# Status of New Physics searches with
${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$ transitions @ LHCb

Simone Bifani
On behalf of the LHCb collaboration  University of Birmingham, School of
Physics and Astronomy, Birmingham, United Kingdom

###### Abstract

Rare decays of heavy-flavoured particles provide an ideal laboratory to look
for deviations from the Standard Model, and explore energy regimes beyond the
LHC reach. Decays proceeding via electroweak penguin diagrams are excellent
probes to search for New Physics, and
${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$ processes are particularly
interesting since they give access to many observables such as branching
fractions, asymmetries and angular observables. Recent results from the LHCb
experiment are reviewed.

##  1 Introduction

Flavour Changing Neutral Current (FCNC) processes, where a quark changes its
flavour without altering its electric charge, are forbidden at tree level in
the Standard Model (SM) and can only occur via loop diagrams. This makes such
transitions rare and, due to the lack of a dominant tree-level SM
contribution, sensitive to new unobserved particles that can show up either as
a sizeable increase or decrease in the rate of particular decays, or as a
change in the angular distribution of the particles in the detector. A good
laboratory to study FCNC are decays of a $b$ quark into an $s$ quark and two
leptons, ${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$, which are described by
the electroweak diagrams shown in Fig. 1 (left).

The LHCb detector [1, 2] is a single arm spectrometer fully instrumented in
the forward region and designed to study heavy-flavoured hadrons. During
Run-1, LHCb collected about 1 and 2 $\mbox{\,fb}^{-1}$ of ${p}{p}$ collisions
at centre-of-mass energies of 7 and 8 $\mathrm{\,Te\kern-1.00006ptV}$,
respectively. Due to the large production cross-section in the forward
direction these data provide unprecedentedly large numbers of $B$ and
${\mathchar 28931\relax}^{0}_{b}$ hadron decays. A flexible trigger system,
excellent momentum and impact parameter resolutions, and the most performant
vertexing and particle identification capabilities at the LHC, make LHCb the
ideal place to look for New Physics (NP) through precise studies of rare
$b$-quark processes. Recent measurements of semileptonic $b$-hadron decays are
discussed.

Figure 1: (left) Electroweak box and penguin diagrams describing the
transition of a $b$ quark into an $s$ quark and two leptons. (right) Sketch of
the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ differential branching
fraction as a function of the dilepton invariant mass squared. Different
Wilson coefficients can be probed in different $q^{2}$ regions.

##  2 Branching fractions

The most basic observable that physics beyond the SM can affect is the rate at
which a particular decay occurs, which motivated the LHCb collaboration to
perform the measurement of the branching fraction of a series of FCNC
processes. All measurements are performed as a function of the dilepton
invariant mass squared, $q^{2}$, and compared against the SM prediction. Final
states with muons are considered as being experimentally easier to
reconstruct. The dependence of the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ differential branching
fraction as a function of $q^{2}$ is shown in Fig. 1 (right). Figure 2 shows
the differential branching fraction of a collection of
${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$ decays of $B$ and ${\mathchar
28931\relax}^{0}_{b}$ hadrons, as well as partner transitions such as
${b}\\!\rightarrow{d}\,{\ell^{+}}{\ell^{-}}$. In all cases, the experimental
uncertainty is dominated by the limited statistics of the samples available in
the Run-1 data set.

In the region below $\sim 6\leavevmode\nobreak\
{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ in $q^{2}$, the SM predictions
consistently overshoot the data, a common trend that is observed both in the
mesonic and in the baryonic sectors. The largest deviation is found in the
decay ${{B}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}\mu^{-}}$ in the region
$1<{q^{2}}<6\leavevmode\nobreak\
{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, where the data are $3.3\sigma$
below the prediction [3]. Decays of ${\mathchar 28931\relax}^{0}_{b}$ hadrons
are also overestimated in the SM, however predictions are currently much less
precise than for $B$ mesons [4]. Finally, although the
${{B}^{+}}\\!\rightarrow{{\pi}^{+}}{\mu^{+}\mu^{-}}$ branching fraction is
generally compatible with the prediction, agreement in the lowest-$q^{2}$ bin
is only achieved when contributions from $\rho$ and $\omega$ resonances are
taken into account [5].

Figure 2: Differential branching fraction of the decay (top)
${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{+}\mu^{-}}$ [6],
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ [7],
${{B}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}\mu^{-}}$ [3] and (bottom)
${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar
28931\relax}{\mu^{+}\mu^{-}}$ [4],
${{B}^{+}}\\!\rightarrow{{\pi}^{+}}{\mu^{+}\mu^{-}}$ [5], compared to SM
predictions.

##  3 Angular analyses

Besides using branching fractions, much stronger constraints to possible
extensions of the SM can be set by studying the angular distribution of the
final state particles of FCNC decays. Depending on the decay mode and on the
size of the available data sample, full or simplified angular analyses have
been performed.

The decay ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ has a complex
angular structure that can be fully described by three angles and $q^{2}$, and
that provides many observables sensitive to different types of NP. The LHCb
collaboration has performed the first full angular analysis of this mode, and
measured the full set of CP-averaged angular terms, their correlations, as
well as the full set of CP-asymmetries [7]. The forward-backward asymmetry of
the dimuon system, $A_{FB}$, and the longitudinal polarisation fraction of the
${K}^{*0}$, $F_{L}$, compared to the SM prediction are presented in Fig. 3
(top). There is general consistency, but, similarly to the branching fraction,
large uncertainties due to the hadronic-matrix elements affect the
predictions. However, it is possible to construct ratios of observables that
are less dependent on the form-factors and that can be theoretically
determined with improved precision [8]. Figure 3 (top right) shows one of such
observables, $P_{5}^{\prime}$, which manifests a local deviation from the SM
in the region between 4 and 8 ${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$
in $q^{2}$ of about $3\sigma$. An angular analysis of the decay
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ in the $q^{2}$ range between
0.002 and 1.120 ${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ has also been
performed by LHCb [9], where all the measured observables are found to be
consistent with the predictions.

