EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2022-100 LHCb-PAPER-2021-049

Direct $C\\!P$ violation in charmless three-body decays of $B^{\pm}$ mesons

LHCb collaboration

Measurements of $C\\!P$ asymmetries in charmless three-body decays of
$B^{\pm}$ mesons are reported using proton-proton collision data collected by
the LHCb detector, corresponding to an integrated luminosity of
5.9$\text{\,fb}^{-1}$. The previously observed $C\\!P$ asymmetry in
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays is confirmed,
and $C\\!P$ asymmetries are observed with a significance of more than five
standard deviations in the
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays, while the
$C\\!P$ asymmetry of
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ decays is
confirmed to be compatible with zero. The distributions of these asymmetries
are also studied as a function of the three-body phase space and suggest
contributions from rescattering and resonance interference processes. An
indication of the presence of the decays
$B^{\pm}\rightarrow\pi^{\pm}\chi_{c0}(1P)$ in both
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays is observed,
as is $C\\!P$ violation involving these amplitudes.

Submitted to Phys. Rev. D

© 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.

##  1 Introduction

The violation of charge-parity ($C\\!P$) symmetry has been observed in
$B^{+}$, $B^{0}$ and $B^{0}_{s}$ decays, and yet our understanding of the
dynamics involved in these asymmetries is incomplete. Some crucial questions
remain to be understood: what is the dynamical origin of the weak phase
associated to the decay amplitudes? Why has $C\\!P$ violation not been
observed so far in baryon decays? What is the primary source of the strong
phase difference in the observed $C\\!P$ asymmetries [1]?

The last question is directly related to the long-standing debate about the
role of short- and long-distance contributions to the generation of the
strong-phase differences, needed for direct $C\\!P$ violation to occur, and
three-body decays of $B$ mesons offer a way of finding the answer. A recent
amplitude analysis found a large $C\\!P$ asymmetry related to the interference
between the S\- and P-wave contributions in
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ decays [2, 3].
In a similar analysis, large $C\\!P$ violation involving $\pi\pi\rightarrow
KK$ rescattering was observed in
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays [4]. The
patterns of localised $C\\!P$ asymmetries in the four charmless decay modes
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$,
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$,
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$, reported in Refs.
[5, 6], can be interpreted as originating from long-distance hadronic
interactions [1].

Heavy meson three-body decays can be understood in terms of a superposition of
intermediate states involving resonances. From this perspective, the phase-
space-integrated $C\\!P$ asymmetries ($A_{{C\\!P}}$) observed in charmless
three-body decays of $B^{\pm}$ decays are the sum of the asymmetries of each
intermediate state. In the
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decay [6], the
observed $C\\!P$ asymmetry would result predominantly from the
$\pi\pi\rightarrow KK$ rescattering amplitude, weighted by its fit fraction
[4]. The same can be seen from the recent amplitude analysis of the
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ decay [2, 3].
The sum of the $C\\!P$ asymmetries for the $B^{\pm}\rightarrow\sigma\pi^{\pm}$
and $B^{\pm}\rightarrow f_{2}(1270)\pi^{\pm}$ decays weighted by their
respective fractions is almost equal to the inclusive $C\\!P$ asymmetry
observed for the ${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$
decay [6].

Another approach considers the global properties of the three-body amplitudes.
The authors of Refs. [7, 8, 9] use U-spin symmetry to relate the $C\\!P$
asymmetries and partial decay widths of $B^{\pm}\rightarrow
h^{\pm}h^{\prime+}h^{\prime-}$ decays. U-spin symmetry predicts that the
ratios $\Delta\Gamma(B^{\pm}\rightarrow
K^{\pm}\pi^{+}\pi^{-})/\Delta\Gamma(B^{\pm}\rightarrow\pi^{\pm}K^{+}K^{-})$
and $\Delta\Gamma(B^{\pm}\rightarrow
K^{\pm}K^{+}K^{-})/\Delta\Gamma(B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-})$,
where $\Delta\Gamma$ is the partial decay width difference between $B^{-}$ and
$B^{+}$, must be close to $-1$. These predictions, based on a general three-
body decay approach, are supported by experimental measurements, within the
uncertainties [6, 8, 10, 11].

These two approaches are complementary and the role of each one must be
determined from experimental results, such as the $C\\!P$ asymmetries and the
branching fractions of charmless three-body ${B}^{\pm}$ decays, recently
presented by the LHCb collaboration [12].

In this article, $C\\!P$ violation is studied in four charmless $B^{\pm}$
meson decays into three charged pseudoscalar particles:
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$,
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$,
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$. The analysis is
based on proton-proton ($pp$) collision data at a centre-of-mass energy of 13
TeV, corresponding to an integrated luminosity of 5.9$\text{\,fb}^{-1}$,
recorded with the LHCb detector between 2015 and 2018. Phase-space integrated
$C\\!P$ asymmetries are measured, as well as $C\\!P$ asymmetries in specific
regions of the Dalitz plots. The analysis uses the decay $B^{\pm}\rightarrow
J/\psi(\rightarrow\mu^{+}\mu^{-})K^{\pm}$ as a control channel, in order to
take into account the production asymmetry of the $B^{\pm}$ mesons.

##  2 LHCb detector, data set and simulation

The LHCb detector [13, 14] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about 4 Tm, and three stations of
silicon-strip detectors and straw drift tubes placed downstream of the magnet.
The tracking system provides a measurement of the momentum, $p$, of charged
particles with a relative uncertainty that varies from 0.5% at low momentum
($\sim$ 2$\text{\,Ge\kern-1.00006ptV\\!/}c$) to 1.0% at
200$\text{\,Ge\kern-1.00006ptV\\!/}c$. The minimum distance of a track to a
primary $pp$ collision vertex (PV), the impact parameter (IP), is measured
with a resolution of (15 + 29/$p_{\mathrm{T}}$) $\upmu$m, where
$p_{\mathrm{T}}$ is the component of the momentum transverse to the beam, in
$\text{\,Ge\kern-1.00006ptV\\!/}c$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors. The
corresponding efficiencies and misidentification rates are given in Ref.[15].
Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
and a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers.

The online event selection is performed by a trigger, which consists of a
hardware stage, based on information from the calorimeter system, followed by
a software stage, which applies a full event reconstruction. At the hardware
trigger stage, events are selected if a transverse energy, typically larger
than 3.5 GeV, is deposited in the calorimeters by at least one hadron from the
$B^{\pm}\rightarrow h^{\pm}h^{\prime+}h^{\prime-}$ candidates. For the control
channel, it is required that the hardware trigger decision must also be due to
at least one muon with high $p_{\mathrm{T}}$. In all final states, events are
also selected if the trigger decision is independent of the signal.

The software trigger requires a two-, three- or four-track vertex with a
significant displacement from any PV. At least one charged particle must have
a large $p_{\mathrm{T}}$ and be inconsistent with originating from any PV. A
multivariate algorithm is used for the identification of displaced vertices
consistent with the decay of a b-hadron [16].

Samples of simulated events are used to model the effects of the detector
efficiency and the selection requirements, to determine the fit models and to
evaluate efficiencies. In the simulation, $pp$ collisions are generated using
Pythia 8 [17] with a specific LHCb configuration [18]. Decays of unstable
particles are described by EvtGen [19], in which final-state radiation is
generated using Photos [20]. The interaction of the generated particles with
the detector, and its response, are implemented using the Geant4 toolkit [21,
22] as described in Ref. [23].

##  3 Candidate selection

The selection of the four ${{B}^{\pm}}\\!\rightarrow
h^{\pm}h^{\prime+}h^{\prime-}$ samples is based on simulations and data-driven
methods. The analysed decay modes share the same topology and similar
kinematics, leading to a common strategy for the selection. Requirements are
made on the quality of the three-track vertices: the decay vertex must be
detached from the PV, and the momentum vector of the reconstructed ${B}^{\pm}$
candidate must point back to the PV. The final-state particles must have a
significant IP with respect to the PV.

The combinatorial background is reduced by a multivariate analysis of a set of
ten topological and kinematic variables, using boosted decision trees (BDT)
[24] as classifiers. Samples of simulated events are used to describe the
signal, whereas candidates satisfying
$m(h^{\pm}h^{{}^{\prime}+}h^{{}^{\prime}-})>5.4\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$,
are used to represent the combinatorial background. The BDT classifier is
trained separately for each decay mode. A requirement on the BDT response for
each mode is chosen to maximise the ratio
$\varepsilon_{\textrm{sim}}/\sqrt{(S+B)_{\textrm{data}}}$, where
$\varepsilon_{\textrm{sim}}$ is the signal efficiency from simulation and
$(S+B)_{\textrm{data}}$ is obtained by counting the candidates from data with
invariant mass in the interval of
$\pm$40$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ centred on the known $B^{+}$
mass [25]. The most discriminating BDT variables are related to the $B^{\pm}$
candidate distance of closest approach to the PV, pointing angle, decay vertex
quality, and IP of each final-state particle.

Particle identification (PID) requirements are made on all three final-state
particles, reducing the contamination from other ${{B}^{\pm}}\\!\rightarrow
h^{\pm}h^{\prime+}h^{\prime-}$ decays to the per cent level. This background
is due mainly to $K-\pi$ misidentification. The analysis is restricted to
particles with momentum below 100$\text{\,Ge\kern-1.00006ptV\\!/}c$, since
above this limit the discrimination power of the PID system is limited. In
order to avoid the contamination from semileptonic decays, a muon veto is
placed on all decay products by applying requirements on track reconstruction
and trigger decisions. Finally, a method based on Artificial Neural Network is
employed to remove electrons [26].

More stringent PID requirements are applied to the
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays. These decays
have lower branching fractions compared to the
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays and are
therefore more severely affected by the cross-feed from other
${{B}^{\pm}}\\!\rightarrow h^{\pm}h^{\prime+}h^{\prime-}$ decays.

There are significant backgrounds from two-body decays of ${B}^{\pm}$ mesons
into a ${D}^{0}$ or ${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ meson
and a pseudoscalar particle, followed by the two-body decay of the ${D}^{0}$
meson into pions and kaons. This background is removed by excluding candidates
for which the two-body invariant masses $m(\pi^{+}\pi^{-})$,
$m(K^{\pm}\pi^{\mp})$ or $m(K^{+}K^{-})$ are within $\pm
30$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known ${D}^{0}$ mass [25].

After the final selection, approximately 0.5% of the events contain more than
one signal candidate. One candidate is chosen randomly in these cases.

The decay
${{B}^{\pm}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{\pm}}$, with
subsequent decay ${J\mskip-3.0mu/\mskip-2.0mu\psi}$
$\rightarrow\mu^{+}\mu^{-}$ is used to measure the ${B}^{\pm}$ production
asymmetry. The sample is selected with the same requirements as the
${{B}^{\pm}}\\!\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}$ decay, with three
exceptions: the particle identification, in which the pion PID is substituted
by that of muon; the use of events for which the hardware trigger decision is
due to a muon with high $p_{\mathrm{T}}$; the requirement of the
${J\mskip-3.0mu/\mskip-2.0mu\psi}$ candidate mass within the interval
$3.05<m(\mu^{+}\mu^{-})<3.15\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$.

##  4 Invariant mass fits

\begin{overpic}[width=212.47617pt]{Fig1a.pdf}
\put(11.0,44.0){\scriptsize{(a)}}
\end{overpic}\begin{overpic}[width=203.80193pt]{Fig1b.pdf}
\put(11.0,47.0){\scriptsize{(b)}} \end{overpic}

\begin{overpic}[width=212.47617pt]{Fig1c.pdf}
\put(11.0,47.0){\scriptsize{(c)}}
\end{overpic}\begin{overpic}[width=212.47617pt]{Fig1d.pdf}
\put(11.0,47.0){\scriptsize{(d)}} \end{overpic}

Figure 1: Invariant mass spectra of (a)
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$, (b)
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$, (c)
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and (d)
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays in the mass
range [5050–5650]$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$. The left panel in
each figure shows the ${{B}^{-}}$ candidates and the right panel shows the
${{B}^{+}}$ candidates. The results of the unbinned maximum likelihood fits
are overlaid.

