# Searching for cosmic string induced stochastic gravitational wave background
with the Parkes Pulsar Timing Array

Ligong Bian<EMAIL_ADDRESS>Department of Physics and Chongqing Key
Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing
401331, China  Center for High Energy Physics, Peking University, Beijing
100871, China  Jing Shu<EMAIL_ADDRESS>School of Physics and State Key
Laboratory of Nuclear Physics and Technology, Peking University, Beijing
100871, China  Center for High Energy Physics, Peking University, Beijing
100871, China  CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China  School of Fundamental Physics and Mathematical
Sciences, Hangzhou Institute for Advanced
Study, University of Chinese Academy of Sciences, Hangzhou 310024, China
International Centre for Theoretical Physics Asia-Pacific, University of
Chinese Academy of Sciences, 100190 Beijing, China  Bo Wang  International
Centre for Theoretical Physics Asia-Pacific, University of Chinese Academy of
Sciences, 100190 Beijing, China  Qiang Yuan<EMAIL_ADDRESS>Key Laboratory of
Dark Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy
of Sciences, Nanjing 210023, China  School of Astronomy and Space Science,
University of Science and Technology of China, Hefei 230026, China  Junchao
Zong  Department of Physics, Nanjing University, Nanjing 210093, China  CAS
Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China

###### Abstract

We search for stochastic gravitational wave background emitted from cosmic
strings using the Parkes Pulsar Timing Array data over 15 years. While we find
that the common power-law excess revealed by several pulsar timing array
experiments might be accounted for by the gravitational wave background from
cosmic strings, the lack of the characteristic Hellings-Downs correlation
cannot establish its physical origin yet. The constraints on the cosmic string
model parameters are thus derived with conservative assumption that the common
power-law excess is due to unknown background. Two representative cosmic
string models with different loop distribution functions are considered. We
obtain constraints on the dimensionless string tension parameter
$G\mu<10^{-11}\sim 10^{-10}$, which is more stringent by two orders of
magnitude than that obtained by the high-frequency LIGO-Virgo experiment for
one model, and less stringent for the other. The results provide the chance to
test the Grand unified theories, with the spontaneous symmetry breaking scale
of $U(1)$ being two-to-three orders of magnitude below $10^{16}$ GeV. The
pulsar timing array experiments are thus quite complementary to the LIGO-Virgo
experiment in probing the cosmic strings and the underlying beyond standard
model physics in the early Universe.

##  I Introduction

Cosmic strings (CS) are one-dimensional topological defects supposed to be
formed during phase transitions where symmetry gets broken spontaneously
Kibble:1976sj ; Hindmarsh:1994re . The CS dynamics can be described by the
Nambu-Goto action for thin and local strings with no internal structures. In
this situation, infinite strings can reach the scaling regime Bennett:1987vf ;
Allen:1990tv ; Sakellariadou:1990nd  and go to loops through the
intercommutation of intersecting string segments Vachaspati:1984dz . The
formed small loops will oscillate and emit gravitational wave (GW) bursts by
the structures of Cusps and Kinks Damour:2001bk ; Damour:2000wa . The
superposition of uncorrelated GW bursts from many CSs form a stochastic
gravitational wave background (SGWB). In the Nambu-Goto string scenario, the
SGWB is characterized by the loop number density and the string tension
($\mu$) Vilenkin:2000jqa . The dimensionless parameter $G\mu$ ($G$ is the
Newtonian constant) that parameterized the gravitational interactions of
strings is usually adopted in literature, which is tightly connect with the
GUT scale ($M_{\rm GUT}\sim 10^{16}$ GeV) as $G\mu\sim(\eta/M_{\rm GUT})^{2}$
since CSs are generally predicted in the symmetry breaking chain of grand
unified theories (GUT). Therefore, the detection of SGWB from CSs provide an
intriguing way to access the beyond-standard-model physics close to the GUT
scale that are inaccessible by high-energy colliders King:2021gmj ;
King:2020hyd ; Buchmuller:2019gfy ; Caldwell:2022qsj , such as leptogenesis
and the type-I seesaw scale Dror:2019syi .

The gravitational waves spectra from CSs span in a wide frequency range
characterized by a plateau in the high-frequency region. The SGWB from CSs is
one of the most promising target of LIGO-Virgo LIGOScientific:2021nrg ;
LIGOScientific:2017ikf , Laser Interferometer Space Antenna (LISA)
Auclair:2019wcv , and pulsar timing arrays (PTA) Yonemaru:2020bmr ;
Sanidas:2012ee . Recently, both LIGO-Virgo and PTA experiments made great
progresses on the search of GWs. The LIGO-Virgo group place stringent
constraints on the GW bursts and the SGWB from CSs in the high-frequency
window LIGOScientific:2021nrg . In the nanohertz range, several PTAs reported
the detection of a mysterious common red process NANOGrav:2020bcs ;
Goncharov:2021oub ; Chen:2021rqp ; Antoniadis:2022pcn , which may be explained
as SGWB from various sources including CSs Ellis:2020ena ; Bian:2020urb ;
Blasi:2020mfx ; Blanco-Pillado:2021ygr . However, the GW interpretation of the
common red process is still suspicious due to the lack of the characteristic
Hellings-Downs (HD) correlation in the data. In this Letter we search for SGWB
signals generated from CSs utilizing the Parkes Pulsar Timing Array (PPTA)
data. We consider two representative SGWB models of CSs with two different
loop distribution functions, and derive the constraints on the dimensionless
$G\mu$ parameter of CS. The results can be translated to constraints on the
spontaneous symmetry breaking scale of the GUT.

##  II SGWB spectra for cosmic string networks

The SGWB from cosmic string networks comes from the uncorrelated
superpositions of GW bursts of three contributions: cusps, kinks, and kink-
kink collision. The SGWB spectrum emitted from cosmic strings is given by

$\Omega_{\rm GW}(t_{0},f)=\frac{f}{\rho_{c}}\ \frac{d\rho_{\rm
GW}}{df}(t_{0},f),$ (1)

where $\rho_{c}=\frac{3H_{0}^{2}}{8\pi G}$ is the critical energy density of
the universe, $\frac{d\rho_{\rm GW}}{df}(t_{0},f)$ is the GWs energy density
per unit frequency today. Considering different modes of loops oscillation
($n$), we have

$\frac{d\rho_{\rm GW}}{df}(t_{0},f)=G\mu^{2}\ \sum_{n}C_{n}(f)\ P_{n}\;,$ (2)

where the $C_{n}(f)$ is a function of loop distributions depending on the
cosmological background. To account for all contributions from cusps, kinks,
and kink-kink collisions, we adopt the $P_{n}$ from numerical simulations
Blanco-Pillado:2017oxo . Analytically, $P_{n}=\frac{\Gamma}{\zeta(q)}n^{-q}$,
$\Gamma\approx 50$ and $\zeta(q)$ is the Riemann zeta function with $q=4/3$,
$5/3$, and $2$ representing for cusps, kinks, and kink-kink collisions,
respectively.

We first consider the SGWB from CSs with the loop production functions for
non-self-intersecting loops being obtained directly from CS networks
simulation in radiation and matter dominated era111It was noted that the
inflation might also dilute the CS networks if CSs were formed before the
onset of the inflation, in which case the nano-hertz GW experiments may not be
sensitive to probe the SGWB from CSs Cui:2019kkd . by Blanco-Pillado, Olum,
Shlaer Blanco-Pillado:2013qja ; Blanco-Pillado:2011egf  (hereafter denoted as
BOS model, which is dubbed as model A by LIGO-Virgo group in Ref.
LIGOScientific:2021nrg ). In the model, one need to take into account
contributions from the radiation dominated era and the matter dominated era
with the loop number density $n(l,t)$ given in the Supplemental Material. For
the SGWB contributions from the radiation era, one has

$C_{n}(f)=\frac{2n}{f^{2}}\int_{z_{\rm eq}}^{z_{\rm
cut}}\frac{dz}{H_{0}\sqrt{\Omega_{r}}(1+z)^{8}}\ n_{r}(l,t),$ (3)

where $z_{\rm eq}$ is the redshift in the radiation-matter equality and
$z_{\rm cut}$ is the cutoff redshift. The subscript $r$ here denotes radiation
dominated era, similarly, in the following $rm$ is for the case where loops
are formed in radiation dominated era but survive past to matter dominated era
and $m$ for the case of matter dominated era. We note that here the
$n_{r}(l,t)$ is the loop distribution function rather than the oscillation
mode $n$ before the integration. The SGWB contributions from matter dominated
era constitute two parts: loops that survive to matter dominated era and loops
formed in matter dominated era. The $C_{n}$ takes the form of

$C_{n}(f)=\frac{2n}{f^{2}}\int_{0}^{z_{\rm
eq}}\frac{dz}{{H_{0}\sqrt{\Omega_{m}}}(1+z)^{15/2}}\ n_{i}(l,t),$ (4)

with $i$ being $rm$ and $m$ for these two cases, respectively.

Following the same procedure, we can calculate the SGWB spectrum with loop
distribution functions being derived analytically by Lorenz, Ringeval, and
Sakellariadou Lorenz:2010sm  (denoted as LRS, which is dubbed as model B by
LIGO-Virgo group in Ref. LIGOScientific:2021nrg ), where the distribution of
non-self-intersecting scaling loops are extracted from simulations
Ringeval:2005kr . In the model, the contribution from the radiation dominated
era takes the form of

$\Omega_{\rm GW}(f)=\frac{64\pi G^{2}\mu^{2}\Omega_{r}}{3}\sum_{n}P_{n}\int
dx\ n(x),$ (5)

and in the matter dominated era, the GW spectrum is

$\Omega_{\rm GW}(f)=\frac{162\pi
G^{2}\mu^{2}}{\Omega_{m}^{-2}H_{0}^{-2}}\frac{1}{f^{2}}\sum_{n}\ P_{n}\
n^{2}\int dx\ n(x).$ (6)

Figure 1: Expected spectra of SGWB from CS for the BOS (LRS) model with
$G\mu=10^{-8}$, $10^{-9}$, and $10^{-10}$, which are indicated by red, orange
and purple solid (dashed) lines. The solid and dashed black lines represent
the $95\%$ C.L. upper limits for free spectrum assumption derived from the
PPTA data for different hypotheses (H2/H3 and H4 in Tables 1 and 2,
respectively). The sensitive frequency regions of PTA and LIGO-Virgo are
denoted by green and blue shaded bands.

Fig. 1 shows the total SGWB spectra of the CS-induced SGWB for the BOS and LRS
models, for a few values of $G\mu$. It is shown that the BOS model predicts
relatively flat spectra (solid lines) and hence the PTA experiment is more
suitable to probe it. On the other hand, the LRS model predicts harder spectra
(dashed lines) which is more optimal for the high-frequency GW experiment such
as LIGO-Virgo. Also shown in Fig. 1 are the upper limits of the free-spectrum
SGWB derived from the PPTA data Xue:2021gyq . We find that the CS models with
$G\mu$ of $10^{-8}\sim 10^{-10}$ are severely constrained by the PPTA data.
The magnitude of the SGWB spectra for the LRS model is much higher than that
of the BOS model at high frequencies due to more small loops in the loop
distribution function dominate in such frequency range.

