi)\,\hat{\otimes}\,\mathfrak{A}\to\ell^{\infty,1}(\Xi,\mathfrak{A})$

The following corollary effectively reduces our concerns to the study of
tensor products, just as in the group case (cf. [4, Theorem 2.4]).

######  Corollary 3.6.

A discrete groupoid $\Xi$ is hypersymmetric if and only if the Banach
∗-algebra
$\ell^{\infty,1}(\Xi,\mathfrak{A})\cong\ell^{\infty,1}(\Xi)\,\hat{\otimes}\,\mathfrak{A}$
is symmetric, for every $C^{*}$-algebra $\mathfrak{A}$.

###### Proof.

Every algebra of the form $\ell^{\infty,1}(\Xi,\mathfrak{A})$ comes from a
Fell bundle (recall Definition 3.2), so it is symmetric if $\Xi$ is
hypersymmetric. On the other hand, because of Lemma 3.4, given any Fell bundle
$\mathscr{C}$, $\ell^{\infty,1}(\Xi\,|\,\mathscr{C})$ may be identified as a
closed ∗-subalgebra of some algebra of the form
$\ell^{\infty,1}(\Xi,\mathfrak{A})$. So it will be symmetric if the latter is
symmetric, by [12, Theorem 11.4.2]. ∎

######  Remark 3.7.

In the group case, this condition has been called ’rigid symmetry’ by many
authors (including myself) and it was introduced by Leptin and Poguntke in
[9]. It can also be seen in [1, 2, 4, 10, 13].

Before going into the following sections, let us simplify some notation. Let
$\mathfrak{A},\mathfrak{B}$ be Banach ∗-algebras. We will denote by
$\mathfrak{A}\hookrightarrow\mathfrak{B}$ the fact that there exists an
isometric ∗-monomorphism $\iota:\mathfrak{A}\to\mathfrak{B}$. In this
language, the conclusion of Lemma 3.4 can be written as
$\ell^{\infty,1}(\Xi\,|\,\mathscr{C})\hookrightarrow\ell^{\infty,1}\big{(}\Xi,C^{*}(\Xi\,|\,\mathscr{C})\big{)}$.
On the other hand, the dual notation
$\mathfrak{A}\twoheadrightarrow\mathfrak{B}$ means that there exists some
contractive ∗-epimorphism $\pi:\mathfrak{A}\to\mathfrak{B}$.

##  4 The result for discrete groupoids

The rest of the paper is devoted to prove Theorem 4.9. We will follow the
strategy detailed in the introduction, starting with a complete (well-known)
characterization of discrete transitive groupoids as group transformation
groupoids.

######  Definition 4.1.

Let $\gamma$ be a continuous action of the (discrete) group ${\sf G}$ on the
topological space $X$, we define the transformation groupoid $\Xi:={\sf
G}\ltimes_{\gamma}\\!X$ associated to it as follows. As a topological space
${\sf G}\ltimes_{\gamma}\\!X$, it is just ${\sf G}\times X$, the maps ${\rm
r},{\rm d}$ are given by ${\rm d}(a,x)=x$ and ${\rm r}(a,x)=\gamma_{a}(x)$.
The composition is $\big{(}b,\gamma_{a}(x)\big{)}(a,x):=(ba,x)$ and inversion
reads $(a,x)^{-1}:=\big{(}a^{-1}\\!,\gamma_{a}(x)\big{)}$ . The unit space is
$\mathcal{U}=\\{{{\sf e}}\\}\times X\equiv X$.

######  Proposition 4.2.

Let $\Xi$ be a discrete transitive groupoid with isotropy group ${\sf G}$.
There is an abelian group structure on $\mathcal{U}$ and an action $\gamma$ of
${\sf G}^{\prime}={\sf G}\times\mathcal{U}$ on $\mathcal{U}$ such that
$\Xi\cong{\sf G}^{\prime}\ltimes_{\gamma}\mathcal{U}$.