The yield and raw asymmetry of each decay channel are extracted from
simultaneous unbinned extended maximum-likelihood fits to the ${{B}^{-}}$ and
${{B}^{+}}$ invariant mass distributions, as shown in Fig. 1. The signal
components in each of the four channels are parameterised by a sum of a
Gaussian and two Crystal Ball functions [27] with a common mean and different
widths and tails on either side of the peak, accounting for asymmetric effects
such as quark final-state radiation. The combinatorial backgrounds is
described by an exponential function. The slopes and charge asymmetry of the
combinatorial background are free parameters in the fit, the latter being
compatible with zero when systematic uncertainties, production and detection
asymmetries are taken into account. The background due to partially
reconstructed four-body $B$ decays, in which one particle is not
reconstructed, is parameterised by an ARGUS function [28] convolved with a
Gaussian resolution function. A contribution from four-body ${B}^{0}_{s}$
decays is also present in the
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ sample.

The dominant peaking backgrounds in the signal regions arise from other three-
body $B$ decays with one or more misidentified final-state particles. For
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ decays, the
largest contribution comes from
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ with the pion
misidentified as kaon, followed by the partially reconstructed decay
${{B}^{\pm}}\rightarrow\eta^{\prime}(\rho^{0}\gamma){{K}^{\pm}}$, in which the
photon is undetected. Conversely, the only peaking contribution to the
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ spectrum comes
from ${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ decays. The
peaking background to ${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$
is the result of single or double misidentification of pions as kaons from
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ decays. The
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decay has the
largest number of significant peaking background contributions due to its
lower branching fraction and both $K\rightarrow\pi$ and $\pi\rightarrow K$
misidentification possibilities, which is why the three other channels can
contribute to its spectrum.

The shapes and fractions of peaking backgrounds relative to the signal yield
are obtained from simulation of the relevant decay modes and fixed in the
fits, whereas the yields of the partially reconstructed background components
vary freely in the fit. The resulting signal yields and raw asymmetries
($A_{\text{raw}}$) are listed in Table 1. The data samples are larger than
those presented in Ref. [6] due to both an increase in the integrated
luminosity and the production cross-section, and the use of a more efficient
selection.

Table 1: Total signal yields, raw asymmetries and average efficiency ratio $R$ of charmless three-body ${B}^{\pm}$ decays for the inclusive data set. Decay mode | Total yield | $A_{\text{raw}}$ | $R=\,<\epsilon^{-}>/<\epsilon^{+}>$
---|---|---|---
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ | $499\,200\pm 900\phantom{0}$ | $+0.006\pm 0.002$ | $1.0038\pm 0.0027$
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ | $365\,000\pm 1000$ | $-0.052\pm 0.002$ | $0.9846\pm 0.0024$
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ | $101\,000\pm 500\phantom{0}$ | $+0.090\pm 0.004$ | $1.0354\pm 0.0037$
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ | $\phantom{3}32\,470\pm 300\phantom{0}$ | $-0.132\pm 0.007$ | $0.9777\pm 0.0032$

##  5 Phase-space integrated $C\\!P$ asymmetries

Neither the selection efficiencies nor the observed raw asymmetries are
uniform across the Dalitz plot, and efficiencies may differ for ${{B}^{+}}$
and ${{B}^{-}}$. Therefore, the efficiency of selecting a signal decay is
parameterised in the two-dimensional square Dalitz plot separately for
${{B}^{+}}$ and ${{B}^{-}}$. Such a transformation improves the resolution in
the regions with high sensitivity by expanding the corners and the borders of
the nominal Dalitz plot relative to the sparsely populated central region.
These maps are primarily obtained using simulated decays; however, effects
arising from the hardware trigger and PID efficiency are determined using data
calibration samples. The efficiency shows a variation from 0.9% to 1.7% for
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$, 0.1% to 0.6% for
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$, 0.1% to 1% for
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and 0.1% to 1.2%
for ${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$. The
efficiency correction applied to the integrated raw asymmetries is determined
from the ratio between the ${{B}^{-}}$ and ${{B}^{+}}$ average efficiencies,
as listed in Table 1. These averages reflect the data distributions across the
phase space by weighting the efficiency maps by the data events.

The phase-space integrated $C\\!P$ asymmetries $A_{{C\\!P}}$ are obtained by
correcting the raw asymmetry for selection efficiency effects to get the
efficiency-corrected raw asymmetries $A_{\rm raw}^{\rm corr}$, and for the
production asymmetry $A_{\rm P}$. The efficiency-corrected raw asymmetry is
defined as

$A_{\rm raw}^{\rm corr}\equiv\dfrac{N_{B^{-}}^{\rm corr}-N_{B^{+}}^{\rm
corr}}{N_{B^{-}}^{\rm corr}+N_{B^{+}}^{\rm
corr}}=\dfrac{(N_{B^{-}}/R)-N_{B^{+}}}{(N_{B^{-}}/R)+N_{B^{+}}},$ (1)

where $N_{B^{-}}^{\rm corr}$ and $N_{B^{+}}^{\rm corr}$ are the number of
${{B}^{-}}$ and ${{B}^{+}}$ signal events obtained from the invariant mass
fit, $N_{B^{-}}$ and $N_{B^{+}}$, corrected for the efficiency. They are
related to the asymmetries by

$\displaystyle N_{{{{B}^{-}}}}^{\rm corr}$ $\displaystyle=$
$\displaystyle\dfrac{N_{{{{B}^{-}}}}^{\rm corr}+N_{{{{B}^{+}}}}^{\rm
corr}}{2}(1+A_{{C\\!P}})(1+A_{\rm P})\,,$ (2) $\displaystyle
N_{{{{B}^{+}}}}^{\rm corr}$ $\displaystyle=$
$\displaystyle\dfrac{N_{{{{B}^{-}}}}^{\rm corr}+N_{{{{B}^{+}}}}^{\rm
corr}}{2}(1-A_{{C\\!P}})(1-A_{\rm P}).$ (3)

Substituting Eqs. 2 and 3 into Eq. 1, the physical $C\\!P$ asymmetry can be
expressed in terms of $A_{\rm raw}^{\rm corr}$ and $A_{\rm P}$:

$A_{{C\\!P}}=\frac{A_{\rm raw}^{\rm corr}-A_{\rm P}}{1-A_{\rm raw}^{\rm
corr}A_{\rm P}}.$ (4)

To obtain $A_{\rm P}$, the decay
${{{B}^{\pm}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{\pm}}}$ is
used as a control channel and its raw asymmetry is obtained from a fit to the
invariant mass distribution. This decay has larger signal yield and lower
background contributions compared to ${{B}^{\pm}}\\!\rightarrow
h^{\pm}h^{\prime+}h^{\prime-}$ decays. Its background is parameterised as the
sum of a combinatorial component, represented by a single exponential, and
partially-reconstructed
$B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}K^{*}(892)^{\pm,0}$ decay,
described by an ARGUS function. The
${{{B}^{\pm}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{\pm}}}$ model
is parameterised with the same function used for the
${{B}^{\pm}}\\!\rightarrow h^{\pm}h^{\prime+}h^{\prime-}$ channels. Systematic
uncertainties of the raw asymmetry are obtained by varying the signal fit
model, leaving the background asymmetry to vary in the fit, and looking at
variations from different trigger samples of the data. The total systematic
uncertainty is taken as the sum in quadrature of the individual uncertainties.
The raw asymmetry of the control channel is measured to be $A_{\rm
raw}({{{B}^{\pm}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{\pm}}})=-0.0118\pm
0.0008\,^{+0.0007}_{-0.0008}$, where the first uncertainty is statistical and
the second systematic.

To obtain the $B^{\pm}$ production asymmetry, this raw asymmetry is corrected
by its efficiency ratio, calculated using a sample of simulated events
produced without any $B^{\pm}$ production asymmetry, to obtain $A_{\rm
raw}^{\rm corr}$ as before, and the world average value of the
${{{B}^{\pm}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{\pm}}}$
$C\\!P$ asymmetry, $0.0018\pm 0.0030$ [25], is subtracted

$A_{\rm P}=A_{\rm raw}^{\rm
corr}({{{B}^{\pm}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{\pm}}})-A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{\pm}}}).$
(5)

The measured $B$ meson production asymmetry is $A_{\rm P}=-0.0070\pm
0.0008\,^{+0.0007}_{-0.0008}\pm 0.0030$, where the last uncertainty is due to
the $C\\!P$ asymmetry of
${{{B}^{\pm}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{\pm}}}$ decays
[25].

Finally, the $C\\!P$ asymmetries of the four ${{B}^{\pm}}\\!\rightarrow
h^{\pm}h^{\prime+}h^{\prime-}$ modes are measured to be

$\displaystyle
A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}})=+0.011\pm
0.002,$ $\displaystyle
A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}})=-0.037\pm
0.002,$ $\displaystyle
A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}})=+0.080\pm
0.004,$ $\displaystyle
A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}})=-0.114\pm
0.007,$

where the statistical uncertainties are obtained from propagation of Eq. 4,
assuming no correlation term.

##  6 Systematic uncertainties and results

Several sources of systematic uncertainties are considered and can be broadly
divided into three groups: potential mismodelling of the invariant mass
distributions, phase-space efficiency corrections and knowledge of the
$B^{\pm}$ production asymmetry. The systematic uncertainties due to the mass
fit models are quantified by taking the difference in the $C\\!P$ asymmetry
resulting from variations of the model. The alternative fits have good quality
and describe the data accurately. To estimate the uncertainty due to the
choice of the signal mass function, the initial model is replaced by an
alternative empirical distribution [27].

The contribution associated with the peaking background fractions reflects the
uncertainties in the expected yields determined from simulation and it is
evaluated by varying the fractions within their statistical uncertainties. In
addition, the systematic uncertainty associated to the fact that the peaking
background asymmetry is fixed to zero is estimated by setting it to the value
obtained in the previous analysis [6], within the corresponding uncertainties.
The uncertainty due to the choice of an exponential function to model the
combinatorial component is estimated by repeating the fit using a second order
polynomial function.

The systematic uncertainty related to the efficiency correction procedure
consists of two parts: the statistical uncertainty in the detection efficiency
due to the finite size of the simulated samples, and the uncertainty due to
the finite size of the bins, which is evaluated by varying the binning used in
the efficiency correction. The former term dominates for all decays.

The systematic uncertainty due to the $B^{\pm}$ production asymmetry
measurement using the
${{{B}^{\pm}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{\pm}}}$
control channel is obtained by summing in quadrature the experimental
statistical and systematic uncertainties of $A_{\rm P}$.

Table 2: Individual components and total systematic uncertainties related to the mass fit model, efficiency corrections for the integrated phase space and $B^{\pm}$ production asymmetry. The total is the sum in quadrature of these components. Source of uncertainty | $K^{\pm}\pi^{+}\pi^{-}$ | $K^{\pm}K^{+}K^{-}$ | $\pi^{\pm}\pi^{+}\pi^{-}$ | $\pi^{\pm}K^{+}K^{-}$
---|---|---|---|---
Signal model | 0.0004 | 0.0007 | 0.0000 | 0.0001
Peaking background fraction | 0.0005 | 0.0010 | 0.0002 | 0.0004
Peaking background asymmetry | 0.0022 | 0.0001 | 0.0005 | 0.0007
Combinatorial model | 0.0002 | 0.0005 | 0.0015 | 0.0025
Efficiency correction | 0.0014 | 0.0016 | 0.0018 | 0.0019
Production asymmetry | 0.0011 | 0.0011 | 0.0011 | 0.0011
Total | 0.0029 | 0.0024 | 0.0027 | 0.0035

The systematic uncertainties are summarised in Table 2, where the total is the
sum in quadrature of the individual contributions. The results for the
integrated $C\\!P$ asymmetries are

$\displaystyle
A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}})=+0.011\pm
0.002\pm 0.003\pm 0.003,$ $\displaystyle
A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}})=-0.037\pm
0.002\pm 0.002\pm 0.003,$ $\displaystyle
A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}})=+0.080\pm
0.004\pm 0.003\pm 0.003,$ $\displaystyle
A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}})=-0.114\pm
0.007\pm 0.003\pm 0.003,$

where the first uncertainty is statistical, the second is systematic and the
third is due to the limited knowledge of the $C\\!P$ asymmetry of the
${{{B}^{\pm}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{\pm}}}$
control channel [25].