##  III Data Analysis

The dataset we used in this work is the second data release (DR2) of the PPTA
Kerr:2020qdo , which is available in the CSIRO pulsar data
archive222https://doi.org/10.4225/08/5afff8174e9b3. Searches for SGWBs and
other fundamental physics problems using these data (or the subsets) were also
carried out Shannon:2015ect ; Porayko:2018sfa ; PPTA:2021uzb ;
Goncharov:2021oub ; Xue:2021gyq . In PTA, to search for such an SGWB signal
from CS is to find spatially correlated time residuals among time-of-arrivals
(ToA) of different pulsars. The residuals are composed of deterministic timing
models, noises, and the hypothetical signal (SGWB from CS here). We use the
TEMPO2 tool Hobbs:2006cd ; Edwards:2006zg  to fit the timing models of
pulsars, the ENTERPRISE333https://github.com/nanograv/enterprise and the
ENTERPRISE_EXTENSIONS444https://github.com/nanograv/enterprise_extensions to
model the noises, and PTMCMCSampler justin_ellis_2017_1037579  to do the
Bayesian analysis. The noise model is based on single pulsar analyses of Ref.
Goncharov:2020krd . Briefly speaking, the stochastic noises mainly include two
parts, the white noise and red noise. The white noise may come from the radio
frequency interference, pulse profile changes or instrumental artifacts. We
use three parameters, EFAC (Error FACtor), EQUAD (Error added in QUADrature)
and ECORR (Error of CORRelation between ToAs in a single epoch), to account
for white noise which can not be subtracted by fitting the timing model. As
usually done Porayko:2018sfa ; PPTA:2021uzb , we fix the white noise
parameters as their maximum likelihood values from the single pulsar analyses
in the Bayesian analysis. The red noises are mainly caused by the
irregularities of the pulsar spin (spin noise) and the dispersion measure of
photons when traveling through the interstellar medium (DM noise). For some
pulsars there are band noise for ToAs of a certain photon frequency band and
chromatic noise which correlates between different photon frequencies. All the
red noises are modeled as power-law forms with amplitude $A$ and slope
$\gamma$. Several analyses revealed that there is a common power-law (CPL)
process in the pulsar ToAs, whose nature is still in debate NANOGrav:2020bcs ;
Goncharov:2021oub ; Chen:2021rqp ; Antoniadis:2022pcn . In this work we will
test different assumptions on the CPL, as either the SGWB signal from CS (see
also Ellis:2020ena ; Bian:2020urb ) or an unknown background. The solar system
ephemeris uncertainties are modeled with a 11-parameter model BayesEphem
implemented in ENTERPRISE, including perturbations in masses of major planets,
the drift rate of the Earth-Moon barycenter orbit, and the perturbations of
the Earth’s orbit from Jupiter’s average orbital elements described by 6
parameters NANOGRAV:2018hou . We summarize the noise and signal parameters
together with their priors adopted in the Bayesian analysis in Table S1 in the
Supplemental Material.

##  IV RESULT

To address how significant the CS-induced SGWB signals in the data, we test
the Bayes factor (BF) of the “signal hypothesis” against the null hypothesis,
following the Savage-Dickey formula 10.2307/2958475

${\rm
BF}_{10}=\frac{P_{1}(\boldsymbol{D})}{P_{0}(\boldsymbol{D})}=\frac{P(\boldsymbol{\phi}=\boldsymbol{\phi_{0}})}{P(\boldsymbol{\phi}=\boldsymbol{\phi_{0}}|\boldsymbol{D})},$
(7)

where ${\rm BF}_{10}$ means the BF of hypothesis H1 against H0,
$\boldsymbol{D}$ is the observational data, $\boldsymbol{\phi}$ is the
parameters of the signal model, and $\boldsymbol{\phi_{0}}$ is the parameters
of the null hypothesis which is a subset of $\boldsymbol{\phi}$. $P_{0}$ and
$P_{1}$ are the evidence of the noise and signal hypotheses. The ${\rm
BF}_{10}$ is equivalent to the ratio of the prior to the posterior
probabilities of the null hypothesis.

The null hypothesis (H0) corresponds to the model with only the pulsar timing
model and noise. For hypothesis H1, we assume an additional CPL process in the
model. The CS model with the HD correlation is included in substitution of the
CPL process is labelled as H2. In addition, in H3 and H4 we consider
simultaneously the CPL process as an unknown systematics and the CS
contribution, in which the auto-correlation of a pulsar’s own ToAs is not
subtracted (H3) and subtracted (H4), respectively. The results of the fittings
for different hypotheses are summarized in Tables 1 and 2.

Table 1: Hypotheses, Bayes factors, and estimated model parameters for the BOS model. Hypothesis | Pulsar | CPL | HD process | Bayes Factors | Parameter Estimation ($1\sigma$ interval)
---|---|---|---|---|---
Noise | Process | CS spectrum | $\quad\quad\log_{10}G\mu\quad\quad$ | $\log_{10}A_{\text{CPL}},\gamma_{\text{CPL}}$
H0:Pulsar Noise | ✓ |  |  |  |  |
H1:CPL | ✓ | ✓ |  |  $10^{3.2}$ (/H0) |  | $-14.48^{+0.62}_{-0.64},3.34^{+1.37}_{-1.53}$
H2:CS | ✓ |  | ✓(full HD) |  $10^{3.1}$ (/H0) | $-10.38^{+0.21}_{-0.21}$ |
H3:CS1 | ✓ | ✓ | ✓(full HD) |  $1.96\ $ (/H1) |  $<-10.02$ (95% C.L.) | $-15.58^{+1.21}_{-1.64},3.11^{+1.95}_{-2.02}$
H4:CS2 | ✓ | ✓ | ✓(no-auto HD) |  $0.60\ $ (/H1) |  $<-10.54$ (95% C.L.) | $-14.61^{+0.58}_{-0.59},3.63^{+1.24}_{-1.40}$
Table 2: Hypotheses, Bayes factors, and estimated model parameters for the LRS model. Hypothesis | Pulsar | CPL | HD process | Bayes Factors | Parameter Estimation ($1\sigma$ interval)
---|---|---|---|---|---
Noise | process | CS spectrum | $\quad\quad\log_{10}G\mu\quad\quad$ | $\log_{10}A_{\text{CPL}},\gamma_{\text{CPL}}$
H0:Pulsar Noise | ✓ |  |  |  |  |
H1:CPL | ✓ | ✓ |  |  $10^{3.2}$ (/H0) |  | $-14.48^{+0.62}_{-0.64},3.34^{+1.37}_{-1.53}$
H2:CS | ✓ |  | ✓(full HD) |  $10^{3.3}$ (/H0) | $-10.89^{+0.14}_{-0.17}$ |
H3:CS1 | ✓ | ✓ | ✓(full HD) |  $1.62\ $ (/H1) |  $<-10.64$ (95% C.L.) | $-15.44^{+1.18}_{-1.74},3.08^{+1.94}_{-1.99}$
H4:CS2 | ✓ | ✓ | ✓(no-auto HD) |  $0.55\ $ (/H1) |  $<-11.04$ (95% C.L.) | $-14.57^{+0.58}_{-0.59},3.54^{+1.24}_{-1.41}$

For hypothesis H1, a clear CPL signal with a BF of $10^{3.2}$ is revealed in
the data, consistent with previous studies NANOGrav:2020bcs ;
Goncharov:2021oub ; Chen:2021rqp ; Antoniadis:2022pcn . The posterior
distributions of the CPL model parameters are shown in Fig. S1 in the
Supplemental Material. Similar signal with comparable BF value is found if we
assume a CS-induced SGWB component in the model rather than the CPL (H2).
Fitting results of the CS parameter $\log_{10}G\mu$ are $-10.38\pm 0.21$ and
$-10.89^{+0.14}_{-0.17}$ for the BOS and LRS model, with $1\sigma$ error bars
presented. The corresponding parameter values are consistent with that
employed to account for the NANOGrav data, but with different model
assumptions Ellis:2020ena . Since the nature of the CPL process is unclear,
and particularly the characteristic HD correlation of GWs is still lack, we
also test the hypotheses that treating the CPL as an unknown background. The
resulting BF of hypotheses H3 or H4 against H1 is close to 1 which means that
the evidence of the CS is not significant in case of a CPL background. We
therefore derive the 95% credible level (C.L.) upper limits on
$\log_{10}G\mu$, as given in Tables 1 and 2. The posterior distributions of
the CPL and CS parameters for H4 are shown in Fig. 2. More results of the
analysis based on different hypotheses can be found in Figs. S2-S3 in the
Supplemental Material. Additional tests assuming that the CPL has an
astrophysical origin from the supermassive binary black hole (SMBBH) give
similar conclusion (see the Supplemental Material).

Figure 2: Posteriors distribution of $\log_{10}G\mu$ and the CPL parameters of
BOS (red) and LRS (blue) models assuming a no-auto HD correlation. The
$1\sigma$ and $2\sigma$ regions are presented in light and dark colors.

Our results can be compared with those obtained by the LIGO-Virgo observations
of high-frequency GWs LIGOScientific:2021nrg , which is displayed in Fig. 3.
We also show the bound from the cosmic microwave background (CMB), while the
less stronger limit from Big Bang nucleosynthese (BBN) is too weak to be shown
in the figure Planck:2013mgr ; LIGOScientific:2021nrg . The LIGO-Virgo upper
limits of $G\mu$ are $10^{-8}-10^{-6}$ and $(4.0-6.3)\times 10^{-15}$ for the
BOS and LRS model (model A and B in LIGOScientific:2021nrg ), respectively.
Compared with LIGO-Virgo results, the PPTA upper limits on $\log_{10}G\mu$
improve by more than two orders of magnitude for the BOS model. For the LRS
model, the LIGO-Virgo result is more stringent because the SGWB spectrum is
harder. The PTA experiment is thus very complementary to the LIGO-Virgo
experiment for the searches of CS-induced SGWB.

Figure 3: Constraints on $G\mu$ at the $95\%$ C.L. from the PPTA data for
different hypotheses, compared with those obtained by LIGO-Virgo experiment
LIGOScientific:2021nrg . Silver boxes represent the ranges reported in the
LIGO-Virgo analysis. The limit from cosmic microwave background (CMB) is also
reported by the black dashed line Planck:2013mgr .

##  V CONCLUSION

The very precise timing measurements of pulsars provide a powerful tool to
probe the fundamental new physics process occurred in the early Universe. In
this work we use more than 15 years of observations of 26 millisecond pulsars
by the PPTA experiment to search for SGWB from CS networks. We show that the
CPL process revealed by several PTAs recently can be explained by the CS-
induced SGWB signal with $\log_{10}G\mu\sim-10.38$ and $-10.89$ for the two
typical classes of CS models, the BOS and LRS models. While the LRS model
explanation would be excluded by the LIGO-Virgo observations at high
frequencies, the BOS model explanation is consistent with the LIGO-Virgo data
LIGOScientific:2021nrg . As a conservative alternative, we also assume the CPL
as a background component and turn to set limits on the CS model parameters.
The PPTA upper limits on $\log_{10}G\mu$ reach about $-10\sim-11$, depending
on the CS models and the analysis methods. These constraints correspond to the
symmetry breaking scale $\eta\sim 10^{13}-10^{14}$ GeV, that challenge the
grand unification theories below the GUT scale already. For the BOS model, the
PPTA limits are more stringent than those from LIGO-Virgo, and for the LRS
model the LIGO-Virgo experiment is more sensitive. This is due to the fact
that, in the LRS model, more small loops are contained in the loop
distribution functions that control the behaviours of the SGWB spectra from CS
networks in high frequency range. We expect that continuous accumulation of
more precise data by PTAs around the world in the near future will critically
test the CS models and more generally, a class of GUTs admitting the U(1)
symmetry breaking in the early Universe.