Although the decay ${{B}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}\mu^{-}}$ has a
reduced number of angular observables that can be accessed compared to
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\ell^{+}}{\ell^{-}}$ modes, a full angular
analysis has also been performed [3]. Figure 3 (bottom left) shows $F_{L}$,
which is found to be in good agreement with the SM prediction. Finally,
because baryonic transitions allow to extract complementary information to
that available in decays of $B$ mesons, LHCb has performed the first angular
analysis of ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar
28931\relax}{\mu^{+}\mu^{-}}$ [4]. Two forward-backward asymmetries, one in
the hadronic and one in the leptonic system, have been measured. While the
former is in good agreement with the SM, the latter is consistently above the
prediction, as reported in Fig. 3 (bottom right).

Figure 3: Angular observables of the decay (top)
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ [7] and (bottom)
${{B}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}\mu^{-}}$ [3], ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$
[4], compared to SM predictions.

##  4 Tests of lepton universality

Due to the equality of the electroweak couplings of gauge bosons to leptons,
the decay rate of processes whose final states only differ by the flavour of
the participating leptons are expected to be the same in the SM, except from
very small Higgs penguin contributions and difference in phase space due to
the lepton masses. In particular, ratios of branching fractions represent a
powerful null test of the SM as theoretical uncertainties largely cancel in
the predictions, and experimental systematics are much reduced.

In the SM the ratio $R_{{K}}=\frac{{\cal
B}({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{+}\mu^{-}})}{{\cal
B}({{B}^{+}}\\!\rightarrow{{K}^{+}}{e^{+}e^{-}})}$ is precisely predicted to
be $1\pm\mathcal{O}(10^{-3})$ [10]. The LHCb collaboration has performed the
most precise test of lepton universality using
${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{+}\mu^{-}}$ and
${{B}^{+}}\\!\rightarrow{{K}^{+}}{e^{+}e^{-}}$ decays to date [11]. The
experimental result in the range $1<{q^{2}}<6\leavevmode\nobreak\
{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ is
$R_{{K}}=0.745^{+0.090}_{-0.074}\mathrm{\,(stat)}\pm 0.036\mathrm{\,(syst)}$,
which manifests a tension with the SM prediction at the level of $2.6\sigma$,
as shown in Fig. 4.

Several other tests of lepton universality are currently being carried out by
the LHCb collaboration, most notably the measurement of the ratio of the
branching fractions of ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ and
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$, $R_{{{K}^{*0}}}$. The
experimental environment in which the detector operates leads to significant
differences in the treatment of decays involving muons or electrons in the
final state. Particularly, the much larger amount of bremsstrahlung emitted by
the electrons and the different trigger thresholds due to the higher occupancy
of the calorimeters than of the muon stations result in a difference in the
reconstruction efficiency of about a factor five in favour of the muonic mode.
Figure 5 shows the distribution of the di-lepton invariant mass squared as a
function of the four-body invariant mass of the ${B}^{0}$ candidates for final
states with muons and electrons, where the second exhibit a more prominent
radiative tail of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}$ resonances as well as a stronger contamination by partially-
reconstructed backgrounds. The efficiency corrections are controlled by
measuring the ratio $\frac{{\cal
B}({{B}^{0}}\\!\rightarrow{{K}^{*0}}{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\\!\rightarrow{\mu^{+}\mu^{-}}))}{{\cal
B}({{B}^{0}}\\!\rightarrow{{K}^{*0}}{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\\!\rightarrow{e^{+}e^{-}}))}$, as well as determining
$R_{{{K}^{*0}}}$ as a double ratio to the resonant modes
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\\!\rightarrow{\ell^{+}}{\ell^{-}})$. A preliminary result has been
presented at a CERN-LHC Seminar [12].

Independent tests of lepton universality are also being carried out using
${b}\\!\rightarrow{c}\ell\nu_{\ell}$ decays with tau leptons in the final
state. The ratio of branching fractions $R_{{{D}^{*}}}=\frac{{\cal B}({{\kern
1.25995pt\overline{\kern-1.25995ptB}{}}{}^{0}}\\!\rightarrow{{D}^{*+}}{\tau^{-}}{{\overline{\nu}}_{\tau}})}{{\cal
B}({{\kern
1.25995pt\overline{\kern-1.25995ptB}{}}{}^{0}}\\!\rightarrow{{D}^{*+}}{\mu^{-}}{{\overline{\nu}}_{\mu}})}$
is measured to be $R_{{{D}^{*}}}=0.336\pm 0.027\mathrm{\,(stat)}\pm
0.030\mathrm{\,(syst)}$, which is $2.1\sigma$ larger than the value expected
in the SM [13].

Figure 4: Measurement of $R_{{K}}=\frac{{\cal
B}({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{+}\mu^{-}})}{{\cal
B}({{B}^{+}}\\!\rightarrow{{K}^{+}}{e^{+}e^{-}})}$ [11], compared to previous
experiments and to the SM prediction.

Figure 5: Di-lepton invariant mass squared, $q^{2}$, as a function of the
four-body invariant mass of the ${B}^{0}$ candidates for
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\ell^{+}}{\ell^{-}}$ final states with
(left) muons and (right) electrons.

##  5 Summary and conclusions

Recent rare decays of $B$ and ${\mathchar 28931\relax}^{0}_{b}$ hadrons
performed by the LHCb collaboration have been presented. While most of the
observables are in good agreement with the SM predictions, some intriguing
tensions have been observed, most notably in branching fractions of
${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$ processes in the low region of
$q^{2}$, in the $P_{5}^{\prime}$ angular observable in
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$, and in the $R_{{K}}$ and
$R_{{{D}^{*}}}$ ratios.

Several attempts to interpret these anomalies have been made by performing
global fits to various ${b}\rightarrow{s}$ observables from different
experiments [14, 15, 16]. All these fits point to a tension between the data
and the SM with a significance of around 3–$4\sigma$. Different models have
been proposed to account for such effects, for example containing a new heavy
gauge boson $Z^{\prime}$ [17, 18, 19, 20, 21, 22] or leptoquarks [23, 24], but
a definitive explanation has yet to be found.

The current status strongly motivates further work both in the theory and in
the experimental side to clarify the present observations. With the upcoming
Run-2 data LHCb will continue to perform analyses of rare $b$-quark decays,
including additional tests of lepton universality such as $R_{\phi}$.