The significance of the $C\\!P$ asymmetries is computed by dividing the
central values by the sum in quadrature of the uncertainties, yielding 8.5
standard deviations ($\sigma$) for
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays, 14.1$\sigma$
for ${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ decays and
13.6$\sigma$ for ${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$
decays. This is the first observation of $C\\!P$ asymmetries in these
channels. For the ${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$
decays, the significance of the $C\\!P$ asymmetry is 2.4$\sigma$, consistent
with $C\\!P$ conservation.

##  7 Localised $C\\!P$ asymmetries

\begin{overpic}[width=212.47617pt]{Fig2a.pdf}
\put(50.0,64.0){\scriptsize{(a)}}
\end{overpic}\begin{overpic}[width=212.47617pt]{Fig2b.pdf}
\put(50.0,64.0){\scriptsize{(b)}}
\end{overpic}\begin{overpic}[width=212.47617pt]{Fig2c.pdf}
\put(50.0,64.0){\scriptsize{(c)}}
\end{overpic}\begin{overpic}[width=212.47617pt]{Fig2d.pdf}
\put(50.0,64.0){\scriptsize{(d)}} \end{overpic}

Figure 2: Dalitz plot distributions for (a)
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$, (b)
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$, (c)
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ and (d)
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays. The colour
scale indicates the number of events.

The two-dimensional phase space referred to as the Dalitz plot, formed by the
invariant masses squared of two of the three possible particle pairs, reflects
directly the dynamics of the decay and allows the study of the different
resonant and nonresonant components to inspect $C\\!P$ asymmetry effects. The
Dalitz plot distributions of signal yield, obtained with the sPlot technique
[29], for ${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$,
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$,
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ are shown in Fig. 2.
In the symmetric channels, the phase space distribution and its projections
are presented with the two axes being the squares of the low-mass
$m_{\textrm{low}}$ and high-mass $m_{\textrm{high}}$ combinations of the
opposite-sign particle pairs, for visualization purposes.

Most of the candidates are concentrated in the low-mass regions, as expected
for charmless decays dominated by resonant contributions. The gap from the
vetoed potential $J/\psi$ contributions is visible in the
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ channel, as well
as the gaps from $D^{0}$ regions excluded in all modes.

In order to visualise localised asymmetries, the $A_{{C\\!P}}$ in bins of the
phase space [30] is constructed. Adaptive binning is employed, such that the
efficiency-corrected signal yield, also obtained with the sPlot technique, is
approximately equal in all bins. There is no specific rule for choosing the
best binning except for requiring a minimum bin occupancy. Figure 3 reveals a
rich pattern of large and localised asymmetries which result from interference
between the contributions, as well as possible $\pi\pi\rightarrow KK$
rescattering, as was observed in the amplitude analyses of Refs. [2, 3, 4] and
elastic scattering experiments [31, 32].

\begin{overpic}[width=212.47617pt]{Fig3a.pdf}
\put(74.0,59.0){\scriptsize{(a)}}
\end{overpic}\begin{overpic}[width=212.47617pt]{Fig3b.pdf}
\put(74.0,59.0){\scriptsize{(b)}}
\end{overpic}\begin{overpic}[width=212.47617pt]{Fig3c.pdf}
\put(74.0,59.0){\scriptsize{(c)}}
\end{overpic}\begin{overpic}[width=212.47617pt]{Fig3d.pdf}
\put(74.0,59.0){\scriptsize{(d)}} \end{overpic}

Figure 3: Asymmetry distribution in bins of the Dalitz plot for (a)
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$, using 400 bins
with approximately 229 events/bin; (b)
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$, using 1728 bins
with approximately 276 events/bin; (c)
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$, using 256 bins with
approximately 127 events/bin; and (d)
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$, using 729 bins with
approximately 461 events/bin.

Different regions of the Dalitz plots in Fig. 3 are defined to study the
localised $C\\!P$ asymmetries. The rescattering region [31] is defined in the
Dalitz plot in the two-kaon or two-pion invariant mass range
1.1–2.25$\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ for
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ due to the presence of
the $\phi(1020)$ resonance and
1.0–2.25$\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ for the other three
channels, as listed in Table 3. The data are studied in intervals of the other
two-body invariant mass. The vertical axis projection depends on the decay
channel, with two regions defined for
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$,
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$, and one for
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$. There are also two
other regions covering higher masses for
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays. All regions
for the different decay channels are defined in Table 3. The invariant-mass
fits used to compute $A_{{C\\!P}}$ in each region follow the same fit
procedure as described in Sec. 4, where the efficiency ratios are computed
separately per region.

Table 3: Definitions of phase-space regions in units of $\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$. In the ${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ channel, the $\phi(1020)$ vector meson is excluded by requiring $m^{2}(K^{+}K^{-})_{\textrm{low}}>1.1\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$. |  | ${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ |
---|---|---|---
Region 1 |  $\quad 1<m^{2}(\pi^{+}\pi^{-})_{\textrm{low}}<$ 2.25 | and | $3.5<m^{2}(\pi^{+}\pi^{-})_{\textrm{high}}<16$
Region 2 |  $\quad 1<m^{2}(\pi^{+}\pi^{-})_{\textrm{low}}<$ 2.25 | and | $\;16<m^{2}(\pi^{+}\pi^{-})_{\textrm{high}}<23$
Region 3 |  $4<m^{2}(\pi^{+}\pi^{-})_{\textrm{low}}<$ 15 | and | $\quad 4<m^{2}(\pi^{+}\pi^{-})_{\textrm{high}}<16$
|  | ${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ |
Region 1 | $1<m^{2}(\pi^{+}\pi^{-})<2.25$ | and | $\;3.5<m^{2}(K^{+}\pi^{-})<19.5$
Region 2 | $1<m^{2}(\pi^{+}\pi^{-})<2.25$ | and | $19.5<m^{2}(K^{+}\pi^{-})<25.5$
|  | ${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ |
Region 1 | $\quad 1<m^{2}(K^{+}K^{-})<2.25$ | and | $4<m^{2}(K^{+}\pi^{-})<19$
Region 2 | $4<m^{2}(K^{+}K^{-})<25$ | and | $3<m^{2}(K^{+}\pi^{-})<16$
|  | ${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ |
Region 1 | $1.1<m^{2}(K^{+}K^{-})_{\textrm{low}}<2.25$ | and | $\;4<m^{2}(K^{+}K^{-})_{\textrm{high}}<17$
Region 2 | $1.1<m^{2}(K^{+}K^{-})_{\textrm{low}}<2.25$ | and | $17<m^{2}(K^{+}K^{-})_{\textrm{high}}<23$

Most of the rescattering regions give $C\\!P$ asymmetry values in excess of at
least five Gaussian standard deviations. For
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ decays, the
rescattering region when analysed in the
$m^{2}(\pi^{+}\pi^{-})_{\textrm{high}}$ projection presents a $C\\!P$
asymmetry that is positive in region 1 and negative in region 2, as shown in
Fig. 4.

For ${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ decays, the
two regions in the $m^{2}(K^{+}\pi^{-})$ axis reveal a flip of the asymmetry
sign as shown in Fig. 5. Figure 6 illustrates the $m^{2}(K^{+}\pi^{-})$ axis
projection for the rescattering region for
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays, where a
nearly constant $C\\!P$ asymmetry is observed. The
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays also reveal
$C\\!P$ asymmetries in the $m^{2}(K^{+}K^{-})_{\textrm{high}}$ projections in
the rescattering region, as shown in Fig. 7. Here, another change in the
asymmetry sign is observed, in an opposite direction with respect to
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ decays. Region 2
of the ${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decay is the
only rescattering projection that shows small asymmetry.

\begin{overpic}[width=212.47617pt]{Fig4a.pdf}
\put(90.0,51.0){\scriptsize{(a)}}
\end{overpic}\begin{overpic}[width=212.47617pt]{Fig4b.pdf}
\put(90.0,47.0){\scriptsize{(b)}}
\end{overpic}\begin{overpic}[width=212.47617pt]{Fig4c.pdf}
\put(90.0,47.0){\scriptsize{(c)}} \end{overpic}

Figure 4: (a) Projection of $m^{2}(\pi^{+}\pi^{-})_{\textrm{high}}$ for the
rescattering region (black region inside the Dalitz plot) with the
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ mass fits for
(b) region 1 and (c) region 2 (${{B}^{-}}$ candidates on the left). Regions
are separated by a black vertical line in (a), see Table 3.

\begin{overpic}[width=221.14381pt]{Fig5a.pdf}
\put(88.0,51.0){\scriptsize{(a)}}
\end{overpic}\begin{overpic}[width=212.47617pt]{Fig5b.pdf}
\put(90.0,47.0){\scriptsize{(b)}}
\end{overpic}\begin{overpic}[width=212.47617pt]{Fig5c.pdf}
\put(90.0,47.0){\scriptsize{(c)}} \end{overpic}

Figure 5: (a) $m^{2}(K^{+}\pi^{-})$ projection for the rescattering region
with the ${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ mass fits
for (b) region 1 and (c) region 2 (${{B}^{-}}$ on the left). Regions are
separated by a black vertical line in (a).

\begin{overpic}[width=203.80193pt]{Fig6a.pdf}
\put(88.0,51.0){\scriptsize{(a)}}
\end{overpic}\begin{overpic}[width=221.14381pt]{Fig6b.pdf}
\put(90.0,47.0){\scriptsize{(b)}} \end{overpic}

Figure 6: (a) $m^{2}(K^{+}\pi^{-})$ projection for the rescattering region
with the ${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ mass fits
for (b) region 1 (${{B}^{-}}$ on the left).

\begin{overpic}[width=221.14381pt]{Fig7a.pdf}
\put(88.0,51.0){\scriptsize{(a)}}
\end{overpic}\begin{overpic}[width=212.47617pt]{Fig7b.pdf}
\put(90.0,47.0){\scriptsize{(b)}}
\end{overpic}\begin{overpic}[width=212.47617pt]{Fig7c.pdf}
\put(90.0,47.0){\scriptsize{(c)}} \end{overpic}

Figure 7: (a) $m^{2}(K^{+}K^{-})_{\textrm{high}}$ projection for the
rescattering region with the
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ mass fits for (b)
region 1 and (c) region 2 (${{B}^{-}}$ on the left). Regions are separated by
a black vertical line in (a).

The high-mass regions of
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ as defined in Table
3 are also analysed. For the latter mode, presented in Fig. 8, no significant
asymmetry is observed in the integrated asymmetry. However a clear
$\chi_{c0}(1P)$ contribution around
$11.6\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ is visible. For
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ decays, a large
asymmetry is observed, shown in Fig. 9, as well as a $\chi_{c0}(1P)$
contribution around $11.6\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$. The
$\chi_{c0}(1P)$ contribution comes from the $B$ decay, as no such structure
was observed in the invariant mass sidebands. The asymmetry results from the
invariant mass fits for all regions are summarised in Table 4.

\begin{overpic}[width=203.80193pt]{Fig8a.pdf}
\put(88.0,51.0){\scriptsize{(a)}}
\end{overpic}\begin{overpic}[width=221.14381pt]{Fig8b.pdf}
\put(90.0,47.0){\scriptsize{(b)}} \end{overpic}

Figure 8: (a) $m^{2}(K^{+}K^{-})$ projection in the high-mass region (region
2) for ${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays
showing the $\chi_{c0}$ contribution. The mass fit for this region
(${{B}^{-}}$ on the left) is shown in (b).

\begin{overpic}[width=203.80193pt]{Fig9a.pdf}
\put(88.0,51.0){\scriptsize{(a)}}
\end{overpic}\begin{overpic}[width=221.14381pt]{Fig9b.pdf}
\put(90.0,47.0){\scriptsize{(b)}} \end{overpic}