##  VI Acknowledgements

This work uses the public data from the Parkes Pulsar Timing Array. We thank
Huai-Ke Guo and Yue Zhao for useful discussions, and appreciate Xiao Xue for
the help on the data analysis in the early stage of this work. L.B. is
supported by the National Key Research and Development Program of China under
Grant No. 2021YFC2203004, the National Natural Science Foundation of China
(NSFC) under Grants No. 12075041 and No. 12047564, the Fundamental Research
Funds for the Central Universities of China under Grants No. 2021CDJQY-011 and
No. 2020CDJQY-Z003, and Chongqing Natural Science Foundation under Grant No.
cstc2020jcyj-msxmX0814. J. S. is supported by the NSFC under Grants No.
12025507, No. 11690022, and No. 11947302, by the Strategic Priority Research
Program and Key Research Program of Frontier Science of the Chinese Academy of
Sciences (CAS) under Grants No. XDB21010200, No. XDB23010000, No. XDPB15, and
No. ZDBS-LY-7003, and by the CAS Project for Young Scientists in Basic
Research under Grant No. YSBR-006. Q.Y. is supported by the Program for
Innovative Talents and Entrepreneur in Jiangsu and the Key Research Program of
CAS under Grant No. XDPB15.

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## Supplemental material

###  VI.1 CS loop distributions

The BOS model. — In this model, the loop production functions are inferred
from Nambu-Goto simulations of CS networks in radiation and matter dominated
era directly. In the radiation dominated era, we take $l/t\leqslant 0.1$ to
consider the cutoff of the maximum size of loops. In the matter dominated era,
as suggested by simulations, plenty of loops formed in the radiation era will
survive till the time of radiation-matter equality and emit GWs in the matter
dominated era, which imposes a constrain on loop size, $l/t<0.09t_{\rm
eq}/t-\Gamma G\mu$. Furthermore, loops can also be produced when the CS
networks reach the scaling regime in the matter dominated era, in which case
the loop size should satisfy $l/t<0.18$. The loop distribution functions of
all the circumstances are

$\displaystyle n_{r}(l,t)$ $\displaystyle=$
$\displaystyle\frac{0.18}{t^{3/2}(l+\Gamma G\mu t)^{5/2}},l/t\leqslant 0.1,$
(S1) $\displaystyle n_{rm}(l,t)$ $\displaystyle=$
$\displaystyle\frac{0.18t_{\rm eq}^{1/2}}{t^{2}\ (l+\Gamma G\mu
t)^{5/2}},\,l/t<0.09t_{\rm eq}/t-\Gamma G\mu,$ (S2) $\displaystyle n_{m}(l,t)$
$\displaystyle=$ $\displaystyle\frac{0.27-0.45(l/t)^{0.31}}{t^{2}(l+\Gamma
G\mu t)^{2}},l/t<0.18\;.$ (S3)

The LRS model. — Different from the BOS model, in the LRS model, the loop
distribution of the non-self-intersecting scaling loops rather than the loop
production function is inferred from CS network simulations Lorenz:2010sm . On
scales $x=l/t\gg\Gamma G\mu\equiv x_{d}$, the simulation gives

$n(x)=\frac{C_{0}}{x^{p}},$ (S4)

where the two constants $C_{0}$ and $p$ in the radiation dominated era and
matter dominated era are

$\displaystyle p=0.60^{+0.21}_{-0.15}\big{|}_{r},$ $\displaystyle\quad
p=0.41^{+0.08}_{-0.07}\big{|}_{m},$ (S5) $\displaystyle
C_{0}=0.21^{-0.12}_{+0.13}\big{|}_{r},$ $\displaystyle\quad
C_{0}=0.09^{-0.03}_{+0.03}\big{|}_{m}.$ (S6)

Note that, not only loops would emit GWs — which decreases their length $l$ —
but the GW emission also back-reacts on the loops. The back-reaction smooths
out the loops on the smallest scales (in particular the kinks), hindering the
formation of smaller loops. To extend the loop distribution given above down
to smaller scales, a further length scale $x_{c}\ll x_{d}$ (the so-called
“gravitational back-reaction scale”) is introduced, which is estimated as
$x_{c}=20(G\mu)^{1+2\chi}$ through matching the loop distribution on scales
$x\gg x_{d}$ (Eq. (S4)), where $\chi_{r}=0.2^{+0.07}_{-0.10}$ for the
radiation dominated era and $\chi_{m}=0.295^{+0.03}_{-0.04}$ for the matter
dominated era Auclair:2019wcv . The specific distributions in all periods are
then given by Lorenz:2010sm

$\displaystyle n(x>x_{d})$ $\displaystyle\simeq$
$\displaystyle\frac{C}{(x+x_{d})^{3-2\chi}},$ (S7) $\displaystyle
n(x_{c}<x<x_{d})$ $\displaystyle\simeq$
$\displaystyle\frac{C(3\nu-2\chi-1)}{2-2\chi}\frac{1}{x_{d}}\frac{1}{x^{2(1-\chi)}},$
(S8) $\displaystyle n(x<x_{c}<x_{d})$ $\displaystyle\simeq$
$\displaystyle\frac{C(3\nu-2\chi-1)}{2-2\chi}\frac{1}{x_{c}^{2(1-\chi)}}\frac{1}{x_{d}},$
(S9)

where $C=C_{0}(1-\nu)^{2-p}$, and $\nu=1/2$ for radiation dominated era and
$\nu=2/3$ for matter dominated era.

###  VI.2 PTA model parameters

Table S1 summarizes the major model parameters and their prior distributions
of the noise and signal components used in the analysis. The timing model
parameters of each pulsar are not listed here.

Table S1: Parameters and their prior distribution in data analysis. U and log-U stand for the uniform and log-uniform distribution. parameter | Description | Prior | Comments
---|---|---|---
Noise parameters($\boldsymbol{\vartheta}$)
EFAC | White-noise modifier per backend | U $[0,10]$ | Fixed for setting limits
EQUAD | Quadratic white noise per backend | log-U $[-10,-5]$ | Fixed for setting limits
ECORR | Correlated-ToAs white noise per backend | log-U $[-10,-5]$ | Fixed for setting limits
$A_{\textrm{SN}}$ | Spin-noise amplitude | log-U $[-20,-6]$ (search) | One parameter per pulsar
|  | U [$10^{-20},10^{-6}$] (limit) |
$\gamma_{\textrm{SN}}$ | Spin-noise spectral index | U $[0,10]$ | One parameter per pulsar
$A_{\textrm{DM}}$ | DM-noise amplitude | log-U $[-20,-6]$ (search) | One parameter per pulsar
|  | U [$10^{-20},10^{-6}$] (limit) |
$\gamma_{\textrm{DM}}$ | DM-noise spectral index | U $[0,10]$ | One parameter per pulsar
$A_{\textrm{BAND}}$ | Band-noise amplitude | log-U $[-20,-6]$ (search) | One parameter partial pulsars
|  | U [$10^{-20},10^{-6}$] (limit) |
$\gamma_{\textrm{BAND}}$ | Band-noise spectral index | U $[0,10]$ | One parameter partial pulsars
| One parameter per pulsar |  |
$A_{\textrm{CHROM}}$ | Chromatic-noise amplitude | log-U $[-20,-6]$ (search) | One parameter partial pulsars
|  | U [$10^{-20},10^{-6}$] (limit) |
$\gamma_{\textrm{CHROM}}$ | Chromatic-noise spectral index | U $[0,10]$ | One parameter partial pulsars
$n_{\textrm{CHROM}}$ | Index of chromatic effects | U $[0,6]$ | Fixed for single pulsar
$A_{\textrm{CPL}}$ | CPL process amplitude | log-U $[-18,-11]$ (search) | One parameter per PTA
|  | U [$10^{-18},10^{-11}$] (limit) |
$\gamma_{\textrm{CPL}}$ | CPL process power index | U $[0,7]$ | One parameter per PTA
Signal parameters ($\boldsymbol{\psi}$)
$G\mu$ | String tension | log-U $[-16,-7]$ (search) | One parameter per PTA
|  | U [$10^{-16},10^{-7}$] (limit) |
BayesEphem parameters ($\boldsymbol{\phi}$)
$z_{\rm drift}$ | Drift-rate of Earth’s orbit about ecliptic $z$-axis | U$[-10^{-9},10^{-9}]\ {\rm rad}\ {\rm yr}^{-1}$ | One parameter per PTA
$\Delta M_{\rm Jupiter}$ | Perturbation of Jupiter’s mass | $N(0,1.5\times 10^{-11})~{}M_{\bigodot}$ | One parameter per PTA
$\Delta M_{\rm Saturn}$ | Perturbation of Saturn’s mass | $N(0,8.2\times 10^{-12})~{}M_{\bigodot}$ | One parameter per PTA
$\Delta M_{\rm Uranus}$ | Perturbation of Uranus’ mass | $N(0,5.7\times 10^{-11})~{}M_{\bigodot}$ | One parameter per PTA
$\Delta M_{\rm Neptune}$ | Perturbation of Neptune’s mass | $N(0,7.9\times 10^{-11})~{}M_{\bigodot}$ | One parameter per PTA
$PCA_{i}$ | Principal components of Jupiter’s orbit | U $[-0.05,0.05]$ | One parameter per PTA

###  VI.3 Results of other Hypotheses

Fig. S1 shows the parameter distributions of the CPL process (H1). The best-
fitting values and $1\sigma$ errors of the amplitude and spectral index are
$\log_{10}A_{\textrm{CPL}}=-14.48^{+0.62}_{-0.64}$ and
$\gamma_{\textrm{CPL}}=3.34^{+1.37}_{-1.53}$. The BF of the hypothesis H1
against the null hypothesis H0 is $10^{3.2}$, indicating a strong evidence of
the exist of such a component.

Figure S1: Posterior distributions of the amplitude
$\log_{10}A_{\textrm{CPL}}$ and power-law index $\gamma_{\textrm{CPL}}$ of the
CPL component for hypothesis H1. Figure S2: Posteriors distribution of
$\log_{10}G\mu$ for hypothesis H2, for the BOS and LRS models.

Figure S3: Posterior distributions of $\log_{10}G\mu$ and the CPL parameters
for hypothesis H3, for the BOS (left) and LRS (right) models.

Then we search for the signal of SGWB from the CS. Fig. S2 shows the one-
dimensional probability distributions of the CS parameter $G\mu$ without
including the CPL process (H2). This test reveals that the CS-induced SGWB can
be used to explain the CPL excess in the data, with similar BFs compared with
the CPL assumption.

Fig. S3 gives the posterior distributions of the CPL parameters and the CS
parameter for hypothesis H3. In this case, the CS signal is not evident, and
we can derive constraints on the CS parameter. At the 95% C.L., we find
$\log_{10}G\mu<-10.02$ ($-10.64$) for the BOS (LRS) model. Similar analysis
but removing the auto-correlation of the pulsars’ ToAs (H4) is presented in
the main text.