###### Proof.

Let us give $\mathcal{U}$ some abelian group structure with additive notation
and define the action of ${\sf G}^{\prime}$ by $\gamma_{(\alpha,w)}(v)=w+v$.
In order to construct an isomorphism $\varphi$, fix some $u\in\mathcal{U}$ and
realize ${\sf G}$ as $\Xi_{u}^{u}$. Since $\Xi$ is transitive, for every
$v\in\mathcal{U}$, there exists an arrow $z_{v}\in\Xi$ such that ${\rm
d}(z_{v})=v$ and ${\rm r}(z_{v})=u$. Then

$\varphi:{\sf G}^{\prime}\ltimes_{\gamma}\mathcal{U}\to\Xi\textup{ defined by
}\varphi\big{(}(x,w),v\big{)}=z^{-1}_{w+v}xz_{v},\textup{ is the required
isomorphism.}$

Let us verify that $\varphi$ is indeed a groupoid isomorphism:

  * (i)

If
$\big{(}(x_{1},w_{1}),w_{2}+v\big{)}\big{(}(x_{2},w_{2}),v\big{)}=\big{(}(x_{1}x_{2},w_{1}+w_{2}),v\big{)}$,
then

$\displaystyle\varphi\big{(}(x_{1}x_{2},w_{1}+w_{2}),v\big{)}$
$\displaystyle=z^{-1}_{w_{1}+w_{2}+v}x_{1}x_{2}z_{v}$
$\displaystyle=z^{-1}_{w_{1}+w_{2}+v}x_{1}z_{w_{2}+v}z_{w_{2}+v}^{-1}x_{2}z_{v}$
$\displaystyle=\varphi\big{(}(x_{1},w_{1}),w_{2}+v\big{)}\varphi\big{(}(x_{2},w_{2}),v\big{)}$

  * (ii)

If $\big{(}(x,w),v\big{)}\in{\sf G}^{\prime}\ltimes_{\gamma}\mathcal{U}$, then
$\big{(}(x,w),v\big{)}^{-1}=\big{(}(x^{-1},-w),w+v\big{)}$ and

$\displaystyle\varphi\big{(}(x^{-1},-w),w+v\big{)}$
$\displaystyle=z^{-1}_{v}x^{-1}z_{w+v}$
$\displaystyle=(z^{-1}_{w+v}xz_{v})^{-1}=\varphi\big{(}(x,w),v\big{)}^{-1}.$

  * (iii)

The inverse function $\varphi^{-1}$ is given by
$\varphi^{-1}(\xi)=\big{(}(z_{{\rm r}(\xi)}\xi z_{{\rm d}(\xi)}^{-1},{\rm
r}(\xi)-{\rm d}(\xi)),{\rm d}(\xi)\big{)}$.

∎

######  Remark 4.3.

It follows from Proposition 4.2 that the pair groupoid
$\mathcal{U}\times\mathcal{U}$ is isomorphic to
$\mathcal{U}\ltimes_{\gamma|_{\mathcal{U}}}\mathcal{U}$. (Here ${\sf
G}=\\{{\sf e}\\}$ is the trivial group)

While the commutativity of $\mathcal{U}$ was not essential for the previous
proof -any group structure would have worked out-, it will be of vital
importance in the future (cf. the proof of Theorem 4.7). That is why it will
be occasionally remarked in the following propositions.

The following lemma requires some notation. Given a $C^{*}$-algebra
$\mathfrak{A}$ and a group action $\gamma:{\sf G}\to{\rm Bij}(\mathcal{U})$,
denote by $\Gamma$ the ${\sf G}$-action on
$\mathcal{C}_{0}(\mathcal{U},\mathfrak{A})$ satisfying
$\Gamma_{x}(f)(u)=f\big{(}\gamma_{x^{-1}}(u)\big{)}$.