Figure 9: (a) $m^{2}(\pi^{+}\pi^{-})_{\textrm{high}}$ projection in the high-mass region (region 3) for ${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ decays. The mass fit for this region (${{B}^{-}}$ on the left) is shown in (b). Table 4: Signal yield ($N_{\text{sig}}$), raw asymmetry ($A_{\text{raw}}$) and $A_{{C\\!P}}$ corrected for production and detection asymmetries. The $A_{{C\\!P}}$ uncertainties are statistical, systematic and associated to the control channel, respectively.  ${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ | $N_{\text{sig}}$ | $A_{\text{raw}}$ | $A_{{C\\!P}}$
---|---|---|---
Region 1 | $14\,330\pm 150$ |  $+$0.309 $\pm$ 0.009 |  $+$0.303 $\pm$ 0.009 $\pm$ 0.004 $\pm$ 0.003
Region 2 | $\,\;4\,850\pm 130$ |  $-$0.287 $\pm$ 0.017 |  $-$0.284 $\pm$ 0.017 $\pm$ 0.007 $\pm$ 0.003
Region 3 | $2\,270\pm 60$ |  $+$0.747 $\pm$ 0.027 |  $+$0.745 $\pm$ 0.027 $\pm$ 0.018 $\pm$ 0.003
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ |  |  |
Region 1 | $41\,980\pm 280$ |  $+$0.201 $\pm$ 0.005 |  $+$0.217 $\pm$ 0.005 $\pm$ 0.005 $\pm$ 0.003
Region 2 | $27\,040\pm 250$ |  $-$0.149 $\pm$ 0.007 |  $-$0.145 $\pm$ 0.007 $\pm$ 0.006 $\pm$ 0.003
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ |  |  |
Region 1 | $11\,430\pm 170$ |  $-$0.363 $\pm$ 0.010 |  $-$0.358 $\pm$ 0.010 $\pm$ 0.014 $\pm$ 0.003
Region 2 | $\,\;2\,600\pm 120$ |  $+$0.075 $\pm$ 0.031 |  $+$0.097 $\pm$ 0.031 $\pm$ 0.005 $\pm$ 0.003
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ |  |  |
Region 1 | $76\,020\pm 350$ |  $-$0.189 $\pm$ 0.004 |  $-$0.178 $\pm$ 0.004 $\pm$ 0.004 $\pm$ 0.003
Region 2 | $37\,440\pm 320$ |  $+$0.030 $\pm$ 0.005 |  $+$0.043 $\pm$ 0.005 $\pm$ 0.004 $\pm$ 0.003

##  8 Partial width difference

A relation between $C\\!P$ violation in
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$, as well as that
between ${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$, based on U-spin
symmetry [7, 8, 9], is investigated. This symmetry is an SU(2) subgroup of the
flavour SU(3) group, under which the ($d$,$s$) pairs of quarks form a doublet,
similar to ($u$,$d$) in isospin symmetry.

U-spin symmetry predicts the relationship between the differences of the
partial decay widths for the decays
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ to be

$\Delta\Gamma({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}})=-\Delta\Gamma({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}).$
(6)

A similar prediction is made between the symmetric decay channels
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$,

$\Delta\Gamma({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}})=-\Delta\Gamma({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}).$
(7)

These relations state that the difference in the number of events associated
to $C\\!P$ violation must be the same, independently of the branching fraction
of each decay.

The above relations can be rewritten in terms of the measured integrated
asymmetries, branching fractions and the $B$ meson lifetime. For the
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decay the result is

$\Delta\Gamma({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}})=\frac{A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}})\mathcal{B}({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}})}{\tau(B^{\pm})}\,,$
(8)

where $\tau(B^{\pm})$ is the $B^{\pm}$ lifetime and $\mathcal{B}$ is the
average branching fraction. Using the measured values of the integrated
asymmetry and branching fractions [12], the U-spin relationship between
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ is

$\frac{A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}})\mathcal{B}({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}})}{A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}})\mathcal{B}({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}})}=-0.92\pm
0.18\,.$

Similarly, the pair of
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays give the result

$\frac{A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}})\mathcal{B}({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}})}{A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}})\mathcal{B}({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}})}=-1.06\pm
0.08\,.$

Both results are consistent with $-1$ (uncertainties are statistical only), as
predicted by the U-spin symmetry approach. Furthermore, Refs. [10, 11] also
predicted the relations
$\Delta\Gamma({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}})=\Delta\Gamma({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}})$
and
$\Delta\Gamma({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}})=\Delta\Gamma({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}})$.
In this work the ratios are measured to be (uncertainties are statistical
only):

$\displaystyle\frac{A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}})\mathcal{B}({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}})}{A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}})\mathcal{B}({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}})}=0.47\pm
0.04\,,$
$\displaystyle\frac{A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}})\mathcal{B}({{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}})}{A_{{C\\!P}}({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}})\mathcal{B}({{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}})}=0.48\pm
0.09\,.$

The results presented here reveal a central value with an opposite sign with
respect to the U-spin prediction of $-1$. However, one may not forget that the
U-spin symmetry prediction occurs in a scenario with momentum-independent
amplitudes, in which no hadronic final-state interaction where two pairs of
channels are coupled through the $\pi\pi\leftrightarrow KK$ interaction is
considered. When those contributions are included, one may find ratios that
differ from $-1$ without necessarily claiming the observation of U-spin
symmetry breaking. All these implications are further discussed in Refs. [10,
11].

##  9 Summary and conclusions

In summary, the inclusive $C\\!P$ asymmetries of the
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$,
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$,
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ charmless three-
body decays are measured with data corresponding to 5.9$\text{\,fb}^{-1}$ of
integrated luminosity, collected by the LHCb detector between 2015 and 2018.
Significant inclusive $C\\!P$ asymmetries are found for the latter three $B$
decay channels, the last two observed for the first time.

The asymmetry in localised regions of phase space shows that the $C\\!P$
asymmetries are not uniformly distributed in the phase space, with positive
and negative $A_{{C\\!P}}$ appearing in the same charged $B$ decay channel.
The results of the previous LHCb analysis [6] are confirmed with high
localised $C\\!P$ asymmetries in several regions of the Dalitz plot. It is
also confirmed that, in the $\pi\pi\rightarrow KK$ rescattering region,
between 1 and 2.25$\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ in both
$\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ invariant masses, significant $C\\!P$
violation is present in the four analysed channels. For the
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$,
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays, an
$A_{{C\\!P}}$ sign change is observed across the phase space.

Additionally, an indication of the $\chi_{c0}(1P)$ resonance is seen in the
high ${\pi}^{+}$ ${\pi}^{-}$ and ${K}^{+}$ ${K}^{-}$ invariant mass regions of
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ and
${{{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}}$ decays for the first
time. There is an indication of $C\\!P$ violation involving this resonance, as
predicted several years ago in Refs. [33, 34]. The combination of the present
$A_{{C\\!P}}$ results with the recent LHCb relative branching fraction results
for these decays, shows good agreement with the U-spin symmetry proposed in
Refs. [7, 8, 9].

## References

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# Linear preservers on idempotents of Fourier algebras

Ying-Fen Lin and Shiho Oi  Mathematical Sciences Research Centre, Queen’s
University Belfast, Belfast, BT7 1NN, U.K.
Email<EMAIL_ADDRESS>Department of Mathematics, Faculty of Science, Niigata
University, Niigata 950-2181, Japan
Email<EMAIL_ADDRESS>
###### Abstract.

In this article, we give a representation of bounded complex linear operators
which preserve idempotent elements on the Fourier algebra of a locally compact
group. When such an operator is moreover positive or contractive, we show that
the operator is induced by either a continuous group homomorphism or a
continuous group anti-homomorphism. If the groups are totally disconnected,
bounded homomorphisms on the Fourier algebra can be realised by the idempotent
preserving operators.

###### Key words and phrases:

Fourier algebras, idempotents, linear preservers, quotient groups

###### 2020 Mathematics Subject Classification:

47B48, 47B49, 43A22

##  1\. Introduction

Let $G$ be a locally compact Hausdorff group. The Fourier-Stieltjes $B(G)$ and
the Fourier algebras $A(G)$ of $G$ were introduced by Eymard in his
celebrating paper [12]. Recall that $B(G)$ is the linear combination of all
continuous positive definite functions on $G$, as a Banach space, $B(G)$ is
naturally isometric to the predual of $W^{*}(G)$, the von Neumann algebras
generated by the universal representations $\omega_{G}$ of $G$. Moreover, it
is a commutative Banach $*$-algebra with respect to pointwise multiplication
and complex conjugation. The Fourier algebra $A(G)$ is the closed ideal of
$B(G)$ generated by the functions with compact supports. As a Banach space,
$A(G)$ is isometric to the predual of the group von Neumann algebra
$\mathop{\rm VN}(G)$, the von Neumann algebra generated by the left regular
representations $\lambda_{G}$ of $G$. It is well known that $A(G)$ is regular,
semisimple, and the Fourier and the Fourier-Stieltjes algebras are both
subalgebras of $C_{b}(G)$, the algebra of continuous bounded functions on $G$.

Takesaki and Tatsuuma in [25] showed that there is a one-to-one correspondence
between compact subgroups of $G$ and non-zero right invariant closed self-
adjoint subalgebras of $A(G)$. As a refinement, Bekka, Lau and Schlichting in
[3] studied non-zero, closed, invariant $*$-subalgebras of $A(G)$. They showed
that these spaces are the Fourier algebras $A(G/K)$ of the quotient group
$G/K$ for some compact normal subgroup $K$ of $G$. On the other hand, Forrest
[13] introduced the Fourier algebra $A(G/K)$ of the left coset space $G/K$,
where $K$ is a compact (not necessary to be normal) subgroup of the locally
compact group $G$. This algebra can simultaneously be viewed as an algebra of
functions on $G/K$ and as the subalgebra of $A(G)$ consisting of functions in
$A(G)$ which are constants on left cosets of $K$. Note that $A(G/K)$ is
regular and semisimple, the spectrum $\sigma(A(G/K))$ is $G/K$, and it is a
norm closed left translation invariant $*$-subalgebra of $A(G)$.

A longstanding question in harmonic analysis is to determine all homomorphisms
of Fourier or Fourier-Stieltjes algebras of any locally compact groups. For
any pair of locally compact abelian groups $G$ and $H$, Cohen [7]
characterised all bounded homomorphisms from the group algebra $L^{1}(G)$ to
the measure algebra $M(H)$. In doing so, he made use of a profound discovery
of his characterisation of idempotent measures on the groups. Cohen’s results
were generalized by Host in [15], where he discovered the general form of
idempotents in the Fourier-Stieltjes algebras, and characterised bounded
homomorphisms from $A(G)$ to $B(H)$ when the group $G$ has an abelian subgroup
of finite index. Further generalizations were made in [16, 17] for any locally
compact amenable group $G$, where completely bounded homomorphisms from $A(G)$
into $B(H)$ were characterised by continuous piecewise affine maps (see also
[8]). Most general results were given by Le Pham in [22], he determined all
contractive homomorphisms from $A(G)$ into $B(H)$ for any locally compact
groups $G$ and $H$.

To describe idempotent elements in the Fourier-Stieltjes and the Fourier
algebras, we first recall some terminologies. Let $G$ be a group and $K$ be a
subgroup of $G$, we see that $Ks=ss^{-1}Ks$ for any $s\in G$, which means that
we need not distinguish between left and right cosets of the group $G$. The
_coset ring_ of $G$, denoted $\Omega(G)$, is the smallest ring of subsets of
$G$ containing all cosets of subgroups of $G$. We denote
$\Omega_{\text{o}}(G)$ the ring of subsets generated by open cosets of $G$,
and similarly, $\Omega_{\text{o}}^{\text{c}}(G)$ the ring of subsets generated
by compact open cosets of $G$. By [15], idempotents in the Fourier-Stieltjes
algebra $B(G)$ are the indicator functions $1_{F}$ of an element $F$ of
$\Omega_{\text{o}}(G)$. Let $I_{B}(G)$ be the set of all idempotent elements
in $B(G)$. We denote the closure of the span of $I_{B}(G)$ by $B_{I}(G)$. From
[18, Proposition 1.1], we have that

$\displaystyle A(G)\cap
I_{B}(G)=\\{1_{Y}:Y\in\Omega_{\text{o}}^{\text{c}}(G)\\},$

which gives rise idempotents in $A(G)$, denoted by $I(G)$. Let $A_{I}(G)$ be
the subalgebra of $A(G)$ generated by $I(G)$. Note that Ilie and Spronk [17]
showed that $1_{F}$ is an idempotent in $B(G)$ with $\|1_{F}\|_{B(G)}=1$ if
and only if $F$ is an open coset in $G$, however, there are idempotents with
small norms [24] or with large norms [2]. Moreover, the existence of
idempotents of arbitrarily large norm implies the existence of homomorphisms
of arbitrarily large norm (see [2] for details). Thus, idempotent elements
play an essential role in the study of homomorphisms on Fourier algebras. It
is of its own interest to study the norms of idempotent elements in Fourier-
Stieltjes and Fourier algebras, for our purpose we will focus solely on
operators which preserve idempotents.