## References

## References

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# On projectional skeletons and the Plichko property in Lipschitz-free Banach
spaces

Antonio J. Guirao  Universitat Politècnica de València. Instituto
Universitario de Matemática Pura y Aplicada, Camino de Vera, s/n 46022
Valencia (Spain<EMAIL_ADDRESS>,  Vicente Montesinos  Universitat
Politècnica de València. Instituto Universitario de Matemática Pura y
Aplicada, Camino de Vera, s/n 46022 Valencia (Spain<EMAIL_ADDRESS>and  Andrés Quilis  Universitat Politècnica de València. Instituto
Universitario de Matemática Pura y Aplicada, Camino de Vera, s/n 46022
Valencia (Spain); and Czech Technical University in Prague, Faulty of
Electrical Engineering. Department of Mathematics, Technická 2, 166 27 Praha 6
(Czech Republic<EMAIL_ADDRESS>
###### Abstract.

We study projectional skeletons and the Plichko property in Lipschitz-free
spaces, relating these concepts to the geometry of the underlying metric
space. Specifically, we identify a metric property that characterizes the
Plichko property witnessed by Dirac measures in the associated Lipschitz-free
space. We also show that the Lipschitz-free space of all $\mathbb{R}$-trees
has the Plichko property witnessed by molecules, and define the concept of
retractional trees to generalize this result to a bigger class of metric
spaces. Finally, we show that no separable subspace of $\ell_{\infty}$
containing $c_{0}$ is an $r$-Lipschitz retract for $r<2$, which implies in
particular that $\mathcal{F}(\ell_{\infty})$ is not $r$-Plichko for $r<2$.

###### Key words and phrases:

Lipschitz retractions, Projectional Skeletons, Plichko

###### 2020 Mathematics Subject Classification:

46B20; 46B08; 46B26

##  1\. Introduction

Let $M$ be a complete metric space with a distinguished point $0\in M$. The
vector space $\text{Lip}_{0}(M)=\\{f\colon M\rightarrow\mathbb{R}\colon
f\text{ is Lipschitz and }f(0)=0\\}$ with the norm given by the Lipschitz
constant is a Banach space, which is moreover a dual space. Its canonical
predual is the subspace
$\mathcal{F}(M)=\overline{\text{span}}\\{\delta(x)\colon x\in
M\\}\subset\text{Lip}_{0}(M)^{*}$, where
$\delta(x)\colon\text{Lip}_{0}(M)\rightarrow\mathbb{R}$ is the functional
satisfying $\langle f,\delta(x)\rangle=f(x)$ for all $f\in\text{Lip}_{0}(M)$.
This space, introduced by Arens and Eells in [AE56], is also commonly known as
the _Lipschitz-free_ space associated to $M$, named by Godefroy and Kalton in
the seminal article [GK03]. The above Dirac map $\delta$ is an isometry
embedding, which allows us to regard metric spaces as subsets of their
Lipschitz-free space. With this identification, every Lipschitz map $F\colon
M\rightarrow N$ between two metric spaces such that $F(0)=0$ can be extended
to a linear and bounded operator $\widehat{F}\colon\mathcal{F}(M)\rightarrow
Y$ preserving the Lipschitz constant. Since the publication of [GK03],
Lipschitz-free spaces have been studied extensively and continue to be a very
active area of research in Functional Analysis. We refer to the survey [God15]
and the monograph [Wea18] for a complete overview of the topic of Lipschitz-
free spaces and spaces of Lipschitz functions.

As a continuation of [HQ22], in this note, we study the linear projectional
structure of Lipschitz-free spaces associated to non-separable metric spaces,
which is closely related to the Lipschitz retractional structure of the
underlying metric space thanks to the universal extension property that
characterizes Lipschitz-free spaces. To this end, we focus on the classical
concept of the Plichko property and some related notions.

For a real number $r\geq 1$, a Banach space $X$ is said to be _$r$ -Plichko_
if there exists a pair $(\Delta,N)$, where $\Delta\subset X$ is a linearly
dense subset of $X$ and $N$ is an $r$-norming subspace of $X^{*}$ such that
for every functional $f\in N$, the set

$S_{\Delta}(f)=\\{x\in\Delta\colon\langle f,x\rangle\neq 0\\}$

is countable. We say that the pair $(\Delta,N)$ is a _witness_ of the Plichko
property in $X$. Since $N$ is essentially determined by the linearly dense set
$\Delta$, we sometimes say, equivalently, that $X$ is Plichko witnessed by the
set $\Delta$.

This concept has been deeply studied in the theory of non-separable Banach
spaces. Plichko Banach spaces enjoy geometric and structural properties such
as the existence of a LUR renorming, a strong Markushevich basis, or the
Separable Complementation Property (SCP). We refer to the survey [Kal00] and
the monograph [Háj+08] for a complete introduction to this property.

We are also interested in the characterization of the Plichko property in
terms of the following concept due to Kubiś:

######  Definition 1.1 ([Kub09]).

Let $X$ be a Banach space. A _projectional skeleton_ on $X$ is a family
$\\{P_{s}\\}_{s\in\Gamma}$ of bounded linear projections on $X$ indexed by a
directed partially ordered set $\Gamma$, such that the following conditions
hold:

  * (i)

$P_{s}X$ is separable for all $s\in\Gamma$.

  * (ii)

$P_{s}P_{t}=P_{t}P_{s}=P_{s}$ whenever $s,t\in\Gamma$ and $s\leq t$.

  * (iii)

If $(s_{n})_{n}$ is an increasing sequence of indices in $\Gamma$, then
$s=\sup_{n\in\mathbb{N}}s_{n}$ exists and
$P_{s}X=\overline{\bigcup_{n\in\mathbb{N}}P_{s_{n}}X}$.

  * (iv)

$X=\bigcup_{s\in\Gamma}P_{s}X$.

If $r\geq 1$, we say that an _$r$ -projectional skeleton_ is a projectional
skeleton where every projection has norm less than or equal to $r$. We say
that a projectional skeleton is commutative if $P_{s}P_{t}=P_{t}P_{s}$ for all
$s,t\in\Gamma$, regardless of whether they are comparable or not.

Indeed, it is proven in [Kub09] that a Banach space is $r$-Plichko if and only
if it admits a commutative $r$-projectional skeleton.

We can define an analogous concept to projectional skeletons in metric spaces
using Lipschitz retractions:

######  Definition 1.2.

Let $M$ be a metric space. A _Lipschitz retractional skeleton_ on $M$ is a
family $\\{R_{s}\\}_{s\in\Gamma}$ of Lipschitz retractions on $M$ indexed by a
directed partially ordered set $\Gamma$, such that the following conditions
hold:

  * (i)

$R_{s}(M)$ is a separable subset of $M$ for every $s\in\Gamma$.