######  Lemma 4.4.

Suppose that the discrete group ${\sf G}$ acts on $\mathcal{U}$ via $\gamma$
and $\mathfrak{A}$ is a $C^{*}$-algebra. Then

$\ell^{1}_{\Gamma}\big{(}{\sf
G},\mathcal{C}_{0}(\mathcal{U},\mathfrak{A})\big{)}\twoheadrightarrow\ell^{\infty,1}({\sf
G}\ltimes_{\gamma}\mathcal{U},\mathfrak{A}).$ (4.1)

###### Proof.

Define $\pi:\ell^{1}_{\Gamma}\big{(}{\sf
G},\mathcal{C}_{0}(\mathcal{U},\mathfrak{A})\big{)}\to\ell^{\infty,1}({\sf
G}\ltimes_{\gamma}\mathcal{U},\mathfrak{A})$ by the formula
$\pi(\Phi)(x,u)=\Phi(x)\big{(}\gamma_{x}(u)\big{)}$. This map is well-defined
and clearly surjective, while it also satisfies

$\displaystyle\lVert\pi(\Phi)\rVert_{\ell^{\infty,1}({\sf
G}\ltimes_{\gamma}\mathcal{U},\mathfrak{A})}$
$\displaystyle=\max\Big{\\{}\sup_{u\in\mathcal{U}}\sum_{x\in{\sf
G}}\\!\parallel\\!\Phi(x)\big{(}\gamma_{x}(u)\big{)}\\!\parallel_{\mathfrak{A}}\,,\,\sup_{u\in\mathcal{U}}\sum_{x\in{\sf
G}}\\!\parallel\\!\Phi(x)(u)\\!\parallel_{\mathfrak{A}}\\!\Big{\\}}$
$\displaystyle\leq\sum_{x\in{\sf
G}}\sup_{u\in\mathcal{U}}\\!\parallel\\!\Phi(x)(u)\\!\parallel_{\mathfrak{A}}=\sum_{x\in{\sf
G}}\lVert\Phi(x)\rVert_{\mathcal{C}_{0}(\mathcal{U},\mathfrak{A})}=\lVert\Phi\rVert_{\ell^{1}_{\Gamma}({\sf
G},\mathcal{C}_{0}(\mathcal{U},\mathfrak{A}))}$

and

$\displaystyle\pi(\Phi*\Psi)(x,u)$
$\displaystyle=\big{[}\Phi*\Psi\big{]}(x)\big{(}\gamma_{x}(u)\big{)}$
$\displaystyle=\Big{[}\sum_{y\in{\sf
G}}\Phi(y)\Gamma_{y}\big{[}\Psi(y^{-1}x)\big{]}\Big{]}\big{(}\gamma_{x}(u)\big{)}$
$\displaystyle=\sum_{y\in{\sf
G}}\Phi(y)\big{(}\gamma_{x}(u)\big{)}\Psi(y^{-1}x)\big{(}\gamma_{y^{-1}x}(u)\big{)}$
$\displaystyle=\sum_{y\in{\sf
G}}\pi(\Phi)(y,\gamma_{y^{-1}x}(u))\pi(\Psi)(y^{-1}x,u)$
$\displaystyle=\big{[}\pi(\Phi)*\pi(\Psi)\big{]}(x,u).$

Finally, we see that

$\displaystyle\pi(\Phi^{*})(x,u)$
$\displaystyle=\Phi^{*}(x)\big{(}\gamma_{x}(u)\big{)}$
$\displaystyle=\Gamma_{x}\big{[}\Phi(x^{-1})^{*}]\big{(}\gamma_{x}(u)\big{)}=\Phi(x^{-1})(u)^{*}=\big{[}\pi(\Phi)(x^{-1},\gamma_{x}(u))\big{]}^{*}=\pi(\Phi)^{*}(x,u).$

Hence $\pi$ is a contractive ∗-epimorphism. ∎

Lemma 4.4 will be used in the following form, which allows us to focus on the
study of $\ell^{1}$-algebras arising from group $C^{*}$-dynamical systems.