In the rich literature of linear preservers, there are many works that study
linear maps $T$ on spaces $X$ which preserve some subsets $S$ of $X$, i.e.,
$T(S)\subset S$. Dieudonné in [9] studied semi-linear maps on
$M_{n}(\mathbb{K})$, the algebra of $n\times n$ matrices over a field
$\mathbb{K}$, which preserve the set of all singular matrices. After that,
many mathematicians considered linear maps on $M_{n}(\mathbb{K})$ that
preserve subsets of matrices with different properties (e.g. [19, 23, 10, 4]
to name a few). In [5], it is shown that every complex linear map $T$ on
$M_{n}(\mathbb{C})$ which preserves the set of all idempotents is either an
inner automorphism or an inner anti-automorphism. In addition, in [6] linear
maps on $M_{n}(\mathbb{C})$ which send potent matrices (that is, matrices $A$
satisfy $A^{r}=A$ for some integer $r\geq 2$) to potent matrices were
characterised. Since then, the studies of idempotent preserving maps have
attracted considerable interest, see, e.g. [11, 14]. Recently, in [21] the
authors proved that every additive map from the rational span of Hermitian
idempotents in a von Neumann algebra into the rational span of Hermitian
idempotents in a C*-algebra can be extended to a Jordan $*$-homomorphism.

In this paper, we study bounded linear operators from $A(G)$ into $B(H)$ which
send idempotents to idempotents. We show that such an operator will give rise
an algebraic homomorphism on $A_{I}(G)$. The algebra $A_{I}(G)$ will be our
main object of study, namely, we will characterise linear mappings defined on
the Fourier algebra $A(G)$ or on $A_{I}(G)$ which preserve $I(G)$. Moreover,
we show that when the groups are totally disconnected, idempotent preserving
operators will recover algebraic homomorphisms on the Fourier algebra.

##  2\. Main results

Let $G$ be a locally compact Hausdorff group and $K$ be a closed subgroup of
$G$. We will denote by $G/K$ the homogeneous space of left cosets of $K$. Let

$\displaystyle B(G:K):=\\{u\in B(G):u(xk)=u(x)\,\text{for all }x\in G,k\in
K\\},$

this is, functions in $B(G)$ which are constant on cosets of $K$, and

$\displaystyle A(G:K):=\\{u\in B(G:K):q(\mathop{\rm supp}(u))\text{ is compact
in }G/K\\}^{-_{B(G)}},$

where $\mathop{\rm supp}(u)$ is the support of $u$ in $G$ and $q$ is the
canonical quotient map from $G$ to $G/K$. If furthermore $K$ is a normal
subgroup, by [13, Proposition 3.2] we have that $B(G:K)$ and $A(G:K)$ are
isometrically isomorphic to the Fourier-Stieltjes and the Fourier algebras
$B(G/K)$ and $A(G/K)$, respectively. Note that $A(G:K)\cap A(G)\neq\\{0\\}$ if
and only if $K$ is compact.

Let $e$ be the identity of the group $G$, we denote the connected component of
$e$ by $G_{e}$ which is a closed normal subgroup of $G$, thus, $G/G_{e}$ is a
totally disconnected locally compact group. The following result about the
algebra $A_{I}(G)$ generated by idempotents of $A(G)$ in relation with
$A(G:G_{e})$ was given in [18], for the completion we give a short proof in
the paper.

######  Proposition 2.1.

[18, Proposition 1.1(ii)] If the connected component $G_{e}$ is compact then
$A_{I}(G)=A(G:G_{e})$, this is, $A_{I}(G)$ consists of all functions in $A(G)$
which are constant on cosets of $G_{e}$. In particular, if $G_{e}$ is not
compact then $A_{I}(G)=\\{0\\}$.

###### Proof.

Let $q_{G}:G\to G/G_{e}$ is the quotient map onto $G/G_{e}$. Since $G_{e}$ is
compact, via $u\mapsto u\circ q_{G}$ we have that $A(G/G_{e})$ is
isometrically isomorphic to $A(G:G_{e})$, which is a closed subalgebra of
$A(G)$. Thus,
$A_{I}(G)=\overline{\operatorname{span}}\\{1_{Y}:Y\in\Omega_{\text{o}}^{\text{c}}(G)\\}\subseteq
A(G:G_{e})$. Conversely, since $G/G_{e}$ is totally disconnected, we have that
the span of the idempotents of $A(G/G_{e})$ is dense [13, Theorem 5.3].
Moreover $A(G/G_{e})$ is isomorphic to $A(G:G_{e})$, thus, $A(G:G_{e})$ is
generated by idempotents of $A(G/G_{e})$, so $A(G:G_{e})\subseteq A_{I}(G)$. ∎

If the Fourier algebra which contains non-trivial idempotents, that is, the
connected component $G_{e}$ is compact, then by Proposition 2.1 there is an
isometric isomorphism from $A_{I}(G)$ onto $A(G/G_{e})$. More precisely, it
induces an isometric isomorphism $\varphi_{G}:A_{I}(G)\to A(G/G_{e})$ as

$\displaystyle\varphi_{G}(f)(q_{G}(a))=f(a)$ (1)

for any $f\in A_{I}(G)$ and $a\in G$, where $q_{G}:G\to G/G_{e}$ is the
quotient map onto $G/G_{e}$.

###  2.1. Idempotent preserving maps with $T(I(G))\subset I_{B}(H)$

Let $G$ and $H$ be two locally compact groups. We consider a bounded complex
linear map $T:A(G)\to B(H)$ which satisfies

$\displaystyle T(I(G))\subset I_{B}(H).$ (2)

For any
$f\in\operatorname{span}\\{1_{Y}:Y\in\Omega_{\text{o}}^{\text{c}}(G)\\}$,
there exists $\alpha_{i}\in\mathbb{C}$ and
$Y_{i}\in\Omega_{\text{o}}^{\text{c}}(G)$ such that
$f=\Sigma_{k=1}^{n}\alpha_{k}1_{Y_{k}}$. Thus, we have
$Tf=\Sigma^{n}_{k=1}\alpha_{k}T1_{Y_{k}}\in\operatorname{span}\\{1_{Y}:Y\in\Omega_{\text{o}}(H)\\}\subset
B(H)$. Let us recall that
$A_{I}(G)=\overline{\operatorname{span}}\\{1_{Y}:Y\in\Omega_{\text{o}}^{\text{c}}(G)\\}$
and
$B_{I}(H)=\overline{\operatorname{span}}\\{1_{Y}:Y\in\Omega_{\text{o}}(H)\\}$.
Since $T$ is a bounded map, we obtain $T(A_{I}(G))\subseteq B_{I}(H)\subset
B(H)$.

Our aim is to obtain a representation of such a map $T$ on $A_{I}(G)$. If
$I(G)=\\{0\\}$, then $A_{I}(G)=\\{0\\}$. Since $T$ is complex linear, we have
$T=0$ on $A_{I}(G)$. Thus without loss of generality, we can assume that the
Fourier algebra $A(G)$ have non-zero idempotent elements. Hence, the connected
component $G_{e}$ is always a compact normal subgroup of $G$. On the other
hand, we define the following map which will be used in sequel.

######  Definition 2.2.

Let $G$ be a locally compact Hausdorff group. Using the axiom of choice, let
$S$ be a set of representatives of the cosets of $\mathop{G/G_{e}}$, that is
$G=\bigsqcup_{a\in S}aG_{e}$. Then we define a map $[\ \cdot\
]_{\mathop{G/G_{e}}}$ from $\mathop{G/G_{e}}$ onto $S$ by

$[aG_{e}]_{\mathop{G/G_{e}}}=a$

for any $a\in S$.

We first have the following observations concerning the operator satisfying
(2).

######  Lemma 2.3.

The map $T$ preserves the disjointness of idempotents. This is, $Tf\cdot Tg=0$
for any $f,g\in I(G)$ with $f\cdot g=0$.

###### Proof.

Let $f,g\in I(G)$ such that $f\cdot g=0$. Then we have $(f+g)^{2}=f+g$. Thus
$f+g\in I(G)$. By the assumption, $Tf$, $Tg$ and $T(f+g)\in I_{B}(H)$. Since
we have $(T(f+g))^{2}=Tf+Tg$, we get $Tf\cdot Tg=0$. ∎

######  Definition 2.4.

We define $\Phi:\mathop{A(G/G_{e})}\to B(H)$ by

$\Phi(f)=T\circ\varphi_{G}^{-1}(f)$

for any $f\in\mathop{A(G/G_{e})}$, where $\varphi_{G}$ is given in (1).

Then $\Phi$ is a bounded complex linear operator from $\mathop{A(G/G_{e})}$
into $B(H)$. In order to achieve our main result, we consider the dual map
$\Phi^{*}:W^{*}(H)\to\mathop{\rm VN}(G/G_{e})$ and have the following lemmas.

######  Lemma 2.5.

Let $\lambda\in\mathop{\rm VN}(\mathop{G/G_{e}})$ and $a\in\mathop{G/G_{e}}$.
Suppose that $a\in\mathop{\rm supp}\lambda$. Then for every neighbourhood $V$
of $a$ in $\mathop{G/G_{e}}$, there exists $h\in I(\mathop{G/G_{e}})$ such
that $\mathop{\rm supp}h\subset V$ and $\langle\lambda,h\rangle\neq 0$.

###### Proof.

Since $\mathop{G/G_{e}}$ is totally disconnected, every neighbourhood of the
identity contains an open compact subgroup. As $a^{-1}V$ is a neighbourhood of
the identity, there exists an open compact subgroup $G_{a}$ in
$\mathop{G/G_{e}}$ such that $G_{a}\subset a^{-1}V$. Thus $aG_{a}\subset V$.
Since $aG_{a}$ is a compact open coset in $\mathop{G/G_{e}}$, we have that
$1_{aG_{a}}\in\mathop{A(G/G_{e})}$ is an idempotent with norm $1$. Since
$a\in\mathop{\rm supp}\lambda$, there is $g\in\mathop{A(G/G_{e})}$ such that
$\mathop{\rm supp}g\subset aG_{a}$ and $\langle\lambda,g\rangle\neq 0$. Put
$\delta=|\langle\lambda,g\rangle|$. As $\varphi_{G}^{-1}(g)\in A_{I}(G)$,
there are $\alpha_{i}\in\mathbb{C}$ and $f_{i}\in I(G)$ such that
$\|\varphi_{G}^{-1}(g)-\sum^{n}_{i=1}\alpha_{i}f_{i}\|<\delta/\|\lambda\|$ for
some $n\in\mathbb{N}$. Since $\varphi_{G}$ is an isometric isomorphism, we
have $\|g-\sum^{n}_{i=1}\alpha_{i}\varphi_{G}(f_{i})\|<\delta/\|\lambda\|$ and
$\varphi_{G}(f_{i})\in I(\mathop{G/G_{e}})$. Then we obtain

$\displaystyle
1_{aG_{a}}(g-\sum^{n}_{i=1}\alpha_{i}\varphi_{G}(f_{i}))=g-\sum^{n}_{i=1}\alpha_{i}1_{aG_{a}}\varphi_{G}(f_{i}),$

thus,

$\displaystyle\|g-\sum^{n}_{i=1}\alpha_{i}1_{aG_{a}}\varphi_{G}(f_{i})\|\leq\|g-\sum^{n}_{i=1}\alpha_{i}\varphi_{G}(f_{i})\|<\frac{\delta}{\|\lambda\|}.$

Suppose for every $1\leq i\leq n$, we have
$\langle\lambda,1_{aG_{a}}\varphi_{G}(f_{i})\rangle=0$. Then

$\begin{split}|\langle\lambda,g\rangle|&=|\langle\lambda,g\rangle-\sum^{n}_{i=1}\alpha_{i}\langle\lambda,1_{aG_{a}}\varphi_{G}(f_{i})\rangle|\\\
&=|\langle\lambda,(g-\sum^{n}_{i=1}\alpha_{i}1_{aG_{a}}\varphi_{G}(f_{i}))\rangle|\\\
&\leq\|\lambda\|\|g-\sum^{n}_{i=1}\alpha_{i}1_{aG_{a}}\varphi_{G}(f_{i})\|\\\
&<\|\lambda\|\frac{\delta}{\|\lambda\|}=\delta.\end{split}$

This implies that $|\langle\lambda,g\rangle|<\delta$, which is a
contradiction. Therefore, there is an $i_{0}\in\\{1,\cdots,n\\}$ such that

$\langle\lambda,1_{aG_{a}}\varphi_{G}(f_{i_{0}})\rangle\neq 0.$

We also have $\mathop{\rm supp}(1_{aG_{a}}\varphi_{G}(f_{i_{0}}))\subset V$
and $1_{aG_{a}}\varphi_{G}(f_{i_{0}})\in I(\mathop{G/G_{e}})$, the proof is
thus completed. ∎

######  Proposition 2.6.

For any $a\in H$, there exist uniquely $b\in G/G_{e}$ and
$\alpha\in\mathbb{C}$ such that
$\Phi^{*}(\omega_{H}(a))=\alpha\lambda_{\mathop{G/G_{e}}}(b)$.