  * (ii)

If $s,t\in\Gamma$ such that $s\leq t$, then $R_{s}\circ R_{t}=R_{t}\circ
R_{s}=R_{s}$.

  * (iii)

If $(s_{n})_{n\in\mathbb{N}}$ is a totally ordered sequence in $\Gamma$, then
$s=\text{sup}_{n\in\mathbb{N}}s_{n}$ exists ($\Gamma$ is $\sigma$-complete)
and moreover $R_{s}(M)=\overline{\bigcup_{n\in\mathbb{N}}R_{s_{n}}(M)}$.

  * (iv)

$M=\bigcup_{s\in\Gamma}R_{s}(M)$.

Given $r\geq 1$, an $r$-Lipschitz retractional skeleton is a Lipschitz
retractional skeleton with Lipschitz constant uniformly bounded by $r$. A
Lipschitz retractional skeleton is _commutative_ is $R_{s}\circ
R_{t}=R_{t}\circ R_{t}$ for every $s,t\in\Gamma$, regardless of whether they
are comparable or not.

The linearization property of Lipschitz-free spaces implies that if a metric
space admits a (commutative) Lipschitz retractional skeleton, the associated
Lipschitz-free space has a (commutative) projectional skeleton. This fact,
together with the characterization of the Plichko property in terms of
commutative projectional skeletons, shows that the Lipschitz-free space
associated to a Plichko space is again Plichko (see Corollary 2.9 in [HQ22]).

In brief, metric spaces with a rich Lipschitz retractional structure yield
Lipschitz-free spaces with a rich linear projectional structure (the converse
is not always true, as we discuss in section 5). It is unknown whether there
exists a metric space whose Lipschitz-free space does not have Plichko. It is
also open whether every Lipschitz-free space has the SCP.

In section 2, we characterize which metric spaces $M$ yield a Lipschitz-free
space with the Plichko property witnessed by a subset of
$\delta(M)\subset\mathcal{F}(M)$. We show that in this special case, the
linear projectional structure of the Lipschitz-free space is strong enough to
induce a Lipschitz retractional structure in the underlying space. We offer
several equivalent properties, some relating to the structure of the
Lipschitz-free space and some relating to the geometry of the underlying
metric space.

In section 3, we study the Lipschitz retractional structure of
$\mathbb{R}$-trees (or metric trees). This class of metric spaces has been
studied in the context of Lipschitz-free spaces in recent years, as it was
shown in [God10] by Godard that the Lipschitz-free space of a complete metric
space $M$ is isometric to a subspace of $L_{1}$ if and only if $M$ is
isometric to a subset of a metric tree. See [APP21] as well for a more recent
characterization of metric spaces whose Lipschitz-free space is isometric to a
subspace of $\ell_{1}$ as those which are isometric to subsets of separable
$\mathbb{R}$-trees where the length measure of the closure of the set of
branching points is $0$. We show in our note that the Lipschitz-free space of
every $\mathbb{R}$-tree is $1$-Plichko witnessed by a set of molecules in
$\mathcal{F}(M)$.

In section 4, we introduce the concept of _retractional trees_ in metric
spaces generalizing the result of the previous section to a bigger class of
metric spaces.

Finally, in section 5, we summarize the relationship between the linear
structure of Lipschitz-free spaces associated to Banach spaces and the
Lipschitz retractional structure of the underlying space, and we use this
connection to show that $\mathcal{F}(\ell_{\infty})$ is not $r$-Plichko for
$r<2$.

We finish the introduction by setting the notation: Unless stated otherwise,
every metric space considered in this note is assumed to be complete and
containing a distinguished point $0$. Given a metric space $M$, a point $p\in
M$ and a positive number $r>0$, we will denote by $B(p,r)$ the open ball of
radius $r$ centered at $p$. Given two different points $p,q\in M$, we will
denote by $m_{p,q}$ the (elementary) molecule
$\frac{\delta(p)-\delta(q)}{d(p,q)}\in\mathcal{F}(M)$.

##  2\. Plichko property witnessed by Dirac measures

Let us introduce the main result of this section:

######  Theorem 2.1.

Let $M$ be a metric space with distinguished point $0\in M$, and let
$\lambda\geq 1$. The following statements are equivalent:

  * (i)

For all $p\in M$ and for all $r<\frac{1}{\lambda}$, the ball $B(p,r\cdot
d(p,0))$ is separable.

  * (ii)

$\mathcal{F}(M)$ is $\lambda$-Plichko witnessed by a subset of $\delta(M)$.

  * (iii)

$\mathcal{F}(M)$ admits a commutative $\lambda$-projectional skeleton
$\\{P_{s}\\}_{s\in\Gamma}$ such that $P_{s}(\delta(p))\in\\{0,\delta(p)\\}$
for all $p\in M$.

  * (iv)

$M$ admits a commutative $\lambda$-Lipschitz retractional skeleton
$\\{R_{s}\\}_{s\in\Gamma}$ such that $R_{s}(p)=\\{0,p\\}$ for all $p\in M$.

  * (v)

The closed subspace $\\{f\in\text{Lip}_{0}(M)\colon\text{supp}(f)\text{ is
separable}\\}$ is a $\lambda$-norming subspace of $\text{Lip}_{0}(M)$.

###### Proof.

Write $S_{0}(M)=\\{f\in\text{Lip}_{0}(M)\colon\text{supp}(f)\text{ is
separable}\\}$ for the rest of the proof.

Let us start by proving that $(i)$ and $(v)$ are equivalent. Indeed, the proof
that $(i)$ implies $(v)$ is already included in [HQ22, Proposition 2.5]. To
show the converse, suppose by contradiction that $S_{0}(M)$ is a
$\lambda$-norming subspace of $\text{Lip}_{0}(M)$ and that there exists a
point $p_{0}\in M$ different from $0$ and $0<r_{0}<\frac{1}{\lambda}$ such
that the open ball $B(p_{0},r_{0}\cdot d(p_{0},0))$ is non-separable.