######  Corollary 4.5.

If $\ell^{1}_{\Gamma}\big{(}{\sf
G},\mathcal{C}_{0}(\mathcal{U},\mathfrak{A})\big{)}$ is a symmetric Banach
∗-algebra, then $\ell^{\infty,1}({\sf
G}\ltimes_{\gamma}\mathcal{U},\mathfrak{A})$ is also symmetric.

###### Proof.

In Lemma 4.4, we showed that $\ell^{1}_{\Gamma}\big{(}{\sf
G},\mathcal{C}_{0}(\mathcal{U},\mathfrak{A})\big{)}\twoheadrightarrow\ell^{\infty,1}({\sf
G}\ltimes_{\gamma}\mathcal{U},\mathfrak{A})$, so the conclusion follows from
[12, Theorem 11.4.2]. ∎

######  Proposition 4.6.

Let $({\sf G}\times{\sf H},\Gamma,\mathfrak{A})$ be a $C^{*}$-dynamical system
and assume that ${\sf G}$ acts trivially on $\mathfrak{A}$. Then one has

$\ell^{1}_{\Gamma}({\sf G}^{\prime},\mathfrak{A})\cong\ell^{1}({\sf
G})\,\hat{\otimes}\,\ell^{1}_{\Gamma}({\sf H},\mathfrak{A}),$ (4.2)

where ${\sf G}^{\prime}:={\sf G}\times{\sf H}$.

###### Proof.

Observe that

$\iota:\ell^{1}_{\Gamma}({\sf G}^{\prime},\mathfrak{A})\to\ell^{1}\big{(}{\sf
G},\ell^{1}_{\Gamma}({\sf H},\mathfrak{A})\big{)}\quad\textup{ defined by
}\quad\iota(\Phi)(x)(y)=\Phi(x,y)$

is an isometric ∗-isomorphism. Indeed, bijectivity is direct, while

$\displaystyle\lVert\iota(\Phi)\rVert_{\ell^{1}({\sf G},\ell^{1}_{\Gamma}({\sf
H},\mathfrak{A}))}$ $\displaystyle=\sum_{x\in{\sf
G}}\lVert\iota(\Phi)(x)\rVert_{\ell^{1}_{\Gamma}({\sf
H},\mathfrak{A})}=\sum_{x\in{\sf G}}\sum_{y\in{\sf
H}}\lVert\Phi(x,y)\rVert_{\mathfrak{A}}=\lVert\Phi\rVert_{\ell^{1}_{\Gamma}({\sf
G}^{\prime},\mathfrak{A})}$

and, identifying $(1_{\sf G},b)\in{\sf G}^{\prime}$ with $b\in{\sf H}$,

$\displaystyle\big{[}\iota(\Phi)*\iota(\Psi)\big{]}(x)(y)$
$\displaystyle=\Big{[}\sum_{a\in{\sf
G}}\iota(\Phi)(a)*\iota(\Psi)(a^{-1}x)\Big{]}(y)$
$\displaystyle=\sum_{a\in{\sf G}}\sum_{b\in{\sf
H}}\iota(\Phi)(a)(b)\Gamma_{b}\big{[}\iota(\Psi)(a^{-1}x)(b^{-1}y)\big{]}$
$\displaystyle=\sum_{(a,b)\in{\sf G}\times{\sf
H}}\Phi(a,b){\Gamma}_{(a,b)}\big{[}\Psi(a^{-1}x,b^{-1}y)\big{]}$
$\displaystyle=\iota(\Phi*\Psi)(x)(y).$

Finally,

$\displaystyle\iota(\Phi^{*})(x)(y)=\Phi^{*}(x,y)=\Gamma_{y}\big{[}\Phi(x^{-1},y^{-1})^{*}\big{]}=\iota(\Phi)(x^{-1})^{*}(y)=\iota(\Phi)^{*}(x)(y).$

So $\ell^{1}_{\Gamma}({\sf G}^{\prime},\mathfrak{A})\cong\ell^{1}\big{(}{\sf
G},\ell^{1}_{\Gamma}({\sf H},\mathfrak{A})\big{)}\cong\ell^{1}({\sf
G})\,\hat{\otimes}\,\ell^{1}_{\Gamma}({\sf H},\mathfrak{A})$. ∎

We can put together the previous lemmas to obtain the following result, which
is Theorem 1.2 for transitive groupoids.