###### Proof.

Suppose there were $b_{1},b_{2}\in\mathop{G/G_{e}}$ such that $b_{1},b_{2}$
were both in $\mathop{\rm supp}(\Phi^{*}(\omega_{H}(a)))$. Since $G_{e}$ is a
closed subgroup of $G$, the quotient group $\mathop{G/G_{e}}$ is Hausdorff.
Thus, there are neighbourhoods $V_{b_{1}}$ and $V_{b_{2}}$ of $b_{1}$ and
$b_{2}$, respectively, in $\mathop{G/G_{e}}$ such that $V_{b_{1}}\cap
V_{b_{2}}=\emptyset$. By Lemma 2.5, there are $h_{i}\in I(\mathop{G/G_{e}})$,
for $i=1,2$, such that $\mathop{\rm supp}h_{i}\subset V_{b_{i}}$ and
$\langle\Phi^{*}(\omega_{H}(a)),h_{i}\rangle\neq 0$. As $V_{b_{1}}\cap
V_{b_{2}}=\emptyset$, we get $h_{1}h_{2}=0$. Since $\varphi_{G}$ is an
isomorphism, we have $\varphi_{G}^{-1}(h_{i})\in I(G)$, for $i=1,2$, and
$\varphi_{G}^{-1}(h_{1})\cdot\varphi_{G}^{-1}(h_{2})=\varphi_{G}^{-1}(h_{1}h_{2})=0$.
By Lemma 2.3, we have $T(\varphi_{G}^{-1}(h_{1}))\cdot
T(\varphi_{G}^{-1}(h_{2}))=0$. On the other hand, we obtain

$0\neq\langle\Phi^{*}(\omega_{H}(a)),h_{1}\rangle=\Phi(h_{1})(a)=T\circ\varphi_{G}^{-1}(h_{1})(a)=T(\varphi_{G}^{-1}(h_{1}))(a),$

and

$0\neq\langle\Phi^{*}(\omega_{H}(a)),h_{2}\rangle=\Phi(h_{2})(a)=T\circ\varphi_{G}^{-1}(h_{2})(a)=T(\varphi_{G}^{-1}(h_{2}))(a).$

Therefore,

$T(\varphi_{G}^{-1}(h_{1}))\cdot T(\varphi_{G}^{-1}(h_{2}))\neq 0,$

this is a contradiction. Since $\mathop{\rm
supp}(\Phi^{*}(\omega_{H}(a)))\neq\emptyset$, there is uniquely
$b\in\mathop{G/G_{e}}$ such that $\mathop{\rm
supp}(\Phi^{*}(\omega_{H}(a)))=\\{b\\}$. Consequently, by [20, Corollary
2.5.9], there is an $\alpha\in\mathbb{C}$ such that
$\Phi^{*}(\omega_{H}(a))=\alpha\lambda_{\mathop{G/G_{e}}}(b)$. ∎

For any $a\in H$, by Proposition 2.6, there are unique $b\in G/G_{e}$ and
$\alpha\in\mathbb{C}$ such that
$\Phi^{*}(\omega_{H}(a))=\alpha\lambda_{\mathop{G/G_{e}}}(b)$, thus we have

$\Phi(f)(a)=\alpha f(b),$

for any $f\in\mathop{A(G/G_{e})}$. We define $\phi:H\to\mathbb{C}$ by
$\alpha=\phi(a)$. We also define $\psi:H\to\mathop{G/G_{e}}$ by $b=\psi(a)$.
Then we get

$\Phi(f)(a)=\phi(a)f(\psi(a)),$ (3)

for any $f\in\mathop{A(G/G_{e})}$ and $a\in H$.

For any $h\in I(G)$, since we have $\Phi(\varphi_{G}(h))=T(h)\in I_{B}(H)$, we
obtain that

$\begin{split}(\Phi(\varphi_{G}(h)))^{2}&=T(h)T(h)=T(h)=\Phi(\varphi_{G}(h)).\\\
\end{split}$

On the other hand, since
$(\varphi_{G}(h))^{2}=\varphi_{G}(h^{2})=\varphi_{G}(h)$ in
$A(\mathop{G/G_{e}})$, we obtain that

$\phi(a)^{2}\varphi_{G}(h)(\psi(a))=\phi(a)\varphi_{G}(h)(\psi(a))$ (4)

for any $h\in I(G)$ and $a\in H$.

######  Lemma 2.7.

For any $a\in H$, there is an idempotent $1_{\psi(a)G_{0}}$ of
$A(\mathop{G/G_{e}})$ where $\psi(a)G_{0}$ is an open compact neighbourhood of
$\psi(a)$.

###### Proof.

As $\mathop{G/G_{e}}$ is totally disconnected, there is an open compact
subgroup $G_{0}$ in $\mathop{G/G_{e}}$. For any $\psi(a)\in\mathop{G/G_{e}}$,
$\psi(a)G_{0}$ is a compact open coset in $\mathop{G/G_{e}}$, hence
$1_{\psi(a)G_{0}}$ is an idempotent of $A(\mathop{G/G_{e}})$ with norm $1$. ∎

######  Lemma 2.8.

The map $\Phi:A(\mathop{G/G_{e}})\to B(H)$ is an algebraic homomorphism.

###### Proof.

Let $a\in H$. By Lemma 2.7, there is an idempotent $1_{\psi(a)G_{0}}$ of
$A(\mathop{G/G_{e}})$. Since $\varphi_{G}:A_{I}(G)\to A(\mathop{G/G_{e}})$ is
surjective, there is $f\in A_{I}(G)$ such that
$\varphi_{G}(f)=1_{\psi(a)G_{0}}$. Moreover, we have that
$f^{2}=(\varphi_{G}^{-1}(1_{\psi(a)G_{0}}))^{2}=\varphi_{G}^{-1}(1_{\psi(a)G_{0}})=f$,
this implies that $f\in I(G)$. Thus by (4), we have

$\phi(a)^{2}=\phi(a)^{2}1_{\psi(a)G_{0}}(\psi(a))=\phi(a)^{2}\varphi_{G}(f)(\psi(a))\\\
=\phi(a)\varphi_{G}(f)(\psi(a))=\phi(a)1_{\psi(a)G_{0}}(\psi(a))=\phi(a).$

Since $a\in H$ is arbitrary, we have

$\phi^{2}=\phi$ (5)

on $H$, thus we get $\phi:H\to\\{0,1\\}$. In addition, for any $f,g\in
A(\mathop{G/G_{e}})$ and $a\in H$, we have

$\Phi(fg)(a)=\phi(a)(fg)(\psi(a))=\phi(a)^{2}(fg)(\psi(a))\\\
=\phi(a)f(\psi(a))\phi(a)g(\psi(a))=(\Phi(f)\Phi(g))(a).$

Hence $\Phi$ is an algebraic homomorphism from $A(\mathop{G/G_{e}})$ into
$B(H)$. ∎

######  Lemma 2.9.

The map $\psi:\phi^{-1}(1)\to\mathop{G/G_{e}}$ is continuous.

###### Proof.

For any $a_{0}\in\phi^{-1}(1)\subset H$, let $U$ be an open neighbourhood of
$\psi(a_{0})$ in $\mathop{G/G_{e}}$. Then there is
$f_{0}\in\mathop{A(G/G_{e})}$ such that

$f_{0}(\psi(a_{0}))=1\quad\text{and}\quad f_{0}(b)=0\,\,\text{ for
}b\in(\mathop{G/G_{e}})\setminus U.$

Let $(a_{\lambda})_{\lambda}\subseteq\phi^{-1}(1)$ be a net such that
$a_{\lambda}\to a_{0}$. As $\Phi(f_{0})\in B(H)$, $\Phi
f_{0}(a_{\lambda})\to\Phi f_{0}(a_{0})=f_{0}(\psi(a_{0}))=1$. There is an
$\lambda_{0}$ such that if $\lambda\geq\lambda_{0}$ then $|\Phi
f_{0}(a_{\lambda})|>\frac{1}{2}$. Since $\Phi
f_{0}(a_{\lambda})=f_{0}(\psi(a_{\lambda}))$, we have $\psi(a_{\lambda})\in U$
provided $\lambda\geq\lambda_{0}$. Thus $\psi$ is continuous on
$\phi^{-1}(1)$. ∎

######  Lemma 2.10.

The set $\phi^{-1}(1)$ is an open subset of $H$.

###### Proof.

Let $a\in\phi^{-1}(1)$ be arbitrary. By Lemma 2.7, there is an idempotent
$1_{\psi(a)G_{0}}$ of $A(\mathop{G/G_{e}})$ where $\psi(a)G_{0}$ is an open
compact neighbourhood of $\psi(a)$. Since $\Phi(1_{\psi(a)G_{0}})\in
B(H)\subset C_{b}(H)$, there exists an open neighbourhood $V$ of $a$ in $H$
such that if $b\in V$ then

$|\Phi(1_{\psi(a)G_{0}})(a)-\Phi(1_{\psi(a)G_{0}})(b)|\leq\frac{1}{2}.$

We have

$|1-\phi(b)1_{\psi(a)G_{0}}(\psi(b))|=|\Phi(1_{\psi(a)G_{0}})(a)-\Phi(1_{\psi(a)G_{0}})(b)|\leq\frac{1}{2}.$

Since either $\phi(b)1_{\psi(a)G_{0}}(\psi(b))=1$ or
$\phi(b)1_{\psi(a)G_{0}}(\psi(b))=0$, this implies that

$\phi(b)1_{\psi(a)G_{0}}(\psi(b))=1.$

Hence we have $\phi(b)=1$ for any $b\in V$. Thus $V\subset\phi^{-1}(1)$. It
follows that $\phi^{-1}(1)$ is an open subset of $H$. ∎

######  Theorem 2.11.

Let $G$ and $H$ be two locally compact Hausdorff groups, and $T:A(G)\to B(H)$
be a bounded complex linear operator. Suppose $T$ satisfies that
$T(I(G))\subset I_{B}(H)$. Then there are an open subset $U$ of $H$ and a
continuous map $\psi$ from $U$ into $\mathop{G/G_{e}}$ such that

$Tf(a)=\begin{cases}f([\psi(a)]_{G_{e}})\quad&\text{if $a\in U$,}\\\
0&\text{if $a\in H\setminus U$,}\end{cases}$ (6)

for any $f\in A_{I}(G)$ and $a\in H$.

###### Proof.

Let $U=\phi^{-1}(1)$. By Lemma 2.10, $U$ is an open subset of $H$. Moreover,
Lemma 2.9 shows that $\psi:U\to\mathop{G/G_{e}}$ is a continuous map. Applying
(3), for any $f\in A_{I}(G)$ and $a\in H$, we have

$Tf(a)=\Phi(\varphi_{G}(f))(a)=\phi(a)\varphi_{G}(f)(\psi(a)).$

Thus, we get

$\begin{split}Tf(a)&=\begin{cases}\varphi_{G}(f)(\psi(a))\quad&\text{if $a\in
U$,}\\\ 0&\text{if $a\in H\setminus U$}\end{cases}\\\
&=\begin{cases}f([\psi(a)]_{G_{e}})\quad&\text{if $a\in U$,}\\\ 0&\text{if
$a\in H\setminus U$,}\end{cases}\\\ \end{split}$

for any $f\in A_{I}(G)$ and any $a\in H$. ∎

The following example shows that the assumption in Theorem 2.11 does not imply
$T(I(G))\subset I(H)$. This observation is in line with the well known fact
that $f\circ\psi$ may not be in the Fourier algebra $A(H)$ in general (see
Remark 2.13).