Consider the function $f_{0}\in\text{Lip}_{0}(\\{0,p_{0}\\})$ defined by
$f_{0}(0)=0$ and $f_{0}(p_{0})=d(p_{0},0)$. This function is clearly
$1$-Lipschitz, and thus by Kalton’s Lemma 3.3 in [Kal04], choosing
$\varepsilon_{0}>0$ such that $r_{0}\cdot(\lambda+\varepsilon_{0})<1$ there
exists a $(\lambda+\varepsilon_{0})$-Lipschitz function $g_{0}$ in
$S_{0}(M)\subset\text{Lip}_{0}(M)$ such that $g_{0}(0)=0$ and
$g_{0}(p_{0})=d(p_{0},0)$. Since we are assuming that $B(p_{0},r_{0}\cdot
d(p_{0},0))$ is non-separable, there must exist a point $x_{0}\in M$ with
$d(p_{0},x_{0})<r_{0}\cdot d(p_{0},0)$ such that $g_{0}(x_{0})=0$. However,
this implies that

$\displaystyle|g_{0}(p_{0})|$
$\displaystyle=|g_{0}(p_{0})-g_{0}(x_{0})|\leq(\lambda+\varepsilon_{0})d(p_{0},x_{0})$
$\displaystyle<r_{0}\cdot(\lambda+\varepsilon_{0})d(p_{0},0)<d(p_{0},0),$

which contradicts the choice of $g_{0}$.

We continue with $(i)$ implies $(ii)$ (This proof is essentially contained in
[HQ22] as well. Still, we include it here for completeness): If a complete
metric space $M$ satisfies property $(i)$, we have just proven that the
subspace $S_{0}(M)$ is a $\lambda$-norming subspace of $\text{Lip}_{0}(M)$.
Hence, to show $(ii)$ it is enough to find a dense subset $D$ of $M$ such that
the intersection of $D$ with every separable subset of $M$ is countable again.
Indeed, it is straightforward to check that in that case, the Lipschitz-free
space $\mathcal{F}(M)$ is $\lambda$-Plichko witnessed by
$(\delta(D),S_{0}(M))$.

To find a dense subset $D$ with this property, we use Zorn’s Lemma to obtain a
maximal family in the set

$\displaystyle Z=\big{\\{}\\{A_{i}\\}_{i\in I}\colon$ $\displaystyle
A_{i}\text{ is non-empty, open and separable,}$ $\displaystyle\text{and
}A_{i}\cap A_{j}=\emptyset\text{ for all }i\neq j\in I\big{\\}}.$

Denote by $F_{0}=\\{A_{i}\\}_{i\in I}$ such a maximal family in $Z$. Since
every point in $M$ different from $0$ has a separable neighborhood, by
maximality, we have necessarily that $M=\overline{\bigcup_{i\in
I}A_{i}}\cup\\{0\\}$. Consider now $D_{i}$ a countable dense subset of $A_{i}$
for each $i\in I$. Define $D=\bigcup_{i\in I}D_{i}$. It is straightforward now
to check that $D$ is dense in $M$ and that the intersection of $D$ with any
separable subset of $M$ is countable.

$(ii)$ implies $(iii)$: Write $\Delta\subset\delta(M)$ to denote the set
witnessing the $\lambda$-Plichko property in $\mathcal{F}(M)$. Then $S_{0}(M)$
is countably supported in $\Delta$, so by Proposition 21 in [Kub09], the
Lipschitz-free space $\mathcal{F}(M)$ admits a commutative
$\lambda$-projectional skeleton $\\{P_{s}\\}_{s\in\Gamma}$ which generates
$S_{0}(M)$. Additionally, since $\Delta$ is a subset of $\delta(M)$ which is
linearly dense in $\mathcal{F}(M)$, there exists a dense subset $D$ of $M$
such that $\Delta=\delta(D)$. Now, by Corollary 20 in [CCS22] we may assume
that $P_{s}(\delta(p))\in\\{0,\delta(p)\\}$ for all $p\in D$ and for all
$s\in\Gamma$. By density of $D$ in $M$ and continuity of the projections in
the projectional skeleton, we conclude that
$P_{s}(\delta(p))\in\\{0,\delta(p)\\}$ for all $p\in M$ and all $s\in\Gamma$.

$(iii)$ implies $(iv)$: Suppose that $\mathcal{F}(M)$ admits a commutative
$\lambda$-projectional skeleton $\\{P_{s}\\}_{s\in\Gamma}$ such that
$P_{s}(\delta(p))\in\\{0,\delta(p)\\}$ for all $p\in M$. For every
$s\in\Gamma$, the image of the subset $\delta(M)\subset\mathcal{F}(M)$ is
contained in $\delta(M)$. Hence, when restricting $P_{s}$ to the subset
$\delta(M)$, we obtain a map
$P_{s|\delta(M)}\colon\delta(M)\rightarrow\delta(M)$ which is
$\lambda$-Lipschitz. Since the map $\delta\colon M\rightarrow\delta(M)$ is an
isometry, this restriction induces a $\lambda$-Lipschitz map

$R_{s}\colon M\rightarrow M$

for every $s\in\Gamma$, defined by $R_{s}(p)=\delta^{-1}(P_{s}(\delta(p)))$.
For every $s\in\Gamma$, the map $R_{s}$ is a retraction onto
$\delta^{-1}(P(\delta(M)))$, which is a closed separable subset of $M$. It is
direct to check that $\\{R_{s}\\}_{s\in\Gamma}$ is a commutative
$\lambda$-Lipschitz retractional skeleton in $M$, and it clearly satisfies
that $R_{s}(p)\in\\{0,p\\}$ for all $p\in M$.

$(iv)$ implies $(v)$: Suppose that $M$ admits a commutative
$\lambda$-Lipschitz retractional skeleton $\\{R_{s}\\}_{s\in\Gamma}$ such that
$R_{s}(p)=\\{0,p\\}$ for all $p\in M$. We show that $S_{0}(M)$ is a
$\lambda$-norming subspace of $M$ once more by using Kalton’s Lemma in
[Kal04]. Suppose that $A\subset M$ is a finite set and that
$F\in\text{Lip}_{0}(A)$ is a $1$-Lipschitz function defined in this set. The
set $A$ is finite and thus separable, so there exists $s_{0}\in\Gamma$ such
that $S_{s_{0}}=R_{s_{0}}(M)$ is a separable set containing $A$. By McShane’s
extension theorem, we may find another $1$-Lipschitz function
$\overline{F}\in\text{Lip}_{0}(S_{s_{0}})$ such that $\overline{F}_{|A}=F$.