######  Theorem 4.7.

Let $\Xi$ be a discrete transitive groupoid with isotropy subgroup ${\sf G}$.
If ${\sf G}$ is symmetric (resp. hypersymmetric), then $\Xi$ is symmetric
(resp. hypersymmetric).

###### Proof.

Let us divide the proof in two cases, the first one being about hypersymmetry.
Because of Proposition 4.2, we may assume that $\Xi={\sf
G}^{\prime}\ltimes_{\gamma}{\sf H}$, with ${\sf G}^{\prime}={\sf G}\times{\sf
H}$ and $\gamma_{(g,h)}(k)=h+k$.

First suppose that ${\sf G}$ is hypersymmetric. Because of corollaries 3.6 and
4.5, it is enough to show that ${\sf G}\times{\sf H}$ is hypersymmetric (or
’rigidly symmetric’, see Remark 3.7). But this follows from the fact that
${\sf H}$ is abelian and [9, Theorem 7].

Now suppose that ${\sf G}$ is only symmetric. Note that $\gamma|_{\sf H}$
coincides with the action of ${\sf H}$ on itself by left translation, so we
will denote it by ${\rm lt}$. Then

$\displaystyle\ell^{1}_{\Gamma}\big{(}{\sf G}^{\prime},\mathcal{C}_{0}({\sf
H})\big{)}$ $\displaystyle\overset{\eqref{eq2}}{\cong}\ell^{1}({\sf
G})\,\hat{\otimes}\,\ell^{1}_{\rm lt}\big{(}{\sf H},\mathcal{C}_{0}({\sf
H})\big{)}$ $\displaystyle\overset{\eqref{eq}}{\hookrightarrow}\ell^{1}({\sf
G})\,\hat{\otimes}\,\ell^{1}({\sf H})\,\hat{\otimes}\,\big{(}{\sf
H}\ltimes_{\rm lt}\mathcal{C}_{0}({\sf H})\big{)}$
$\displaystyle\cong\ell^{1}({\sf H})\,\hat{\otimes}\,\ell^{1}\big{(}{\sf
G},\mathcal{K}(\ell^{2}({\sf H})\big{)}).$

The final isomorphism holds because of the Stone-von Neumann theorem: ${\sf
H}\ltimes_{\rm lt}\mathcal{C}_{0}({\sf H})\cong\mathcal{K}(\ell^{2}({\sf
H})\big{)}$. $\ell^{1}\big{(}{\sf G},\mathcal{K}(\ell^{2}({\sf H})\big{)})$ is
symmetric because of [5, Theorem 1], hence $\ell^{1}({\sf
H})\,\hat{\otimes}\,\ell^{1}\big{(}{\sf G},\mathcal{K}(\ell^{2}({\sf
H})\big{)})$ is symmetric because of [8, Theorem 5]. We conclude that
$\ell^{1}_{\Gamma}\big{(}{\sf G}^{\prime},\mathcal{C}_{0}({\sf H})\big{)}$ and
thus $\ell^{\infty,1}(\Xi)$ are symmetric. ∎

######  Corollary 4.8.

The pair groupoid over a discrete set $\mathcal{U}$ is hypersymmetric.

Now we will proceed to upgrade Theorem 4.7 to the case of general discrete
groupoids.

######  Theorem 4.9.

A discrete groupoid $\Xi$ is symmetric (resp. hypersymmetric) if and only if
its isotropy subgroups are symmetric (resp. hypersymmetric).