Finally we consider the possibility that the CPL has an astrophysical origin
from the supermassive binary black holes (SMBBH). The spectrum of the SMBBH
SGWB signal is given by

$S(f)=\frac{A_{\textrm{SMBBH}}^{2}}{12\pi^{2}}\left(\frac{f}{{\rm
yr}^{-1}}\right)^{\gamma_{\textrm{SMBBH}}}~{}{\rm yr}^{3},$ (S10)

where the power-law index $\gamma_{\textrm{SMBBH}}$ is fixed to be 13/3. The
prior of $\log_{10}A_{\textrm{SMBBH}}$ is assumed to be uniformly distributed
in $[-18,-14]$. We test three model assumptions, with H5 being the SMBBH-only
hypothesis and H6 (H7) being the SMBBH + CS for the BOS (LRS) model.

Figure S4: Posterior distribution of $\log_{10}A_{\textrm{SMBBH}}$ for the
SMBBH-only hypothesis.

The constraints on the model parameters of hypotheses H5 - H7 are shown in
Fig. S4, Fig. S5, and Table S2. The CPL process may also be explained by the
SMBBH signal, with $\log_{10}A_{\textrm{SMBBH}}=-14.89^{+0.10}_{-0.12}$. For
hypotheses H6 and H7, we reach similar conclusion with H3 or H4 where the CPL
is treated as a background but with slight changes of the 95% C.L. upper
limits of $\log_{10}G\mu$.

Table S2: Hypotheses, Bayes factors, and estimated model parameters of different model assumptions for the case with the CPL substituted by the SMBBH. Hypothesis | Pulsar | SMBBH | HD process | Bayes Factors | Parameter Estimation ($1\sigma$ interval)
---|---|---|---|---|---
Noise | process | CS spectrum | $\quad\quad\log_{10}G\mu\quad\quad$ | $\log_{10}A_{\text{SMBBH}}$
H5:SMBBH | ✓ | ✓ |  |  $\ 10^{3.3}$ (/H0) |  | $-14.89^{+0.10}_{-0.12}$
H6:CS3 | ✓ | ✓ | ✓(full hd) |  $\ 1.21$ (/H5) |  $<-10.05$ ($95\%$ C.L.) | $-15.11^{+0.25}_{-1.88}$
H7:CS4 | ✓ | ✓ | ✓(full hd) |  $\ 1.07$ (/H5) |  $<-10.66$ ($95\%$ C.L.) | $-15.09^{+0.24}_{-1.87}$

Figure S5: Posterior distributions of $\log_{10}G\mu$ and
$\log_{10}A_{\textrm{SMBBH}}$ for the BOS (left) and LRS (right) model.


# the automatic additivity of $\xi-$Lie derivations on von Neumann algebras

Zhaofang Bai  School of Mathematical Sciences, Xiamen University, Xiamen,
361005, P. R. China<EMAIL_ADDRESS>,  Shuanping Du∗ School of
Mathematical Sciences, Xiamen University, Xiamen, 361005, P. R. China.
<EMAIL_ADDRESS>and  Yu Guo  Department of Mathematics, Shanxi Datong
University, Datong, 037009, P. R. China<EMAIL_ADDRESS>
###### Abstract.

Let ${\mathcal{M}}$ be a von Neumann algebra with no central summands of type
I1. It is shown that every nonlinear $\xi-$Lie derivation ($\xi\neq 1$) on
$\mathcal{M}$ is an additive derivation.

2000 Mathematical Subject Classification. Primary 47B47, 47B49

Key words and phrases. $\xi-$Lie derivation, Derivation, von Neumann algebra

This work was supported partially by National Natural Science Foundation of
China (11071201, 11001230), and the Fundamental Research Funds for the Central
Universities (2010121001).

This paper is in final form and no version of it will be submitted for
publication elsewhere.

∗ Correspondence author

##  1\. Introduction and main results

Let $\mathcal{A}$ be an associate ring (or an algebra over a field
$\mathbb{F}$). Then ${\mathcal{A}}$ is a Lie ring (Lie algebra) under the
product $[x,y]=xy-yx$, i.e., the commutator of $x$ and $y$. Recall that an
additive (linear) map $\delta:{\mathcal{A}}\rightarrow{\mathcal{A}}$ is called
an additive (linear) derivation if $\delta(xy)=\delta(x)y+x\delta(y)$ for all
$x,y\in{\mathcal{A}}$. Derivations are very important maps both in theory and
in applications, and have been studied intensively (see [8, 20, 21, 22] and
the references therein). More generally, an additive (linear) map $L$ from
${\mathcal{A}}$ into itself is called an additive (linear) Lie derivation if
$L([x,y])=[L(x),y]+[x,L(y)]$ for all $x,y\in{\mathcal{A}}$. The questions of
characterizing Lie derivations and revealing the relationship between Lie
derivations and derivations have received many mathematicians’ attention
recently (see [4, 9, 12, 16]). Very roughly speaking, additive (linear) Lie
derivations in the context prime rings (operator algebras) can be decomposed
as $\sigma+\tau$, where $\sigma$ is an additive (linear) derivation and $\tau$
is an additive (linear) map sending commutators into zero. Similarly,
associated with the Jordan product $xy+yx$. we have the conception of Jordan
derivation which is also studied intensively (see [5, 6, 9] and the references
therein).

Note that an important relation associated with the Lie product is the
commutativity. Two elements $x,y$ in an algebra ${\mathcal{A}}$ are
commutative if $xy=yx$, that is, their Lie product is zero. More generally, if
$\xi$ is a scalar and if $xy=\xi yx$, we say that $x$ commutes with $y$ up to
a factor $\xi$. The notion of commutativity up to a factor for pairs of
operators is also important and has been studied in the context of operator
algebras and quantum groups (Refs. [7, 11]). Motivated by this, the authors
introduce a binary operation $[x,y]_{\xi}=xy-\xi yx$, called $\xi$-Lie product
of $x,y$ (Ref. [17]). This product is found playing a more and more important
role in some research topics, and its study has recently attracted many
authors attention (for example, see [17, 18]). Then it is natural to introduce
the concept of $\xi$-Lie derivation. An additive (linear) map $L$ from
${\mathcal{A}}$ into itself is called a $\xi-$Lie derivation if
$L([x,y]_{\xi})=[L(x),y]_{\xi}+[x,L(y)]_{\xi}$ for all $x,y\in{\mathcal{A}}$.
This concept unifies several well-known notions. It is clear that a $\xi-$Lie
derivation is a derivation if $\xi=0$; is a Lie derivation if $\xi=1$; is a
Jordan derivation if $\xi=-1$. In [18], Qi and Hou characterized the additive
$\xi-$Lie derivation on nest algebras.

Let $\Phi:{\mathcal{A}}\rightarrow{\mathcal{A}}$ be a map (without the
additivity or linearity assumption). We say that $\Phi$ is a nonlinear
$\xi-$Lie derivation if
$\Phi([x,y]_{\xi})=[\Phi(x),y]_{\xi}+[x,\Phi(y)]_{\xi}$ for all
$x,y\in{\mathcal{A}}$. Recently, Yu and Zhang [24] described nonlinear Lie
derivation on triangular algebras. The aim of this note is to investigate
nonlinear $\xi-$Lie derivations on von Neumann algebras ($\xi\neq 1$) and to
reveal the relationship between such nonlinear $\xi-$Lie derivations and
additive derivations. Due to vital importance of derivations, we firstly
investigate nonlinear derivations. To our surprising, nonlinear derivations
are automatically additive. Our main results read as follows.

Theorem 1.1. Let ${\mathcal{M}}$ be a von Neumann algebra with no central
summands of type I1. If $\Phi:{\mathcal{M}}\rightarrow{\mathcal{M}}$ is a
nonlinear derivation, then $\Phi$ is an additive derivation.

The following result reveals the relationship between general nonlinear
$\xi-$Lie derivations and additive derivations.

Theorem 1.2. Let ${\mathcal{M}}$ be a von Neumann algebra with no central
summands of type I1. If $\xi$ is a scalar not equal $0,1$ and
$\Phi:{\mathcal{M}}\rightarrow{\mathcal{M}}$ is a nonlinear $\xi-$Lie
derivation, then $\Phi$ is an additive derivation and $\Phi(\xi T)=\xi\Phi(T)$
for all $T\in{\mathcal{M}}$.

It is worth mentioning that, as it turns out from Theorems 1.1 and Theorem
1.2, the additive structure and $\xi-$Lie multiplicative structure of von
Neumann algebra with no central summands of type I1 are very closely related
to each other. We remark that the question when a multiplicative map is
necessary additive is important in quantum mechanics and mathematics, and was
discussed for associative rings in the purely algebraic setting ([14], for a
recent systematic account, see [2]). In recent years, there is a growing
interest in studying the automatic additivity of maps determined by the action
on the product (see [1, 2, 13, 19, 23] and the references therein). We also
remark that if $\xi=1$, then $\xi-$Lie derivation is in fact a Lie derivation,
while Lie derivation is not necessary additive. For example, let $\sigma$ is
an additive derivation of ${\mathcal{M}}$ and $\tau$ is a mapping of
${\mathcal{M}}$ into its center ${\mathcal{Z}}_{\mathcal{M}}$ which maps
commutators into zero. Then $\sigma+\tau$ is a Lie derivation and such Lie
derivation is not additive in general.

##  2\. Notations and Preliminaries

Before embarking on the proof of our main results, we need some notations and
preliminaries about von Neumann algebras. A von Neumann algebra $\mathcal{M}$
is a weakly closed, self-adjoint algebra of operators on a Hilbert space $H$
containing the identity operator I. The set
${\mathcal{Z}}_{\mathcal{M}}=\\{S\in{\mathcal{M}}\mid ST=TS\text{ for all
}T\in{\mathcal{M}}\\}$ is called the center of ${\mathcal{M}}$. For
$A\in{\mathcal{M}}$, the central carrier of $A$, denoted by $\overline{A}$, is
the intersection of all central projections $P$ such that $PA=A$. It is well
known that the central carrier of $A$ is the projection with the range
$[{\mathcal{M}}A(H)]$, the closed linear span of $\\{MA(x)\mid
M\in{\mathcal{M}},x\in H\\}$. For each self-adjoint operator
$A\in{\mathcal{M}}$, we define the central core of $A$, denoted by
$\underline{A}$, to be $\sup\\{S\in{\mathcal{Z}}_{\mathcal{M}}\mid
S=S^{*},S\leq A\\}$. Clearly, one has $A-\underline{A}\geq 0$. Further if
$S\in{\mathcal{Z}}_{\mathcal{M}}$ and $A-\underline{A}\geq S\geq 0$ then
$S=0$. If $P$ is a projection it is clear that $\underline{P}$ is the largest
central projection $\leq P$. We call a projection core-free if
$\underline{P}=0$. It is easy to see that $\underline{P}=0$ if and only if
$\overline{I-P}=I$, here $\overline{I-P}$ denotes the central carrier of
$I-P$. We use [10] as a general reference for the theory of von Neumann
algebras.

In the following, there are several fundamental properties of von Neumann
algebras from [3, 15] which will be used frequently. For convenience, we list
them in a lemma.

Lemma 2.1. Let $\mathcal{M}$ be a von Neumann algebra.