Define now $f_{s_{0}}\colon M\rightarrow\mathbb{R}$ by
$f_{s_{0}}(p)=\overline{F}(R_{s_{0}}(p))$ for all $p\in M$. The map
$f_{s_{0}}$ is $\lambda$-Lipschitz because $R_{s_{0}}$ is a
$\lambda$-Lipschitz retraction. Since $R_{s_{0}}(p)=0$ if $p\notin S_{s_{0}}$
and $\overline{F}(0)=0$, we obtain that the support of $f_{s_{0}}$ is
contained in the separable subset $S_{s_{0}}$, which implies that $f_{s_{0}}$
belongs to $S_{0}(M)$. We conclude that $S_{0}(M)$ is $\lambda$-norming by
Lemma 3.3 in [Kal04].

Since the equivalence between $(i)$ and $(v)$ has already been discussed, we
finish the proof of the theorem. ∎

The geometric condition in $(i)$ of the previous theorem is simple enough to
allow us to construct many non-separable metric spaces whose Lipschitz-free
spaces have the Plichko property witnessed by Dirac measures. The following
example shows that such metric spaces can be found in the Banach spaces
$\ell_{p}(\Gamma)$ for every $1\leq p\leq\infty$ and every uncountable
cardinal $\Gamma$.

######  Example 2.2.

Let $p\in[1,\infty]$ and let $\Gamma$ be uncountable. The Banach space
$\ell_{p}(\Gamma)$ contains a complete metric space $M_{p}$ of density
character $\Gamma$ such that $\mathcal{F}(M_{p})$ is $1$-Plichko witnessed by
Dirac measures.

To show this, fix $p\in[1,\infty]$. For each $\gamma\in\Gamma$, denote
$E_{\gamma}=[0,e_{\gamma}]=\\{(x_{\nu})_{\nu\in\Gamma}\in\ell_{p}(\Gamma)\colon
x_{\gamma}\in[0,1],~{}\text{and }x_{\nu}=0~{}\text{for }\nu\neq\gamma\\}$. Put
$M_{p}=\bigcup_{\gamma\in\Gamma}E_{\gamma}$. Then $M_{p}$ is a complete metric
space of density character $\Gamma$, and it is easy to check that for every
$x\in M_{p}$ different from $0$, the open ball $B(x,d(x,0))$ is contained in
the separable segment $E_{\gamma}$ such that $x\in E_{\gamma}$. Hence, $M_{p}$
satisfies property $(i)$ for $\lambda=1$ in Theorem 2.1 and we conclude that
$\mathcal{F}(M_{p})$ is $1$-Plichko witnessed by Dirac measures.

In the previous example, the constructed metric space $M_{p}$ has a stronger
property than the geometric condition we demand: Every point $x$ in $M_{p}$ is
contained in a separable subset $E_{x}$ such that for every point $y\in E_{x}$
the open ball $B(y,d(y,0))$ is contained in $E_{x}$, and for every point
$z\notin E_{x}$ the open ball $B(z,d(z,0))$ is contained in $M\setminus
E_{x}$. Informally, these metric spaces can be seen as the union of
(uncountably many) open and closed separable components “glued” at the
distinguished point $0$ in such a way that the distance between points in two
different components is always greater than the distance of each individual
point to $0$. This fact may suggest that the geometric condition in $(i)$ of
Theorem 2.1 is only satisfied by metric spaces that are very close to being
separable in the previous sense. However, we show in the next example that
this is not the case:

######  Example 2.3.

There exists a non-separable metric space $N_{2}$ isometric to a subset of
$\ell_{2}(\Gamma)$ for uncountable $\Gamma$ containing $e_{\gamma}$ for all
$\gamma\in\Gamma$, whose Lipschitz-free space is $1$-Plichko witnessed by
Dirac measures with the following property:

For every $\gamma\in\Gamma$ and every separable subset $S_{\gamma}$ in $N_{2}$
containing $e_{\gamma}$, there exists a point $x\in N_{2}\setminus S$ such
that $e_{\gamma}$ belongs to the open ball $B(x,d(x,0))$.

To construct such a space, consider $M_{2}\subset\ell_{2}(\Gamma)$ of Example
2.2. Define
$N_{2}=M_{2}\cup\\{e_{\gamma}+e_{\nu}\colon\gamma\neq\nu\in\Gamma\\}$. As
before, for every $x\in M_{2}$, the open ball $B(x,d(x,0))$ in $N_{2}$ is
contained in the separable set $E_{\gamma}$ such that $x\in E_{\gamma}$.
Additionally, for every pair of indices $\gamma\neq\nu\in\Gamma$, we have that
$d(e_{\gamma}+e_{\nu},0)=\sqrt{2}$, and thus the open ball
$B(e_{\gamma}+e_{\nu},d(e_{\gamma}+e_{\nu},0))$ in $N_{2}$ is contained in the
union of $E_{\gamma}\cup E_{\nu}\cup\\{e_{\gamma}+e_{\nu}\\}$, which is
separable as well. The metric space $N_{2}$ satisfies condition $(i)$ with
$\lambda=1$ in Theorem 2.1, and we have that $\mathcal{F}(N_{2})$ is
$1$-Plichko witnessed by Dirac measures.

Now, for every $\gamma\in\Gamma$, given any separable subset $S_{\gamma}$ in
$N_{2}$ containing $e_{\gamma}$, there exists $\nu\in\Gamma$ such that
$e_{\gamma}+e_{\nu}$ does not belong to $S_{\gamma}$ (since that would
contradict the separability of $S_{\gamma}$). It is simple to verify that the
point $e_{\gamma}$ belongs to the open ball
$B(e_{\gamma}+e_{\nu},d(e_{\gamma}+e_{\nu},0))$.