###### Proof.

Is obvious that symmetry (resp. hypersymmetry) of $\Xi$ implies the symmetry
(resp. hypersymmetry) of its isotropy groups
($\ell^{1}(\Xi_{u}^{u},\mathfrak{A})\hookrightarrow\ell^{\infty,1}(\Xi,\mathfrak{A})$
in a natural way). As the ’only if’ part is clear, let us prove the ’if’ part.

Let $\mathfrak{A}$ be a $C^{*}$-algebra and $\Phi$ be an arbitrary section in
$\ell^{\infty,1}(\Xi,\mathfrak{A})$. $\Phi$ has a countable support (see
Remark 2.2), and since every discrete groupoid can be decomposed as a disjoint
union of discrete transitive subgroupoids, we can find a countable number of
disjoint transitive subgroupoids $\\{\Xi(i)\\}_{i\in\mathbb{N}}$, such that
${\rm supp}(\Phi)\subset\bigcup_{i=1}^{\infty}\Xi(i)$. Define
$\Phi_{n}\in\ell^{\infty,1}(\Xi,\mathfrak{A})$ as

$\Phi_{n}(x):=\left\\{\begin{array}[]{ll}\Phi(x)&\textup{if\
}x\in\bigcup_{i=1}^{n}\Xi(i),\\\ 0_{\mathfrak{A}}&\textup{if\
}x\not\in\bigcup_{i=1}^{n}\Xi(i).\\\ \end{array}\right.$

We have that $\lim_{n}\Phi_{n}=\Phi$ in $\ell^{\infty,1}(\Xi,\mathfrak{A})$,
but more importantly,

${\rm
Spec}_{\ell^{\infty,1}(\Xi,\mathfrak{A})}(\Phi)=\bigcup_{i=1}^{\infty}{\rm
Spec}_{\ell^{\infty,1}(\Xi,\mathfrak{A})}(\Phi_{n}).$ (4.3)

So, to conclude, its enough to prove that ${\rm
Spec}_{\ell^{\infty,1}(\Xi,\mathfrak{A})}(\Phi_{n})\subset\mathbb{R}$,
whenever $\Phi^{*}=\Phi$. Indeed, if $\Phi^{*}=\Phi$, after fixing $n$ we see
that $\Phi_{n}^{*}=\Phi_{n}$ and $\ell^{\infty,1}(\Xi,\mathfrak{A})$ contains
an isometric ∗-isomorphic copy of
$\ell^{\infty,1}(\bigcup_{i=1}^{n}\Xi(i),\mathfrak{A})$, obtained by extending
the sections in the latter algebra with zeros outside its original domain. So
by definition, one has that
$\Phi_{n}\in\ell^{\infty,1}(\bigcup_{i=1}^{n}\Xi(i),\mathfrak{A})\subset\ell^{\infty,1}(\Xi,\mathfrak{A})$,
which implies that

${\rm Spec}_{\ell^{\infty,1}(\Xi,\mathfrak{A})}(\Phi_{n})\subset{\rm
Spec}_{\ell^{\infty,1}(\bigcup_{i=1}^{n}\Xi(i),\mathfrak{A})}(\Phi_{n}).$

But
$\ell^{\infty,1}(\bigcup_{i=1}^{n}\Xi(i),\mathfrak{A})\cong\bigoplus_{i=1}^{n}\ell^{\infty,1}(\Xi(i),\mathfrak{A})$
and since the (finite) direct sum of symmetric Banach ∗-algebras is symmetric,
${\rm
Spec}_{\ell^{\infty,1}(\bigcup_{i=1}^{n}\Xi(i),\mathfrak{A})}(\Phi_{n})\subset\mathbb{R}$
and the result follows. ∎

######  Example 4.10.

Let $\Xi={\sf G}\ltimes_{\gamma}\\!X$ be a transformation groupoid (see
Definition 4.1), where $X$ is discrete. In this case, $\mathcal{U}=X$ and
$\Xi_{u}^{u}=\\{g\in G\mid\gamma_{g}(u)=u\\}\times\\{u\\}$, which is
identificable with the stabilizer of $u$, namely ${\rm
Stab}_{\gamma}(u)\leq{\sf G}$.