######  Example 2.12.

Let $G=\\{0\\}$ be the trivial group. Then we define a bounded linear operator
$T:A(G)\to B(\mathbb{Z})$ by $T(1_{G})=1_{\mathbb{Z}}$. Then it satisfies
$T(I(G))\subset I_{B}(H)$. Note that in this case, we have $U=\mathbb{Z}$ and
the continuous map $\psi:\mathbb{Z}\to\mathop{G/G_{e}}$ is $\psi(n)=0$ for any
$n\in\mathbb{Z}$. On the other hand, since $1_{\mathbb{Z}}\notin
A(\mathbb{Z})$, we have $T(I(G))\nsubseteq I(H)$.

######  Remark 2.13.

In general the converse statement of above theorem may not hold since we do
not know if $Tf\in A(H)$ for any $f\in A_{I}(G)$, even $T$ has a
representation of the form (6). If we only have $\psi:U\subseteq H\to G$ being
continuous, then $f\mapsto f\circ\psi$ maps $A(G)$ into $\ell^{\infty}(H)$ in
general. For abelian groups $G$ and $H$, Cohen [7] showed that $f\mapsto
f\circ\psi$ maps $A(G)$ to $B(H)$ if and only if $\psi$ is a continuous
piecewise affine map from a set in the open coset ring of $H$ into $G$. This
characterisation was extended by Host [15] to the case when $G$ has an abelian
subgroup of finite index and $H$ is arbitrary, and by [22] to general groups.

Under extra assumptions on $T$, we obtain algebraic structures for the open
set $U$ and algebraic properties on the map $\psi$. Let us first recall
positive operators on the Fourier algebra.

A bounded linear operator $T:A_{I}(G)\to B(H)$ is said to be positive if
$T(u)$ is positive definite whenever $u\in A_{I}(G)$ is a positive definite
function.

######  Corollary 2.14.

Let $G$ and $H$ be two locally compact groups. Let $T:A_{I}(G)\to B(H)$ be a
positive bounded complex linear operator. If $T$ satisfies that
$T(I(G))\subset I_{B}(H)$ then there exists an open subgroup $U$ of $H$ and a
continuous group homomorphism or anti-homomorphism $\psi$ from the open
subgroup $U$ of $H$ into $\mathop{G/G_{e}}$ such that

$Tf(a)=\begin{cases}f([\psi(a)]_{G_{e}})\quad&\text{if $a\in U$,}\\\
0&\text{if $a\in H\setminus U$,}\end{cases}$

for any $f\in A_{I}(G)$ and $a\in G$.

###### Proof.

Since isometric isomorphism $\varphi_{G}$ preserves positivity, $u\in
A_{I}(G)$ is a positive definite function if and only if $\varphi_{G}(u)$ is
positive definite. This implies that $T$ is positive if and only if $\Phi$ is
positive, thus, $\Phi:A(\mathop{G/G_{e}})\to B(H)$ is a positive homomorphism
by Lemma 2.8. It follows from [22, Theorem 4.3] that there exists an open
subgroup $U$ of $H$ and a continuous group homomorphism or anti-homomorphism
$\psi$ from $U$ into $\mathop{G/G_{e}}$ such that for any $f\in
A(\mathop{G/G_{e}})$, $\Phi f$ is either equal to $f\circ\psi$ in $U$, or $0$
otherwise. Thus, we have

$Tf(a)=\begin{cases}f([\psi(a)]_{G_{e}})\quad&\text{if $a\in U$,}\\\
0&\text{if $a\in H\setminus U$,}\end{cases}$

for any $f\in A_{I}(G)$ and $a\in H$. ∎

######  Corollary 2.15.

Let $G$ and $H$ be two locally compact groups, and $T:A_{I}(G)\to B(H)$ be a
contractive complex linear operator. If $T$ satisfies that $T(I(G))\subset
I_{B}(H)$ then there exists an open subgroup $U$ of $H$, a continuous group
homomorphism or anti-homomorphism $\psi$ from $U$ into $\mathop{G/G_{e}}$, and
elements $b\in G$ and $c\in H$ such that

$Tf(a)=\begin{cases}f(b[\psi(ca)]_{G_{e}})\quad&\text{if $a\in c^{-1}U$,}\\\
0&\text{if $a\in H\setminus c^{-1}U$.}\end{cases}$

###### Proof.

Since $\varphi_{G}$ is an isometric isomorphism, if $T$ is contractive then
$\Phi$ is also a contractive operator. By Lemma 2.8, $\Phi$ is a contractive
homomorphism from $A(\mathop{G/G_{e}})$ into $B(H)$. It follows from [22,
Theorem 5.1] that there exists an open subgroup $U$ of $H$, a continuous group
homomorphism or anti-homomorphism $\psi$ from $U$ into $\mathop{G/G_{e}}$, and
elements $bG_{e}\in\mathop{G/G_{e}}$ and $c\in H$ such that for any $f\in
A(\mathop{G/G_{e}})$ and $a\in H$, $\Phi f(a)=f(bG_{e}\psi(ca))$ provided
$a\in c^{-1}U$, otherwise, $\Phi f(a)=0$. By recalling the definition of
$\Phi$, we have the characterisation of $T$. ∎

###  2.2. Idempotent preserving maps with $T(I(G))\subset I(H)$

Let us assume that the bounded linear operator $T:A(G)\to B(H)$ satisfies
$T(I(G))\subset I(H)$. Then naturally we obtain $T(A_{I}(G))\subseteq
A_{I}(H)$.

We define $T_{q}:\mathop{A(G/G_{e})}\to\mathop{A(H/H_{e})}$ by

$T_{q}(f)=\varphi_{H}\circ T\circ\varphi_{G}^{-1}(f)=\varphi_{H}\circ\Phi(f),$

for any $f\in\mathop{A(G/G_{e})}$, where
$\varphi_{H}:A_{I}(H)\to\mathop{A(H/H_{e})}$ is an isometric isomorphism
defined similarly as in (1). Note that $T_{q}$ is an algebraic homomorphism.

######  Lemma 2.16.

Let $a\in\phi^{-1}(1)\subset H$ and $b\in H$ such that $a^{-1}b\in H_{e}$.
Then $\phi(b)=1$ and $\psi(a)=\psi(b)$.

###### Proof.

Suppose that $\psi(a)\neq\psi(b)$. By (3), we have
$\Phi:\mathop{A(G/G_{e})}\to B(H)$ such that for any
$f\in\mathop{A(G/G_{e})}$,

$\Phi(f)(a)=f(\psi(a))$

and

$\Phi(f)(b)=\phi(b)f(\psi(b)).$

Since $\mathop{G/G_{e}}$ is Hausdorff, there are disjoint open neighbourhoods
$V_{a}$ and $V_{b}$ of $\psi(a)$ and $\psi(b)$, respectively, in
$\mathop{G/G_{e}}$. By Lemma 2.5, for $\lambda_{\mathop{G/G_{e}}}(\psi(a))\in
VN(\mathop{G/G_{e}})$, there is $h\in I(\mathop{G/G_{e}})$ such that
$\mathop{\rm supp}h\subset V_{a}$ and $h(\psi(a))\neq 0$. Since
$a\in\phi^{-1}(1)$, we get

$\Phi(h)(a)=h(\psi(a))\neq 0$

and

$\Phi(h)(b)=\phi(b)h(\psi(b))=0.$

By the assumption that $T(I(G))\subset I(H)$ and $\varphi_{G}^{-1}(h)\in
I(G)$, we have $\Phi(h)=T(\varphi_{G}^{-1}(h))\in I(H)$, an idempotent in
$A(H)$. Hence, there is $Y\in\Omega_{\text{o}}^{\text{c}}(H)$ such that
$1_{Y}=\Phi(h)$. Since $1_{Y}(a)=\Phi(h)(a)=h(\psi(a))\neq 0$, we have $a\in
Y$. In addition, $Y$ is a clopen subset of $H$ and $H_{e}$ is a connected
component containing $e$, thus $aH_{e}\subset Y$. This implies that
$b=aa^{-1}b\in Y$. It follows that

$1=1_{Y}(b)=\Phi(h)(b)=0.$

This is a contradiction. Thus we have $\psi(a)=\psi(b)$. Furthermore, suppose
that $\phi(b)=0$. There is an $h\in I(\mathop{G/G_{e}})$ such that
$h(\psi(b))\neq 0$. Thus there is $Y\in\Omega_{\text{o}}^{\text{c}}(H)$ such
that $1_{Y}=\Phi(h)$. By a similar argument, we have
$1_{Y}(a)=\Phi(h)(a)=h(\psi(a))\neq 0$, $a\in Y$ and $b\in Y$. We obtain that

$1=1_{Y}(b)=\Phi(h)(b)=\phi(b)h(\psi(b))=0.$

This is a contradiction. Therefore, $\phi(b)=1$ and $\psi(a)=\psi(b)$. ∎

For any $a,b\in H$, the condition $a^{-1}b\in H_{e}$ induces an equivalence
relation on $H$. Lemma 2.16 shows that $\phi:H\to\\{0,1\\}$ and
$\psi:H\to\mathop{G/G_{e}}$ are constant functions on each equivalence class.
Thus these induce maps $\phi^{\prime}:\mathop{H/H_{e}}\to\\{0,1\\}$ and
$\psi^{\prime}:\phi^{\prime-1}(1)\to\mathop{G/G_{e}}$ by

$\phi^{\prime}(aH_{e})=\phi(a)\quad\text{for any $a\in H$}$

and

$\psi^{\prime}(aH_{e})=\psi(a)\quad\text{for any
$aH_{e}\in\phi^{\prime-1}(1)$}.$

By Lemma 2.9, the map $\psi:\phi^{-1}(1)\to\mathop{G/G_{e}}$ is continuous. As
we have $\phi^{\prime-1}(1)=q_{H}(\phi^{-1}(1))$, we obtain that
$\psi^{\prime}:\phi^{\prime-1}(1)\to\mathop{G/G_{e}}$ is continuous.

######  Theorem 2.17.

Let $G$ and $H$ be two locally compact Hausdorff groups and $T:A(G)\to B(H)$
be a bounded complex linear operator. Suppose that $T$ satisfies
$T(I(G))\subset I(H)$. Then there exists an open subset $U$ of $H$ and a
continuous map $\psi^{\prime}$ from an open subset $q_{H}(U)$ of
$\mathop{H/H_{e}}$ into $\mathop{G/G_{e}}$ such that

$Tf(a)=\begin{cases}f([\psi^{\prime}(aH_{e})]_{G_{e}})\quad&\text{if $a\in
U$,}\\\ 0&\text{if $a\in H\setminus U$,}\end{cases}$ (7)

for any $f\in A_{I}(G)$.