In this last example, we may also intuitively identify separable components
whose union forms the whole complete metric space (each set of the form
$E_{\gamma}$ or $E_{\gamma,\nu}$ for every $\gamma,\nu\in\Gamma$). However, as
we have proven, given a point $p$ in one component, there may be points in
different components closer to $p$ than the value $d(p,0)$. Let us formalize
this intuition by characterizing metric spaces with the Plichko property
witnessed by Dirac measures in terms of their metric structure:

Let $M$ be a complete metric space, and let $\lambda\geq 1$. We say that a
collection $\mathcal{S}$ of subsets of $M$ is a separable $\lambda$-slab
decomposition if for all $N\in\mathcal{S}$, the set $N\setminus\\{0\\}$ is an
open separable set, and for all $p\in N$ there exists a countable subfamily
$\mathcal{S}_{p}\subset\mathcal{S}$ such that $B(p,\lambda\cdot
d(p,0))\cap\bigcup_{N\in\mathcal{S}}N$ is contained in $B(p,\lambda\cdot
d(p,0))\cap\bigcup_{N\in\mathcal{S}_{p}}N$.

We say that a separable $\lambda$-slab decomposition is total if
$M=\overline{\bigcup_{N\in\mathcal{S}}N}$.

######  Proposition 2.4.

Let $M$ be a complete metric space and let $\lambda\geq 1$. The following
statements are equivalent:

  * (i)

$\mathcal{F}(M)$ is $\lambda$-Plichko witnessed by a subset of $\delta(M)$.

  * (ii)

For every separable $\lambda$-slab decomposition $\mathcal{S}$ of $M$ there
exists a total separable $\lambda$-slab decomposition $\mathcal{S}^{\prime}$
such that $\mathcal{S}\subset\mathcal{S}^{\prime}$.

  * (iii)

$M$ admits a total separable $\lambda$-slab decomposition.

###### Proof.

We start by showing that $(i)$ implies $(ii)$. Assume first that
$\mathcal{F}(M)$ is $\lambda$-Plichko witnessed by a set
$\Delta\subset\delta(M)$, and fix a separable $\lambda$-slab decomposition
$\mathcal{S}$. Consider the following family:

$\Omega:=\big{\\{}\mathcal{D}=\\{N_{i}\\}_{i\in
I}\colon\mathcal{S}\subset\mathcal{D},~{}\mathcal{D}\text{ is a separable
$\lambda$-slab decomposition}\big{\\}},$

which can be partially ordered by inclusion. Using Zorn’s Lemma, we obtain a
maximal element $\mathcal{S}^{\prime}\in\Omega$. Let us prove that
$\mathcal{S}^{\prime}$ is total.

Indeed, suppose by contradiction that there exists a point $p\in
M\setminus\\{0\\}$ such that $p$ lays outside the closure of the set
$\bigcup_{N\in\mathcal{S}^{\prime}}N$. Then, applying condition $(i)$ of
Theorem 2.1 we can find an open and separable set $A_{p}$ containing $p$ such
that $A_{p}\cap N=\emptyset$ for all $N\in\mathcal{S}^{\prime}$. Since for
every point $x\in A_{p}$ the ball $B(x,\lambda\cdot d(x,0))$ is separable and
$N\setminus\\{0\\}$ is open for all $N\in\mathcal{S}^{\prime}$, it follows
that $\mathcal{S}_{p}=\mathcal{S}^{\prime}\cup\\{A_{p}\cup\\{0\\}\\}$ is a
separable $\lambda$-slab decomposition which contains $\mathcal{S}$,
contradicting the maximality of $\mathcal{S}^{\prime}$ in $\Omega$.

Since $\mathcal{S}=\\{0\\}$ is trivially a separable $\lambda$-slab
decomposition, it is clear that $(ii)$ implies $(iii)$.

Finally, to show that $(iii)$ implies $(i)$, suppose there exists a total
separable $\lambda$-slab decomposition $\mathcal{S}$ in $M$. We will prove
that condition $(i)$ in Theorem 2.1 holds for the dense subset
$\bigcup_{N\in\mathcal{S}}N$, which is enough to prove that $M$ is
$\lambda$-Plichko witnessed by Dirac measures.

Consider a separable set $N\in\mathcal{S}$ and fix $p\in N\setminus\\{0\\}$.
First, since $\mathcal{S}$ is total, it follows that

$B(p,\lambda\cdot d(p,0))=\overline{\bigcup_{N\in\mathcal{S}}N\cap
B(p,\lambda\cdot d(p,0))}.$

Next, since $\mathcal{S}$ is a separable $\lambda$-slab decomposition, there
exists a countable subfamily $\mathcal{S}_{0}\subset S$ such that

$B(p,\lambda\cdot d(p,0))=\overline{\bigcup_{N\in\mathcal{S}_{0}}N\cap
B(p,\lambda\cdot d(p,0))}.$

This shows that $B(p,\lambda\cdot d(p,0))$ is the closure of a countable union
of separable sets, thus separable itself. ∎

We finish this section by mentioning an example of a complete metric space
whose Lipschitz-free space is $1$-Plichko, but which fails to have a subset of
Dirac measures witnessing such property: Consider any $\mathbb{R}$-tree with
at least two branching points with uncountably many branches at each one of
them. Then, whichever choice of the distinguished point we make, there will be
a branching point with no separable neighborhood. This implies that condition
$(i)$ in Theorem 2.1 does not hold, and the conclusion follows. However, as we
will see in the next section, every $\mathbb{R}$-tree does have the
$1$-Plichko property, though witnessed by a set of molecules instead.

##  3\. The $1$-Plichko property in the Lipschitz-free space of an
$\mathbb{R}$-tree

In this section, we prove that the Lipschitz-free space associated to any
$\mathbb{R}$-tree has the $1$-Plichko property witnessed by molecules.

Let us start by formally defining an $\mathbb{R}$-tree. A metric space $(T,d)$
is an _$\mathbb{R}$ -tree_ if for every pair of points, $x\neq y\in T$ there
exists a unique arc $[x,y]\subset T$, and moreover, the arc $[x,y]$ is
isometric to the real line segment $[0,d(x,y)]$.

A subset of $T$ will be called an _$\mathbb{R}$ -subtree_ if it is an
$\mathbb{R}$-tree which contains the distinguished point $0\in T$

Given any $\mathbb{R}$-tree $T$, and given any closed $\mathbb{R}$-subtree $S$
of $T$, it holds that for every point $t\in T$ there exists a unique point
$s\in S$ such that the distance from $t$ to $S$ is attained at only the point
$s$. This defines a retraction $P_{S}\colon T\rightarrow S$, where $P_{S}(t)$
is the unique point in $S$ such that $d(t,S)=d(t,P_{S}(t))$, called the
_metric projection of $T$ onto $S$_. It is known that $P_{S}$ is a
$1$-Lipschitz retraction for every closed $\mathbb{R}$-subtree $S$ of $T$. We
will also use the fact that given two closed $\mathbb{R}$-subtrees $A$ and $B$
of $T$ it holds that $P_{A}\circ P_{B}=P_{B}\circ P_{A}=P_{A\cap B}$. Notice
that this implies that given two closed $\mathbb{R}$-subtrees $A$ and $B$ of
$T$, and a point $p\in A$, the point $P_{B}(p)$ belongs to $A\cap B$.