######  Remark 4.11.

We will finish this paper with a bit of wishful thinking: Is reasonable to
believe that the groupoid analog of [2, Theorem 3.3] could be true. For a
general locally compact groupoid one can define symmetry and hypersymmetry for
$\Xi$ as the symmetry of the Hahn algebras $L^{\infty,1}(\Xi)$ and
$L^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$, respectively. So we may pose two
problems:

  1. $(i)$

Is the (hyper)symmetry of a Hausdorff locally compact groupoid $\Xi$ implied
by the (hyper)symmetry of its discretization $\Xi^{\rm dis}$?

  2. $(ii)$

Is Corollary 3.6 still valid for étale groupoids?

In view of Theorem 1.2, $(i)$ is equivalent to

  1. $(i^{\prime})$

Is the (hyper)symmetry of a Hausdorff locally compact groupoid $\Xi$ implied
by the (hyper)symmetry of its discretized isotropy groups $(\Xi_{u}^{u})^{\rm
dis}$?

## References

  * [1] A. Austad and E. Ortega: _Groupoids and Hermitian Banach ∗-Algebras_, arXiv:2105.14793 [math.FA].
  * [2] F. Flores, D. Jauré and M. Măntoiu: _Symmetry for algebras associated to Fell bundles over groups and groupoids_ , to appear in Journal of Operator Theory.
  * [3] F. Flores and M. Măntoiu: _Topological dynamics for groupoid actions_ , to appear in Groups, Geometry and Dynamics.
  * [4] D. Jauré and M. Măntoiu: _Symmetry and Spectral Invariance for Topologically Graded $C^{*}$-Algebras and Partial Action Systems_, Bull. London Math. Soc. 54(4), 1448–1469, (2022).
  * [5] W. Kugler: _On the Symmetry of Generalized $L^{1}$-Algebras_, Math. Z. 168(3), 241–262, (1979).
  * [6] A. Kumjian:_Fell Bundles over Groupoids_ , Proc. Amer. Math. Soc. 126(4), 1115–1125, (1998).
  * [7] H. Leptin: _Symmetrie in Banachschen Algebren_ , Arch. Math. (Basel), 27(4), 394–400, (1976).
  * [8] H. Leptin: _Ideal Theory in Group Algebras of Locally Compact Groups_. Inventiones mathematicae 31, 259-278, (1976).
  * [9] H. Leptin and D. Poguntke: _Symmetry and Nonsymmetry for Locally Compact Groups_ , J. Funct. Anal. 33(2), 119–134, (1979).
  * [10] M. Măntoiu: _Symmetry and Inverse Closedness for Banach $C^{*}$-Algebras Associated to Discrete Groups_, Banach Journal of Mathematical Analysis, 9(2), 289–310, (2015).
  * [11] P. Muhly and D. Williams: _Equivalence and Disintegration Theorems for Fell Bundles and Their $C^{*}$-Algebras_, Dissertationes Math. (Rozprawy Mat.), 456, 1–57, (2008).
  * [12] T.W. Palmer: _Banach Algebras and the General Theory of ∗-Algebras_, Vol. II. ∗-Algebras, Encyclopedia of Mathematics and its Applications, 69. Cambridge University Press, Cambridge, 2001.
  * [13] D. Poguntke: _Rigidly Symmetric $L^{1}$-Group Algebras_, Seminar Sophus Lie, 2, 189–197(1992).

ADDRESS

F. Flores

Department of Mathematics, University of Virginia,

114 Kerchof Hall. 141 Cabell Dr,

Charlottesville, Virginia, United States

E-mail<EMAIL_ADDRESS>