###### Proof.

Define $U=\phi^{-1}(1)$. Since $q_{H}:H\to\mathop{H/H_{e}}$ is the quotient
map and $U$ is a open subset of $H$ by Lemma 2.10. By (3), for any
$f\in\mathop{A(G/G_{e})}$ and $a\in H$, we have

$T_{q}(f)(aH_{e})=\varphi_{H}\circ\Phi(f)(aH_{e})=\Phi(f)(a)=\phi^{\prime}(aH_{e})f(\psi(a)).$
(8)

We shall show that $\phi^{\prime-1}(1)$ is an open subset of
$\mathop{H/H_{e}}$. Let $a\in\phi^{\prime-1}(1)$. By Lemma 2.7, there is an
idempotent $1_{\psi^{\prime}(a)G_{0}}$ of $A(\mathop{G/G_{e}})$ where
$\psi^{\prime}(a)G_{0}$ is an open compact neighbourhood of
$\psi^{\prime}(a)$. Since $T_{q}(1_{\psi^{\prime}(a)G_{0}})\in
A(\mathop{H/H_{e}})\subset C_{0}(\mathop{H/H_{e}})$, the space of all
continuous functions on $\mathop{H/H_{e}}$ vanishing at infinity, there exists
an open neighbourhood $V$ of $a$ in $\mathop{H/H_{e}}$ such that if $b\in
q_{H}^{-1}(V)$ then

$|T_{q}(1_{\psi^{\prime}(a)G_{0}})(a)-T_{q}(1_{\psi^{\prime}(a)G_{0}})(bH_{e})|\leq\frac{1}{2}.$

We have

$|1-\phi^{\prime}(bH_{e})1_{\psi^{\prime}(a)G_{0}}(\psi(b))|=|T_{q}(1_{\psi^{\prime}(a)G_{0}})(a)-T_{q}(1_{\psi^{\prime}(a)G_{0}})(bH_{e})|\leq\frac{1}{2}.$

Since either $\phi^{\prime}(bH_{e})1_{\psi^{\prime}(a)G_{0}}(\psi(b))=1$ or
$\phi^{\prime}(bH_{e})1_{\psi^{\prime}(a)G_{0}}(\psi(b))=0$, this implies that

$\phi^{\prime}(bH_{e})1_{\psi^{\prime}(a)G_{0}}(\psi(b))=1.$

Hence we have $\phi^{\prime}(bH_{e})=1$ for any $b\in q_{H}^{-1}(V)$. Thus
$V\subset\phi^{\prime-1}(1)$. It follows that $\phi^{\prime-1}(1)$ is an open
subset of $\mathop{H/H_{e}}$. Let us recall that
$\psi^{\prime}:q_{H}(U)\to\mathop{G/G_{e}}$ is a continuous map. Applying (8),
we have

$\begin{split}T_{q}(f)(aH_{e})&=\phi^{\prime}(aH_{e})f(\psi(a))\\\
&=\begin{cases}f(\psi^{\prime}(aH_{e}))\quad&\text{if $a\in U$,}\\\ 0&\text{if
$a\in H\setminus U$,}\end{cases}\\\ \end{split}$

for any $f\in A(\mathop{G/G_{e}})$ and $a\in H$. As we have

$T_{q}(\varphi_{G}(f))(aH_{e})=\varphi_{H}\circ
T\circ\varphi_{G}^{-1}(\varphi_{G}(f))(aH_{e})=\varphi_{H}\circ
T(f)(aH_{e})=Tf(a)$

for any $f\in A_{I}(G)$ and $a\in H$, we get

$\begin{split}Tf(a)&=\begin{cases}\varphi_{G}(f)(\psi^{\prime}(aH_{e}))\quad&\text{if
$a\in U$,}\\\ 0&\text{if $a\in H\setminus U$.}\end{cases}\\\ \end{split}$

∎

##  3\. Idempotent preserving bijections on $A_{I}(G)$

In this section we assume furthermore that the bounded linear operator
$T:A(G)\to B(H)$ satisfies that $T(I(G))\subset I(H)$ and $T|_{A_{I}(G)}$ is a
bijection onto $A_{I}(H)$.

######  Theorem 3.1.

Let $G$ and $H$ be two locally compact groups, and $T:A(G)\to B(H)$ be a
bounded complex linear operator. Suppose the operator $T$ satisfies that
$T(I(G))\subset I(H)$ and $T|_{A_{I}(G)}:A_{I}(G)\to A_{I}(H)$ is bijective.
Then there exists a homeomorphism $\psi:\mathop{H/H_{e}}\to\mathop{G/G_{e}}$
such that

$Tf(a)=f([\psi(aH_{e})]_{G_{e}})$

for all $f\in A_{I}(G)$ and $a\in H$.

###### Proof.

Since $T|_{A_{I}(G)}$ is a bijective linear map, $\varphi_{G}$ and
$\varphi_{H}$ are isometric isomorphisms, by the proof of Proposition 2.1 we
have $T_{q}:=\varphi_{H}\circ T|_{A_{I}(G)}\circ\varphi_{G}^{-1}$ is an
isomorphism from $A(\mathop{G/G_{e}})$ onto $A(\mathop{H/H_{e}})$.

Applying Theorem 2.17, there is an open subset $U$ of $H$ such that and a
continuous map $\psi$ from an open subset $q_{H}(U)$ of $\mathop{H/H_{e}}$
into $\mathop{G/G_{e}}$ such that

$T_{q}(f)(a)=\begin{cases}f(\psi(a))\quad&\text{if $a\in q_{H}(U)$,}\\\
0&\text{if $a\in(\mathop{H/H_{e}})\setminus q_{H}(U)$,}\end{cases}$

for any $f\in A(\mathop{G/G_{e}})$. Since $T_{q}:A(\mathop{G/G_{e}})\to
A(\mathop{H/H_{e}})$ is surjective and the Fourier algebra
$A(\mathop{H/H_{e}})$ separates the points in $\mathop{H/H_{e}}$, we have
$q_{H}(U)=\mathop{H/H_{e}}$. Thus $U=H$ and we have

$T_{q}(f)(a)=f(\psi(a))$

for every $f\in A(\mathop{G/G_{e}})$ and $a\in\mathop{H/H_{e}}$. For any $h\in
I(H)$, there exists $h_{q}\in A(\mathop{H/H_{e}})$ with $h_{q}^{2}=h_{q}$ such
that

$\varphi_{H}(h)=h_{q}.$

Since $T_{q}$ is bijective, there exists $f_{q}\in A(\mathop{G/G_{e}})$ such
that

$T_{q}(f_{q})=h_{q}.$

Moreover, since $T_{q}$ is an algebraic homomorphim, we have
$T_{q}(f_{q}^{2})=(T_{q}(f_{q}))^{2}=h_{q}^{2}=h_{q}=T_{q}(f_{q})$. By the
injectivity of $T_{q}$, we get $f_{q}^{2}=f_{q}$. On the other hand, as
$\varphi_{G}$ is an isometric isomorphism from $A_{I}(G)$ onto
$A(\mathop{G/G_{e}})$, there exists $f\in I(G)$ such that

$\varphi_{G}(f)=f_{q}.$

Hence, we have

$T(f)=(\varphi_{H}^{-1}\circ T_{q}\circ\varphi_{G})(f)=\varphi_{H}^{-1}\circ
T_{q}(f_{q})=\varphi_{H}^{-1}(h_{q})=h.$

This implies that $T(I(G))=I(H)$. In particular, we have $T^{-1}(I(H))\subset
I(G)$. Thus we can apply similar arguments to $T|_{A_{I}(G)}^{-1}:A_{I}(H)\to
A_{I}(G)$ and to $T_{q}^{-1}=\varphi_{G}\circ
T|_{A_{I}(G)}^{-1}\circ\varphi_{H}^{-1}$ on $A(\mathop{H/H_{e}})$, we can then
define a continuous map $\tilde{\psi}:\mathop{G/G_{e}}\to\mathop{H/H_{e}}$
such that

$T_{q}^{-1}(g)(b)=g(\tilde{\psi}(b))$

for any $g\in\mathop{A(H/H_{e})}$ and $b\in\mathop{G/G_{e}}$.

For any $g\in\mathop{A(H/H_{e})}$ and $a\in\mathop{H/H_{e}}$, we have

$g(a)=T_{q}(T_{q}^{-1}g)(a)=g(\tilde{\psi}(\psi(a))).$

Since the Fourier algebra $\mathop{A(H/H_{e})}$ separates points in
$\mathop{H/H_{e}}$, we get

$a=\tilde{\psi}(\psi(a))\quad\text{for}\quad a\in\mathop{H/H_{e}}.$ (9)

Moreover, we obtain

$f(b)=T_{q}^{-1}(T_{q}f)(b)=f(\psi(\tilde{\psi}(b))),$

for any $f\in\mathop{A(G/G_{e})}$ and $b\in\mathop{G/G_{e}}$. Similarly, as
$\mathop{A(G/G_{e})}$ separates points in $\mathop{G/G_{e}}$, we have

$b=\psi(\tilde{\psi}(b))\quad\text{for}\quad b\in\mathop{G/G_{e}}.$ (10)

By (9) and (10), we have that $\psi:\mathop{H/H_{e}}\to\mathop{G/G_{e}}$ is a
bijection and $\tilde{\psi}=\psi^{-1}$. Let us recall that $\psi$ and
$\tilde{\psi}$ are continuous on $\mathop{H/H_{e}}$ and $\mathop{G/G_{e}}$,
respectively. As $\tilde{\psi}=\psi^{-1}$, we have that $\psi$ is a
homeomorphism. In addition, we obtain

$T_{q}(f)(a)=f(\psi(a))\quad\text{for
}\,f\in\mathop{A(G/G_{e})},\,a\in\mathop{H/H_{e}}.$

Since $T=\varphi_{H}^{-1}\circ T_{q}\circ\varphi_{G}$, we get

$Tf(a)=f([\psi(aH_{e})]_{G_{e}})$

for all $f\in A_{I}(G)$ and $a\in H$. ∎

Note that the assumption of bijectivity in above theorem is needed for the
function $\psi:\mathop{H/H_{e}}\to\mathop{G/G_{e}}$ to be a homeomorphism.

######  Example 3.2.

Let $G=\\{1,2\\}$ be a multiplicative group equipped with the discrete
topology. Let $H=\\{0\\}$ be the trivial group. We define $T:A(G)\to A(H)$ by
$Tf(0)=f(1)$ for any $f\in A(G)$. Then $T$ is a bounded complex linear
operator on $A(G)$ and for any $1_{Y}\in A(G)$, $T(1_{Y})=1_{H}$ if $1\in Y$,
otherwise $T(1_{Y})=0$. Thus $T(I(G))=I(H)$. On the other hand,
$T(1_{\\{1\\}})=1_{H}=T(1_{G})$, this implies that $T|_{A_{I}(G)}:A_{I}(G)\to
A_{I}(H)$ is not injective. In addition,
$\psi:\mathop{H/H_{e}}=H\to\mathop{G/G_{e}}=G$ satisfying

$\psi(0)=1.$

is not a homeomorphism.

With extra assumptions on $T$ as in Section 2.1, we obtain a characterisation
of linear idempotent preserving maps between two Fourier algebras. Note that
since the continuous map $\psi$ in the following two corollaries is either a
group isomorphism or an anti-isomorphism, we naturally have $f\circ\psi\in
A_{I}(H)$ for any $f\in A_{I}(G)$ (see [26]), thus, we obtain a necessary and
sufficient condition for the idempotent preserving operator $T$ on $A_{I}(G)$.

######  Corollary 3.3.

Let $G$ and $H$ be two locally compact groups. A surjective complex linear
contraction $T:A_{I}(G)\to A_{I}(H)$ satisfies $T(I(G))\subset I(H)$ if and
only if there exists a continuous group isomorphism or anti-isomorphism
$\psi:\mathop{H/H_{e}}\to\mathop{G/G_{e}}$ and an element $b\in G$ such that

$Tf(a)=f(b[\psi(aH_{e})]_{G_{e}})$

for all $f\in A_{I}(G)$ and $a\in H$.

######  Corollary 3.4.

Let $G$ and $H$ be two locally compact groups. A positive bounded complex
linear bijection $T:A_{I}(G)\to A_{I}(H)$ satisfies $T(I(G))\subset I(H)$ if
and only if there exists a continuous group isomorphism or anti-isomorphism
$\psi:\mathop{H/H_{e}}\to\mathop{G/G_{e}}$ such that

$Tf(a)=f([\psi(aH_{e})]_{G_{e}})$

for all $f\in A_{I}(G)$ and $a\in H$.

We will end our paper with a special case when the groups are totally
disconnected. In such case, $A_{I}(G)$ is isometrically algebraic isomorphic
to $A(G)$. Thus the _idempotent preserving_ operators recover the results of
algebraic homomorphisms.

######  Remark 3.5.

Suppose that $G$ and $H$ are totally disconnected locally compact groups. Let
$T:A(G)\to A(H)$ be a bounded complex linear operator satisfying
$T(I(G))\subset I(H)$. Then there exists a continuous map $\psi$ from an open
subset $U$ of $H$ into $G$ such that

$Tf(a)=\begin{cases}f(\psi(a))\quad&\text{if $a\in U$,}\\\ 0&\text{if $a\in
H\setminus U$,}\end{cases}$

for any $f\in A(G)$ and $a\in H$. In addition, if $T$ is a surjective
contraction or $T$ is a positive bijection, then it is equivalent to
$Tf=f\circ(b\psi)$ for some $b\in G$ or $Tf=f\circ\psi$, respectively, for all
$f\in A(G)$ where $\psi:H\to G$ is a continuous group isomorphism or group
anti-isomorphism; in particular, $T$ is an algebraic homomorphism.

### Acknowledgments

The second author was supported by JSPS KAKENHI Grant Numbers JP21K13804.

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