(i) ([15, Lemma 4]) If ${\mathcal{M}}$ has no summands of type I1, then each
nonzero central projection of ${\mathcal{M}}$ is the central carrier of a
core-free projection of ${\mathcal{M}}$;

(ii) ([3, Lemma 2.6]) If ${\mathcal{M}}$ has no summands of type I1, then
$\mathcal{M}$ equals the ideal of $\mathcal{M}$ generated by all commutators
in $\mathcal{M}$.

By Lemma 2.1(i), one can find a non-trivial core-free projection with central
carrier $I$, denoted by $P_{1}$. Throughout this paper, $P_{1}$ is fixed.
Write $P_{2}=I-P_{1}$. By the definition of central core and central carrier,
$P_{2}$ is also core-free and $\overline{P_{2}}=I$. According to the two-side
Pierce decomposition of $\mathcal{M}$ relative $P_{1}$, denote
${\mathcal{M}}_{ij}=P_{i}{\mathcal{M}}P_{j}$, $i,j=1,2$, then we may write
${\mathcal{M}}={\mathcal{M}}_{11}+{\mathcal{M}}_{12}+{\mathcal{M}}_{21}+{\mathcal{M}}_{22}$.
In all that follows, when we write $T_{ij}$, $S_{ij}$, $M_{ij}$, it indicates
that they are contained in ${\mathcal{M}}_{ij}$. A conclusion which is used
frequently is $TM_{ij}=0$ for every $M_{ij}\in{\mathcal{M}}_{ij}$ implies that
$TP_{i}=0$. Indeed $TP_{i}MP_{j}=0$ for all $M\in{\mathcal{M}}$ together with
$\overline{P_{j}}=I$ gives $TP_{i}=0$. Similarly, if $M_{ij}T=0$ for every
$M_{ij}\in{\mathcal{M}}_{ij}$, then $T^{*}M_{ij}^{*}=0$ and so $P_{j}T=0$. If
$Z\in{\mathcal{Z}}_{\mathcal{M}}$ and $ZP_{i}=0$, then $ZMP_{i}=0$ for all
$M\in{\mathcal{M}}$ which implies $Z=0$.

The next lemma is technical which plays an important role in the proof of
Theorem 1.2.

Lemma 2.2. Let $T\in{\mathcal{M}}$, $\xi\neq 0,1$. Then
$T\in{\mathcal{M}}_{ij}+(\xi P_{i}+P_{j}){\mathcal{Z}}_{\mathcal{M}}$ ($1\leq
i\neq j\leq 2$) if and only if $[T,M_{ij}]_{\xi}=0$ for every
$M_{ij}\in{\mathcal{M}}_{ij}$;

Proof. The necessity is clear. Conversely, assume $[T,M_{ij}]_{\xi}=0$ for
every $M_{ij}\in{\mathcal{M}}_{ij}$. Write $T=\sum_{i,j=1}^{2}T_{ij}$. It
follows that $T_{ii}M_{ij}+T_{ji}M_{ij}=\xi(M_{ij}T_{jj}+M_{ij}T_{ji})$. Thus

$None$ $T_{ii}M_{ij}=\xi M_{ij}T_{jj}$

and $T_{ji}M_{ij}=0$. Noting that $\overline{P_{j}}=I$, we obtain

$T_{ji}=0.$

For every $M_{ii}\in{\mathcal{M}}_{ii}$, $M_{jj}\in{\mathcal{M}}_{jj}$,
$M_{ii}M_{ij},M_{ij}M_{jj}\in{\mathcal{M}}_{ij}$ and so $TM_{ii}M_{ij}=\xi
M_{ii}M_{ij}T$ and $TM_{ij}M_{jj}=\xi M_{ij}M_{jj}T$. From
$[T,M_{ij}]_{\xi}=0$, it follows that $TM_{ii}M_{ij}=M_{ii}TM_{ij}$, that is
$(TM_{ii}-M_{ii}T)M_{ij}=0$. Using $\overline{P_{j}}=I$ again, we have
$T_{ii}M_{ii}-M_{ii}T_{ii}=0$, i.e.,
$T_{ii}\in{\mathcal{Z}}_{P_{i}{\mathcal{M}}P_{i}}$. Thus

$T_{ii}=Z_{i}P_{i}$

for some central element $Z_{i}\in{\mathcal{Z}}_{\mathcal{M}}$. Similarly,
combining $TM_{ij}M_{jj}=\xi M_{ij}M_{jj}T$ and $[T,M_{ij}]_{\xi}=0$, we can
obtain

$T_{jj}=Z_{j}P_{j}$

for some central element $Z_{j}\in{\mathcal{Z}}_{\mathcal{M}}$. Now equation
(1) implies that $(Z_{i}-\xi Z_{j})M_{ij}=0$. From $\overline{P_{j}}=I$ and
$M_{ij}$ is arbitrary, it follows that $(Z_{i}-\xi Z_{j})P_{i}=0$. Since
$Z_{i}-\xi Z_{j}\in{\mathcal{Z}}_{\mathcal{M}}$, $M(Z_{i}-\xi
Z_{j})P_{i}=(Z_{i}-\xi Z_{j})MP_{i}=0$ for all $M\in\mathcal{M}$. By
$\overline{P_{i}}=I$, it follows that $Z_{i}=\xi Z_{j}$. So $T=T_{ij}+(\xi
P_{i}+P_{j})Z_{j}\in{\mathcal{M}}_{ij}+(\xi
P_{i}+P_{j}){\mathcal{Z}}_{\mathcal{M}}$.

##  3\. Proofs of main results

In the following, we are firstly aimed to prove Theorem 1.1.

Proof of Theorem 1.1. In what follows,
$\Phi:{\mathcal{M}}\rightarrow{\mathcal{M}}$ is a nonlinear derivation. We
will prove that $\Phi$ is additive, that is, for all $T,S\in{\mathcal{M}}$,
$\Phi(T+S)=\Phi(T)+\Phi(S)$. It is clear that $\Phi(0)=\Phi(0)0+0\Phi(0)=0$.
Note that $\Phi(P_{1}P_{2})=\Phi(P_{1})P_{2}+P_{1}\Phi(P_{2})=0$, multiplying
by $P_{2}$ from the both sides of this equation, we get
$P_{2}\Phi(P_{1})P_{2}=0$. Similarly, multiplying by $P_{1}$ from the both
sides of this equation, we have $P_{1}\Phi(P_{2})P_{1}=0$. For every
$M_{12}\in{\mathcal{M}}_{12}$,
$\Phi(M_{12})=\Phi(P_{1}M_{12})=\Phi(P_{1})M_{12}+P_{1}\Phi(M_{12})$ and so
$P_{1}\Phi(P_{1})M_{12}=0$. Hence $P_{1}\Phi(P_{1})P_{1}=0$. Similarly, from
$\Phi(M_{12})=\Phi(M_{12}P_{2})$, one can obtain $P_{2}\Phi(P_{2})P_{2}=0$.

Denote $T_{0}=P_{1}\Phi(P_{1})P_{2}-P_{2}\Phi(P_{1})P_{1}$. Define
$\Psi:{\mathcal{M}}\rightarrow{\mathcal{M}}$ by $\Psi(T)=\Phi(T)-[T,T_{0}]$
for every $T\in{\mathcal{M}}$. Then it is easy to see that $\Psi$ is also a
nonlinear derivation and $\Psi(P_{1})=\Psi(P_{2})=0$. Note that for every
$T\in{\mathcal{M}}:T\mapsto[T,T_{0}]$ is an additive derivation of
${\mathcal{M}}$. Therefore, without loss of generality, we may assume
$\Phi(P_{1})=\Phi(P_{2})=0$. Then for every $T_{ij}\in{\mathcal{M}}_{ij}$,
$\Phi(T_{ij})=P_{i}\Phi(M_{ij})P_{j}\in{\mathcal{M}}_{ij}$ ($i,j=1,2$).

Let $T$ be in $\mathcal{M}$, write $T=T_{11}+T_{12}+T_{21}+T_{22}$. In order
to prove the additivity of $\Phi$, we only need to show $\Phi$ is additive on
${\mathcal{M}}_{ij}(1\leq i,j\leq 2)$ and
$\Phi(T_{11}+T_{12}+T_{21}+T_{22})=\Phi(T_{11})+\Phi(T_{12})+\Phi(T_{21})+\Phi(T_{22})$.
We will complete the proof by checking two claims.

Claim 1. $\Phi$ is additive on ${\mathcal{M}}_{ij}(1\leq i,j\leq 2)$.

Set $T_{ij},S_{ij},M_{ij}\in{\mathcal{M}}_{ij}$. From
$(T_{11}+T_{12})M_{12}=T_{11}M_{12}$, it follows that

$\Phi(T_{11}+T_{12})M_{12}+(T_{11}+T_{12})\Phi(M_{12})=\Phi(T_{11})M_{12}+T_{11}\Phi(M_{12}).$

Note that $\Phi(M_{12})\in{\mathcal{M}}_{12}$, so
$(\Phi(T_{11}+T_{12})-\Phi(T_{11}))M_{12}=0.$ Then
$(\Phi(T_{11}+T_{12})-\Phi(T_{11}))P_{1}=0$. This implies

$(\Phi(T_{11}+T_{12})-\Phi(T_{11})-\Phi(T_{12}))P_{1}=0.$

Similarly, from $(T_{11}+T_{12})M_{21}=T_{12}M_{21}$, we have
$(\Phi(T_{11}+T_{12})-\Phi(T_{11})-\Phi(T_{12}))M_{21}=0.$ Then

$(\Phi(T_{11}+T_{12})-\Phi(T_{11})-\Phi(T_{12}))P_{2}=0.$

Thus

$\Phi(T_{11}+T_{12})=\Phi(T_{11})+\Phi(T_{12}).$

Similarly, $\Phi(T_{12}+T_{22})=\Phi(T_{12})+\Phi(T_{22}).$ Since
$T_{12}+S_{12}=(P_{1}+T_{12})(P_{2}+S_{12})$, we have that

$\displaystyle\Phi(T_{12}+S_{12})$
$\displaystyle=\Phi(P_{1}+T_{12})(P_{2}+S_{12})+(P_{1}+T_{12})\Phi(P_{2}+S_{12})$
$\displaystyle=\Phi(T_{12})+\Phi(S_{12}).$

In the same way, one can show that
$\Phi(T_{21}+S_{21})=\Phi(T_{21})+\Phi(S_{21}).$ That is, $\Phi$ is additive
on ${\mathcal{M}}_{12},{\mathcal{M}}_{21}$.

From $(T_{11}+S_{11})M_{12}=T_{11}M_{12}+S_{11}M_{12}$, it follows that

$\begin{array}[]{rl}&\Phi(T_{11}+S_{11})M_{12}+(T_{11}+S_{11})\Phi(M_{12})\\\
=&\Phi(T_{11}M_{12})+\Phi(S_{11}M_{12})\\\
=&\Phi(T_{11})M_{12}+T_{11}\Phi(M_{12})+\Phi(S_{11})M_{12}+S_{11}\Phi(M_{12}).\end{array}$

Thus $(\Phi(T_{11}+S_{11})-\Phi(T_{11})-\Phi(S_{11}))M_{12}=0$. This yields

$(\Phi(T_{11}+S_{11})-\Phi(T_{11})-\Phi(S_{11}))P_{1}=0.$

Note that
$\Phi(T_{11}+S_{11})-\Phi(T_{11})-\Phi(S_{11})\in{\mathcal{M}}_{11}$. So

$\Phi(T_{11}+S_{11})=\Phi(S_{11})+\Phi(T_{11}).$

Similarly ,$\Phi(T_{22}+S_{22})=\Phi(T_{22})+\Phi(S_{22}).$ That is, $\Phi$ is
additive on ${\mathcal{M}}_{11},{\mathcal{M}}_{22}$, as desired.

Claim 2.
$\Phi(T_{11}+T_{12}+T_{21}+T_{22})=\Phi(T_{11})+\Phi(T_{12})+\Phi(T_{21})+\Phi(T_{22})$

From $(T_{11}+T_{12}+T_{21}+T_{22})M_{12}=(T_{11}+T_{21})M_{12}$, we have

$\begin{array}[]{rl}&\Phi(T_{11}+T_{12}+T_{21}+T_{22})M_{12}+(T_{11}+T_{12}+T_{21}+T_{22})\Phi(M_{12})\\\
=&\Phi(T_{11}M_{12})+\Phi(T_{21}M_{12})\\\
=&\Phi(T_{11})M_{12}+T_{11}\Phi(M_{12})+\Phi(T_{21})M_{12}+T_{21}\Phi(M_{12}).\end{array}$

Then

$\begin{array}[]{ll}&(\Phi(T_{11}+T_{12}+T_{21}+T_{22})-\Phi(T_{11})-\Phi(T_{12})-\Phi(T_{21})-\Phi(T_{22}))M_{12}\\\
=&(\Phi(T_{11}+T_{12}+T_{21}+T_{22})-\Phi(T_{11})-\Phi(T_{21}))M_{12}=0.\end{array}$

This gives

$(\Phi(T_{11}+T_{12}+T_{21}+T_{22})-\Phi(T_{11})-\Phi(T_{12})-\Phi(T_{21})-\Phi(T_{22}))P_{1}=0.$

From $(T_{11}+T_{12}+T_{21}+T_{22})M_{21}=(T_{12}+T_{22})M_{21}$, it follows
that

$(\Phi(T_{11}+T_{22}+T_{12}+T_{21})-\Phi(T_{11})-\Phi(T_{12})-\Phi(T_{21})-\Phi(T_{22}))P_{2}=0.$

So
$\Phi(T_{11}+T_{12}+T_{21}+T_{22})=\Phi(T_{11})+\Phi(T_{12})+\Phi(T_{21})+\Phi(T_{22})$.

Now, we turn to prove Theorem 1.2.

Proof of Theorem 1.2. We will finish the proof of the Theorem 1.2 by checking
several claims.

Claim 1. $\Phi(0)=0$ and there is $T_{0}\in{\mathcal{M}}$ such that
$\Phi(P_{i})=[P_{i},T_{0}]$ ($i=1,2$).

It is clear that
$\Phi(0)=\Phi([0,0]_{\xi})=[\Phi(0),0]_{\xi}+[0,\Phi(0)]_{\xi}=0$.

For every $M_{12}$,

$None$
$\begin{array}[]{ll}\Phi(M_{12})&=\Phi([P_{1},M_{12}]_{\xi})=[\Phi(P_{1}),M_{12}]_{\xi}+[P_{1},\Phi(M_{12})]_{\xi}\\\
&=\Phi(P_{1})M_{12}-\xi
M_{12}\Phi(P_{1})+P_{1}\Phi(M_{12})-\xi\Phi(M_{12})P_{1}.\end{array}$

Multiplying by $P_{1},P_{2}$ from the left and the right in equation (2)
respectively, we have

$P_{1}\Phi(P_{1})P_{1}M_{12}=\xi M_{12}P_{2}\Phi(P_{1})P_{2}.$

That is $[P_{1}\Phi(P_{1})P_{1}+P_{2}\Phi(P_{1})P_{2},M_{12}]_{\xi}=0$. Now
Lemma 2.2 yields that $P_{1}\Phi(P_{1})P_{1}+P_{2}\Phi(P_{1})P_{2}\in(\xi
P_{1}+P_{2}){\mathcal{Z}}_{\mathcal{M}}$. For every $M_{21}$,

$None$
$\begin{array}[]{ll}\Phi(M_{21})&=\Phi([P_{2},M_{21}]_{\xi})=[\Phi(P_{2}),M_{21}]_{\xi}+[P_{2},\Phi(M_{21})]_{\xi}\\\
&=\Phi(P_{2})M_{21}-\xi
M_{21}\Phi(P_{2})+P_{2}\Phi(M_{21})-\xi\Phi(M_{21})P_{2}.\end{array}$

Multiplying by $P_{2},P_{1}$ from the left and the right in equation (3)
respectively, we obtain

$P_{2}\Phi(P_{2})P_{2}M_{21}=\xi M_{21}P_{1}\Phi(P_{2})P_{1}.$

That is $[P_{2}\Phi(P_{2})P_{2}+P_{1}\Phi(P_{2})P_{1},M_{21}]_{\xi}=0$. Using
Lemma 2.2 again, we get
$P_{2}\Phi(P_{2})P_{2}+P_{1}\Phi(P_{2})P_{1}\in(P_{1}+\xi
P_{2}){\mathcal{Z}}_{\mathcal{M}}$. Assume
$P_{1}\Phi(P_{1})P_{1}+P_{2}\Phi(P_{1})P_{2}=(\xi P_{1}+P_{2})Z_{1}$ and
$P_{2}\Phi(P_{2})P_{2}+P_{1}\Phi(P_{2})P_{1}=(P_{1}+\xi P_{2})Z_{2}$,
$Z_{1},Z_{2}\in{\mathcal{Z}}_{\mathcal{M}}$. From $[P_{1},P_{2}]_{\xi}=0$, it
follows that

$\begin{array}[]{ll}&\Phi([P_{1},P_{2}]_{\xi})=[\Phi(P_{1}),P_{2}]_{\xi}+[P_{1},\Phi(P_{2})]_{\xi}\\\
=&\Phi(P_{1})P_{2}-\xi
P_{2}\Phi(P_{1})+P_{1}\Phi(P_{2})-\xi\Phi(P_{2})P_{1}\\\
=&(1-\xi)P_{1}\Phi(P_{2})P_{1}+(1-\xi)P_{2}\Phi(P_{1})P_{2}+P_{1}\Phi(P_{1})P_{2}\\\
&+P_{1}\Phi(P_{2})P_{2}-\xi P_{2}\Phi(P_{2})P_{1}-\xi P_{2}\Phi(P_{1})P_{1}\\\
=&0.\end{array}$

Then

$None$
$P_{1}\Phi(P_{2})P_{1}=P_{2}\Phi(P_{1})P_{2}=P_{1}\Phi(P_{1})P_{2}+P_{1}\Phi(P_{2})P_{2}=P_{2}\Phi(P_{1})P_{1}+P_{2}\Phi(P_{2})P_{1}=0.$

A direct computation shows that $[(\xi
P_{1}+P_{2})Z_{1},P_{2}]_{\xi}=[P_{1}\Phi(P_{1})P_{1}+P_{2}\Phi(P_{1})P_{2},P_{2}]_{\xi}=0$.
And so $(1-\xi)P_{2}Z_{1}=0$. Then $Z_{1}MP_{2}=0$ for all
$M\in{\mathcal{M}}$. Noting that ${\overline{P}_{2}}=I$, we have $Z_{1}=0$.
That is $P_{1}\Phi(P_{1})P_{1}+P_{2}\Phi(P_{1})P_{2}=0$. Similarly,
$P_{2}\Phi(P_{2})P_{2}+P_{1}\Phi(P_{2})P_{1}=0$. By (4),
$\Phi(P_{1})+\Phi(P_{2})=0$. Denote
$T_{0}=P_{1}\Phi(P_{1})P_{2}-P_{2}\Phi(P_{1})P_{1}$. Then it is easy to check
that $T_{0}$ is the desired.

Obviously, $T\mapsto[T,T_{0}]$ is an additive derivation. Without loss of
generality, we may assume that $\Phi(P_{1})=\Phi(P_{2})=0$.

If $\Phi$ is additive, then $\Phi(I)=\Phi(P_{1})+\Phi(P_{2})=0$.
$\Phi((1-\xi)T)=\Phi([I,T]_{\xi})=[I,\Phi(T)]_{\xi}=(1-\xi)\Phi(T)$ for all
$T\in{\mathcal{M}}$. So $\Phi(\xi T)=\xi\Phi(T)$ for all $T\in{\mathcal{M}}$.
Taking $T,S\in{\mathcal{M}}$ and noting that
$(1-\xi)[S,T]_{-1}=[S,T]_{\xi}+[T,S]_{\xi}$, we obtain that

$\begin{array}[]{ll}&\Phi((1-\xi)[S,T]_{-1})=\Phi([S,T]_{\xi})+\Phi([T,S]_{\xi})\\\
=&\Phi(S)T-\xi T\Phi(S)+S\Phi(T)-\xi\Phi(T)S+\Phi(T)S-\xi
S\Phi(T)+T\Phi(S)-\xi\Phi(S)T\\\
=&(1-\xi)(\Phi(S)T+S\Phi(T)+\Phi(T)S+T\Phi(S)).\end{array}$

Note that $\Phi((1-\xi)T)=(1-\xi)\Phi(T)$ for all $T\in{\mathcal{M}}$, it
follows that

$\Phi([S,T]_{-1})=[\Phi(S),T]_{-1}+[S,\Phi(T)]_{-1}$

for all $T,S\in{\mathcal{M}}$. Hence $\Phi$ is an additive Jordan derivation.
By [5], $\Phi$ is an additive derivation which is the conclusion of our
Theorem 1.2. Now we only need to show $\Phi$ is additive. For every
$T\in\mathcal{M}$, it has the form $T=T_{11}+T_{12}+T_{21}+T_{22}$. Just like
the proof of Theorem 1.1, we will show $\Phi$ is additive on
${\mathcal{M}}_{ij}(1\leq i,j\leq 2)$ and
$\Phi(T_{11}+T_{12}+T_{21}+T_{22})=\Phi(T_{11})+\Phi(T_{12})+\Phi(T_{21})+\Phi(T_{22})$.
We divide the proof into several steps.

Claim 2. $\Phi(M_{ij})\in{\mathcal{M}}_{ij}$ for every
$M_{ij}\in{\mathcal{M}}_{ij}$ ($1\leq i\neq j\leq 2$).

We only treat the case $i=1,j=2$. The other case can be treated similarly.
Noting $[P_{1},M_{12}]_{\xi}=M_{12}$, we have

$\displaystyle\Phi(M_{12})$
$\displaystyle=\Phi([P_{1},M_{12}]_{\xi})=[\Phi(P_{1}),M_{12}]_{\xi}+[P_{1},\Phi(M_{12})]_{\xi}$
$\displaystyle=[P_{1},\Phi(M_{12})]_{\xi}=P_{1}\Phi(M_{12})-\xi\Phi(M_{12})P_{1}.$

Then

$None$ $P_{2}\Phi(M_{12})P_{2}=P_{1}\Phi(M_{12})P_{1}=0.$

Furthermore, $P_{2}\Phi(M_{12})P_{1}=0$, if $\xi\neq-1$, i.e.,
$\Phi(M_{12})\in{\mathcal{M}}_{12}$.

Next we treat the case $\xi=-1$. For every $M_{11}$,

$\displaystyle\Phi(M_{11}M_{12})$
$\displaystyle=\Phi([M_{11},M_{12}]_{-1})=[\Phi(M_{11}),M_{12}]_{-1}+[M_{11},\Phi(M_{12})]_{-1}$
$\displaystyle=\Phi(M_{11})M_{12}+M_{12}\Phi(M_{11})+M_{11}\Phi(M_{12})+\Phi(M_{12})M_{11}.$

By (5), we have

$P_{2}\Phi(M_{11}M_{12})P_{1}=\Phi(M_{12})M_{11}.$

Then for every $N_{11}$,
$P_{2}\Phi(N_{11}M_{11}M_{12})P_{1}=\Phi(M_{12})N_{11}M_{11}$. On the other
hand,

$P_{2}\Phi(N_{11}M_{11}M_{12})P_{1}=\Phi(M_{11}M_{12})N_{11}=\Phi(M_{12})M_{11}N_{11}.$

Thus $\Phi(M_{12})[N_{11},M_{11}]=0$. For every $R_{11}$,

$\Phi(M_{12})R_{11}[N_{11},M_{11}]=P_{2}\Phi(R_{11}M_{12})P_{1}[N_{11},M_{11}]=0.$

By Lemma 2.1(ii), $\Phi(M_{12})P_{1}=0$ which finishes the proof.

Claim 3. $\Phi(M_{ii})\in{\mathcal{M}}_{ii}$ for every
$M_{ii}\in{\mathcal{M}}_{ii}$ $(i=1,2)$.

Proof. Without loss of generality, we only treat the case $i=1$.

$\begin{array}[]{ll}\Phi(P_{1})&=\Phi([I,\frac{1}{1-\xi}P_{1}]_{\xi})=[\Phi(I),\frac{1}{1-\xi}P_{1}]_{\xi}+[I,\Phi(\frac{1}{1-\xi}P_{1})]_{\xi}\\\
&=\frac{1}{1-\xi}\Phi([I,P_{1}]_{\xi})+[I,\Phi(\frac{1}{1-\xi}P_{1})]_{\xi}\\\
&=\frac{1}{1-\xi}\Phi((1-\xi)P_{1})+(1-\xi)\Phi(\frac{1}{1-\xi}P_{1})=0.\end{array}$

Note that $\Phi((1-\xi)P_{1})=\Phi([P_{1},P_{1}]_{\xi})=0$, so
$\Phi(\frac{1}{1-\xi}P_{1})=0$.

$\displaystyle\Phi(M_{11})$
$\displaystyle=\Phi([\frac{1}{1-\xi}P_{1},M_{11}]_{\xi})=[\frac{1}{1-\xi}P_{1},\Phi(M_{11})]_{\xi}$
$\displaystyle=\frac{1}{1-\xi}(P_{1}\Phi(M_{11})-\xi\Phi(M_{11})P_{1}).$

This implies $\Phi(M_{11})\in{\mathcal{M}}_{11}$.

Claim 4. For every $T_{ii}$, $T_{ji}$ and $T_{ij}$ $(1\leq i\neq j\leq 2)$,
$\Phi(T_{ii}+T_{ij})=\Phi(T_{ii})+\Phi(T_{ij})$,
$\Phi(T_{ii}+T_{ji})=\Phi(T_{ii})+\Phi(T_{ji})$.

Assume $i=1,j=2$. For every $M_{12}\in{\mathcal{M}}_{12}$,
$[T_{11}+T_{12},M_{12}]_{\xi}=[T_{11},M_{12}]_{\xi}$, by Claim 2,

$[\Phi(T_{11}+T_{12}),M_{12}]_{\xi}+[T_{11}+T_{12},\Phi(M_{12})]_{\xi}=[\Phi(T_{11}),M_{12}]_{\xi}+[T_{11},\Phi(M_{12})]_{\xi},$
$[\Phi(T_{11}+T_{12})-\Phi(T_{11}),M_{12}]_{\xi}=0.$

From Lemma 2.2,

$\Phi(T_{11}+T_{12})-\Phi(T_{11})=P_{1}(\Phi(T_{11}+T_{12})-\Phi(T_{11}))P_{2}+(\xi
P_{1}+P_{2})Z$

for some central element $Z\in Z_{\mathcal{M}}.$ By computing,

$\displaystyle\Phi(T_{12})$
$\displaystyle=\Phi([P_{1},[T_{11}+T_{12},P_{2}]_{\xi}]_{\xi})$
$\displaystyle=[P_{1},[\Phi(T_{11}+T_{12}),P_{2}]_{\xi}]_{\xi}$
$\displaystyle=P_{1}\Phi(T_{11}+T_{22})P_{2}+{\xi}^{2}P_{2}\Phi(T_{11}+T_{22}).$

From Claim 2 and Claim 3, we know that
$\Phi(T_{12})=P_{1}\Phi(T_{11}+T_{12})P_{2}$ and $P_{1}\Phi(T_{11})P_{2}=0$.
Thus

$\Phi(T_{11}+T_{12})-\Phi(T_{11})=\Phi(T_{12})+(\xi P_{1}+P_{2})Z.$

Note that

$\displaystyle\Phi([T_{11}+T_{12},P_{2}]_{\xi})$
$\displaystyle=[\Phi(T_{11}+T_{12}),P_{2}]_{\xi}$
$\displaystyle=[\Phi(T_{11})+\Phi(T_{12})+(\xi P_{1}+P_{2})Z,P_{2}]_{\xi}.$

On the other hand,
$\Phi([T_{11}+T_{12},P_{2}]_{\xi})=\Phi([T_{12},P_{2}]_{\xi})=[\Phi(T_{12}),P_{2}]_{\xi}$.
Combining this with Claim 3, we have $[(\xi P_{1}+P_{2})Z,P_{2}]_{\xi}=0$ and
so $ZP_{2}=0$ which implies $Z=0$. Similarly,
$\Phi(T_{11}+T_{21})=\Phi(T_{11})+\Phi(T_{21})$. The rest goes similarly.

Claim 5. $\Phi$ is additive on ${\mathcal{M}}_{12}$ and ${\mathcal{M}}_{21}$.

Let $T_{12},S_{12}\in{\mathcal{M}}_{12}$. Since
$T_{12}+S_{12}=[P_{1}+T_{12},P_{2}+S_{12}]_{\xi}$, we have that

$\begin{array}[]{rl}&\Phi(T_{12}+S_{12})=[\Phi(P_{1}+T_{12}),P_{2}+S_{12}]_{\xi}+[P_{1}+T_{12},\Phi(P_{2}+S_{12})]_{\xi}\\\
=&[\Phi(P_{1})+\Phi(T_{12}),P_{2}+S_{12}]_{\xi}+[P_{1}+T_{12},\Phi(P_{2})+\Phi(S_{12})]_{\xi}\\\
=&\Phi(T_{12})+\Phi(S_{12}).\end{array}$

Similarly, $\Phi$ is additive on ${\mathcal{M}}_{21}$.

Claim 6. For every $T_{11}\in{\mathcal{M}}_{11}$,
$T_{22}\in{\mathcal{M}}_{22}$,
$\Phi(T_{11}+T_{22})=\Phi(T_{11})+\Phi(T_{22})$.

For every $M_{12}\in{\mathcal{M}}_{12}$,
$[T_{11}+T_{22},M_{12}]_{\xi}=T_{11}M_{12}-\xi M_{12}T_{22}$. From Claim 5, it
follows that

$\begin{array}[]{rl}&[\Phi(T_{11}+T_{22}),M_{12}]_{\xi}+[T_{11}+T_{22},\Phi(M_{12})]_{\xi}=\Phi([T_{11}+T_{22},M_{12}]_{\xi})\\\
=&\Phi(T_{11}M_{12})+\Phi(-\xi
M_{12}T_{22})=\Phi([T_{11},M_{12}]_{\xi})+\Phi([T_{22},M_{12}]_{\xi})\\\
=&[\Phi(T_{11}),M_{12}]_{\xi}+[T_{11},\Phi(M_{12})]_{\xi}+[\Phi(T_{22}),M_{12}]_{\xi}+[T_{22},\Phi(M_{12})]_{\xi}.\end{array}$

Thus $[\Phi(T_{11}+T_{22})-\Phi(T_{11})-\Phi(T_{22}),M_{12}]_{\xi}=0$. By
Lemma 2.2,

$\Phi(T_{11}+T_{22})-\Phi(T_{11})-\Phi(T_{22})\in{\mathcal{M}}_{12}+(\xi
P_{1}+P_{2}){\mathcal{Z}}_{\mathcal{M}}.$

On the other hand, $[T_{11}+T_{22},\frac{P_{1}}{1-\xi}]_{\xi}=T_{11}$. From
the proof of Claim 3, one can see $\Phi(\frac{P_{1}}{1-\xi})=0$. Hence
$[\Phi(T_{11}+T_{22}),\frac{P_{1}}{1-\xi}]_{\xi}=\Phi(T_{11})$, i.e.,
$(1-\xi)\Phi(T_{11})=\Phi(T_{11}+T_{22})P_{1}-\xi P_{1}\Phi(T_{11}+T_{22})$.
Multiplying by $P_{1}$ and $P_{2}$ from the left and the right in the above
equation, we have $P_{1}\Phi(T_{11}+T_{22})P_{2}=0$. So

$\Phi(T_{11}+T_{22})-\Phi(T_{11})-\Phi(T_{22})=(\xi P_{1}+P_{2})Z$

for some central element $Z\in Z_{\mathcal{M}}$. Combining $\Phi(P_{1})=0$ and
Claim 3, we conclude

$\begin{array}[]{ll}&\Phi([T_{11},P_{1}]_{\xi})=\Phi([T_{11}+T_{22},P_{1}]_{\xi})\\\
=&[\Phi(T_{11}+T_{22}),P_{1}]_{\xi}+[T_{11}+T_{22},\Phi(P_{1})]_{\xi}\\\
=&[\Phi(T_{11})+(\xi P_{1}+P_{2})Z,P_{1}]_{\xi}.\end{array}$

Thus $[(\xi P_{1}+P_{2})Z,P_{1}]_{\xi}=0$ which implies $Z=0$. This gives
$\Phi(T_{11}+T_{22})=\Phi(T_{11})+\Phi(T_{22})$.

Claim 7. For every $T_{ii},S_{ii}\in{\mathcal{M}}_{ii}$ $(i=1,2)$,
$\Phi(T_{ii}+S_{ii})=\Phi(T_{ii})+\Phi(S_{ii})$.

Assume $i=1$. For every $M_{12}\in{\mathcal{M}}_{12}$,
$[T_{11}+S_{11},M_{12}]_{\xi}=T_{11}M_{12}+S_{11}M_{12}$. From Claim 5, it
follows that

$\begin{array}[]{rl}&[\Phi(T_{11}+S_{11}),M_{12}]_{\xi}+[T_{11}+S_{11},\Phi(M_{12})]_{\xi}=\Phi([T_{11}+S_{11},M_{12}]_{\xi})\\\
=&\Phi(T_{11}M_{12})+\Phi(S_{11}M_{12})=\Phi([T_{11},M_{12}]_{\xi})+\Phi([S_{11},M_{12}]_{\xi})\\\
=&[\Phi(T_{11}),M_{12}]_{\xi}+[T_{11},\Phi(M_{12})]_{\xi}+[\Phi(S_{11}),M_{12}]_{\xi}+[S_{11},\Phi(M_{12})]_{\xi}.\end{array}$

Thus $[\Phi(T_{11}+S_{11})-\Phi(T_{11})-\Phi(S_{11}),M_{12}]_{\xi}=0$. By
Lemma 2.2,

$\Phi(T_{11}+S_{11})-\Phi(T_{11})-\Phi(S_{11})\in{\mathcal{M}}_{12}+(\xi
P_{1}+P_{2}){\mathcal{Z}}.$

On the other hand, Claim 3 tells us that
$P_{1}(\Phi(T_{11}+S_{11})-\Phi(T_{11})-\Phi(S_{11}))P_{2}=0$. So

$\Phi(T_{11}+S_{11})-\Phi(T_{11})-\Phi(S_{11})=(\xi P_{1}+P_{2})Z$

for some $Z\in{\mathcal{Z}}_{\mathcal{M}}$. This further indicates

$\displaystyle 0$
$\displaystyle=\Phi([T_{11}+S_{11},P_{2}]_{\xi})=[\Phi(T_{11}+S_{11}),P_{2}]_{\xi}$
$\displaystyle=[\Phi(T_{11})+\Phi(S_{11})+(\xi P_{1}+P_{2})Z,P_{2}]_{\xi}$
$\displaystyle=[(\xi P_{1}+P_{2})Z,P_{2}]_{\xi}.$

Then $P_{2}Z=0$, consequently, $Z=0$. That is, $\Phi$ is additive on
${\mathcal{M}}_{11}$. Similarly, $\Phi$ is additive on ${\mathcal{M}}_{22}$.

Claim 8. For every $T_{ii},T_{jj},T_{ij}$, $(1\leq i\neq j\leq 2)$
$\Phi(T_{ii}+T_{jj}+T_{ij})=\Phi(T_{ii})+\Phi(T_{jj})+\Phi(T_{ij})$ .

Assume $i=1,j=2$. For every $M_{12}\in{\mathcal{M}}_{12}$,
$[T_{11}+T_{22}+T_{12},M_{12}]_{\xi}=[T_{11}+T_{22},M_{12}]_{\xi}$. By Claim
6, it follows that

$[\Phi(T_{11}+T_{22}+T_{12}),M_{12}]_{\xi}+[T_{11}+T_{22}+T_{12},\Phi(M_{12})]_{\xi}=[\Phi(T_{11})+\Phi(T_{22}),M_{12}]_{\xi}+[T_{11}+T_{22},\Phi(M_{12})]_{\xi}.$

Thus $[\Phi(T_{11}+T_{22}+T_{12})-\Phi(T_{11})-\Phi(T_{22}),M_{12}]_{\xi}=0.$
From Lemma 2.2 and Claim 3, we obtain

$\begin{array}[]{ll}&\Phi(T_{11}+T_{22}+T_{12})-\Phi(T_{11})-\Phi(T_{22})\\\
=&P_{1}(\Phi(T_{11}+T_{22}+T_{12})-\Phi(T_{11})-\Phi(T_{22}))P_{2}+(\xi
P_{1}+P_{2})Z\\\ =&P_{1}\Phi(T_{11}+T_{22}+T_{12})P_{2}+(\xi
P_{1}+P_{2})Z\end{array}$

for some central element $Z$. A direct computation shows that

$\displaystyle\Phi(T_{12})$
$\displaystyle=\Phi(P_{1}(T_{11}+T_{22}+T_{12})P_{2})$
$\displaystyle=\Phi([P_{1},[T_{11}+T_{22}+T_{12},P_{2}]_{\xi}]_{\xi})$
$\displaystyle=[P_{1},[\Phi([T_{11}+T_{22}+T_{12},P_{2}]_{\xi})]_{\xi}$
$\displaystyle=P_{1}\Phi(T_{11}+T_{22}+T_{12})P_{2}.$

Thus

$\Phi(T_{11}+T_{22}+T_{12})=\Phi(T_{11})+\Phi(T_{22})+\Phi(T_{12})+(\xi
P_{1}+P_{2})Z.$

It is easy to see

$\begin{array}[]{ll}&[\Phi(T_{11}+T_{22}+T_{12}),P_{2}]_{\xi}=\Phi([T_{11}+T_{22}+T_{12},P_{2}]_{\xi})\\\
&=\Phi([T_{12}+T_{22},P_{2}]_{\xi})=[\Phi(T_{12})+\Phi(T_{22}),P_{2}]_{\xi}\end{array}.$

Then $[(\xi P_{1}+P_{2})Z,P_{2}]=0$, $ZP_{2}=0$ which implies $Z=0$. That is
$\Phi(T_{11}+T_{22}+T_{12})=\Phi(T_{11})+\Phi(T_{22})+\Phi(T_{12})$. The rest
goes similarly.

Claim 9. For every $T_{11},T_{12},T_{21},T_{22}$,

$\Phi(T_{11}+T_{12}+T_{21}+T_{22})-\Phi(T_{11})-\Phi(T_{12})-\Phi(T_{21})-\Phi(T_{22})\in(\xi
P_{1}+P_{2}){\mathcal{Z}}_{\mathcal{M}}\cap(P_{1}+\xi
P_{2}){\mathcal{Z}}_{\mathcal{M}}.$

Consequently, if $\xi\neq-1$,
$\Phi(T_{11}+T_{12}+T_{21}+T_{22})=\Phi(T_{11})+\Phi(T_{12})+\Phi(T_{21})+\Phi(T_{22})$.

For every $M_{12}\in{\mathcal{M}}_{12}$,
$[T_{11}+T_{12}+T_{21}+T_{22},M_{12}]_{\xi}=[T_{11}+T_{21}+T_{22},M_{12}]_{\xi}$.
From Claim 8, it follows that

$\begin{array}[]{ll}&[\Phi(T_{11}+T_{12}+T_{21}+T_{22}),M_{12}]_{\xi}+[T_{11}+T_{12}+T_{21}+T_{22},\Phi(M_{12})]_{\xi}\\\
=&[\Phi(T_{11})+\Phi(T_{21})+\Phi(T_{22}),M_{12}]_{\xi}+[T_{11}+T_{21}+T_{22},\Phi(M_{12})]_{\xi}.\end{array}$

Thus

$[\Phi(T_{11}+T_{12}+T_{21}+T_{22})-\Phi(T_{11})-\Phi(T_{21})-\Phi(T_{22}),M_{12}]_{\xi}=0.$

Since $\Phi(T_{12})\in{\mathcal{M}}_{12}$, we have

$[\Phi(T_{11}+T_{12}+T_{21}+T_{22})-\Phi(T_{11})-\Phi(T_{12})-\Phi(T_{21})-\Phi(T_{22}),M_{12}]_{\xi}=0.$

Similarly, from
$[T_{11}+T_{12}+T_{21}+T_{22},M_{21}]_{\xi}=[T_{11}+T_{12}+T_{22},M_{21}]_{\xi},$
we can obtain

$[\Phi(T_{11}+T_{12}+T_{21}+T_{22})-\Phi(T_{11})-\Phi(T_{12})-\Phi(T_{21})-\Phi(T_{22}),M_{21}]_{\xi}=0.$

From Lemma 2.2, it follows that

$\Phi(T_{11}+T_{12}+T_{21}+T_{22})-\Phi(T_{11})-\Phi(T_{12})-\Phi(T_{21})-\Phi(T_{22})\in(\xi
P_{1}+P_{2}){\mathcal{Z}}_{\mathcal{M}}\cap(P_{1}+\xi
P_{2}){\mathcal{Z}}_{\mathcal{M}}.$

Note that if $\xi\neq-1$, $(\xi
P_{1}+P_{2}){\mathcal{Z}}_{\mathcal{M}}\cap(P_{1}+\xi
P_{2}){\mathcal{Z}}_{\mathcal{M}}=\\{0\\}$. Thus

$\Phi(T_{11}+T_{12}+T_{21}+T_{22})=\Phi(T_{11})+\Phi(T_{12})+\Phi(T_{21})+\Phi(T_{22}).$

Claim 10. If $\xi=-1$,
$\Phi(T_{11}+T_{12}+T_{21}+T_{22})=\Phi(T_{11})+\Phi(T_{12})+\Phi(T_{21})+\Phi(T_{22})$
holds true, too.

By Claim 9, we may assume
$\Phi(T_{12}+T_{21})=\Phi(T_{12})+\Phi(T_{21})+(-P_{1}+P_{2})Z_{1}$,
$\Phi(T_{11}+T_{12}+T_{21})=\Phi(T_{11})+\Phi(T_{12})+\Phi(T_{21})+(-P_{1}+P_{2})Z_{2}$
and
$\Phi(T_{11}+T_{12}+T_{21}+T_{22})=\Phi(T_{11})+\Phi(T_{12})+\Phi(T_{21})+\Phi(T_{22})+(-P_{1}+P_{2})Z_{3}$.
The following is devoted to showing $Z_{1}=Z_{2}=Z_{3}=0$. Since
$\Phi(T_{12}+T_{21})=\Phi([T_{12}+T_{21},P_{1}]_{-1})=[\Phi(T_{12}+T_{21}),P_{1}]_{-1}$,
substituting
$\Phi(T_{12}+T_{21})=\Phi(T_{12})+\Phi(T_{21})+(-P_{1}+P_{2})Z_{1}$ into above
equation, we have
$(-P_{1}+P_{2})Z_{1}=[(-P_{1}+P_{2})Z_{1},P_{1}]_{-1}=-2P_{1}Z_{1}$. Then
$Z_{1}P_{1}=Z_{1}P_{2}=0$ and so $Z_{1}=0$. From

$\begin{array}[]{ll}&[\Phi(T_{11})+\Phi(T_{12})+\Phi(T_{21})+(-P_{1}+P_{2})Z_{2},P_{2}]_{-1}\\\
=&[\Phi(T_{11}+T_{12}+T_{21}),P_{2}]_{-1}=\Phi([T_{11}+T_{12}+T_{21},P_{2}]_{-1})\\\
=&\Phi(T_{12}+T_{21})=\Phi(T_{12})+\Phi(T_{21})\\\
=&[\Phi(T_{11})+\Phi(T_{12})+\Phi(T_{21}),P_{2}]_{-1},\end{array}$

it follows that $[(-P_{1}+P_{2})Z_{2},P_{2}]=0$. Thus $Z_{2}=0$. At last,

$\begin{array}[]{ll}&[\Phi(T_{11})+\Phi(T_{12})+\Phi(T_{21})+\Phi(T_{22})+(-P_{1}+P_{2})Z_{3},P_{1}]_{-1}\\\
=&[\Phi(T_{11}+T_{12}+T_{21}+T_{22}),P_{1}]_{-1}=\Phi([T_{11}+T_{12}+T_{21}+T_{22},P_{1}]_{-1})\\\
=&\Phi([T_{11}+T_{12}+T_{21},P_{1}]_{-1})=[\Phi(T_{11})+\Phi(T_{12})+\Phi(T_{21})+\Phi(T_{22}),P_{1}]_{-1}.\end{array}$

So $[(-P_{1}+P_{2})Z_{3},P_{1}]_{-1}=0$ which implies $Z_{3}=0$. Hence
$\Phi(T_{11}+T_{12}+T_{21}+T_{22})=\Phi(T_{11})+\Phi(T_{12})+\Phi(T_{21})+\Phi(T_{22})$,
as desired.

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