######  Theorem 3.1.

Let $T$ be an $\mathbb{R}$-tree. Then $\mathcal{F}(T)$ is $1$-Plichko
witnessed by a pair $(\Delta,N)$ where $\Delta$ is a linearly dense set of
molecules.

###### Proof.

Let $D\subset T$ be a dense set in $T$ such that $D\cap[0,p]$ is countable for
every $p\in T$. We will define inductively an increasing sequence of families
$\\{G_{\alpha}\\}_{\alpha\leq\kappa}$, where $\kappa$ is a big enough cardinal
and $G_{\alpha}$ is a family of closed separable $\mathbb{R}$-subtrees of $T$
for every $\alpha\leq\kappa$, such that $T=\bigcup_{S\in G_{\kappa}}S$ and the
following properties are satisfied for every $\alpha\leq\kappa$:

  * (a)

For every $H\in G_{\alpha}$, the family $\\{S\in G_{\alpha}\colon S\subset
H\\}$ is totally ordered for inclusion. That is: for every $S_{1},S_{2}\in
G_{\alpha}$ with $S_{1},S_{2}\subset H$, either $S_{1}\subset S_{2}$ or
$S_{2}\subset S_{1}$.

  * (b)

$G_{\alpha}$ is closed for finite intersections. Moreover, for every pair
$H_{1},H_{2}\in G_{\alpha}$ with $H_{i}\in G_{\beta_{i}}$ for some
$\beta_{i}\leq\alpha$, and such that $H_{1}\cap H_{2}\subsetneq H_{i}$ for
$i=1,2$, there exists $\beta<\min\\{\beta_{1},\beta_{2}\\}$ such that
$H_{1}\cap H_{2}\in G_{\beta}$. In other words, any two incomparable sets in
$G_{\alpha}$ meet at a set which belongs to a strictly lower generation than
both sets.

  * (c)

For every family $\\{S_{n}\\}_{n\in\mathbb{N}}$ of $\mathbb{R}$-subtrees such
that $S_{n}\in G_{\alpha}$ and $S_{n}\subset S_{n+1}$ for all
$n\in\mathbb{N}$, the set $\overline{\bigcup_{n\in\mathbb{N}}S_{n}}$ is in
$G_{\alpha}$.

Put $G_{0}=\left\\{\\{0\\}\right\\}$, which trivially satisfies the
assumptions. Suppose we have constructed $G_{\beta}$ for all $\beta$ smaller
than a limit ordinal $\alpha$.

Then put

$G_{\alpha}=\bigcup_{\beta<\alpha}G_{\beta}\cup\left\\{\overline{\bigcup_{n\in\mathbb{N}}S_{n}}\colon\text{
where }S_{n}\in G_{\beta}\text{ for some }\beta_{n}<\alpha,\text{ and
}S_{n}\subset S_{n+1}\text{ for all }n\in\mathbb{N}\right\\}.$

It is direct that condition $(c)$ holds in this case. Condition $(a)$ follows
easily from the inductive hypothesis.

Finally, while it is clear that $G_{\alpha}$ is closed for finite
intersections, to show the second part of $(b)$, consider two sets $H_{1}$ and
$H_{2}$ in $G_{\alpha}$ such that $H_{i}$ is not a subset of $H_{j}$ for
$i\neq j\in\\{1,2\\}$. For $i\in\\{1,2\\}$, we may assume that
$H_{i}=\bigcup_{n\in\mathbb{N}}S^{i}_{n}$ with $S^{i}_{n}\in
G_{\beta^{i}_{n}}$ for some $\beta_{n}^{i}<\alpha$ and $S^{i}_{n}\subset
S^{i}_{n+1}$ for all $n\in\mathbb{N}$. Then, there exists $n_{0}\in\mathbb{N}$
such that $S^{i}_{n_{0}}$ is not a subset of $S^{j}_{n_{0}}$ for $i\neq
j\in\\{1,2\\}$. Applying $(a)$ to $H_{1}$, we obtain that $H_{1}\cap
S^{2}_{n_{0}}=S^{1}_{n_{0}}\cap S^{2}_{n_{0}}$. Similarly, we obtain that
$H_{1}\cap H_{2}=S^{1}_{n_{0}}\cap S^{2}_{n_{0}}$. Condition $(b)$ for the
pair $S^{1}_{n_{0}}$ and $S^{2}_{n_{0}}$ yields the desired result.

For the successor ordinal case, assume $G_{\alpha}$ is defined for a given
ordinal $\alpha$. For every $p\in\bigcup_{S\in G_{\alpha}}S$ we may define the
ordinal $\text{height}(p)$ as the least ordinal $\beta\leq\alpha$ such that
$p\in H$ for some $H\in G_{\beta}$. Using minimality, such a set is unique by
the second part of condition $(b)$. Hence, we may define $H(p)\in
G_{\text{height}(p)}$ as the (unique) smallest $\mathbb{R}$-subtree in
$G_{\alpha}$ containing $p$.

If $T=\bigcup_{S\in G_{\alpha}}S$ we stop the inductive process with
$\kappa=\alpha$. Otherwise, consider the collection of families of separable
$\mathbb{R}$-subtrees given by:

$\displaystyle\Omega_{\alpha}=\big{\\{}\\{H_{i}\\}_{i\in I}\colon$
$\displaystyle H_{i}\text{ is a separable $\mathbb{R}$-subtree and there
exists a family }\\{S_{i}\\}_{i\in I}\subset G_{\alpha}\text{ such that}$
$\displaystyle\text{for every }i\in I,\text{ the set }S_{i}\text{ is strictly
contained in }H_{i}$ $\displaystyle\text{and }H_{i}\cap S=S_{i}\cap S\text{
for all }S\in G_{\alpha},$ $\displaystyle\text{and such that }H_{i}\cap
H_{j}=S_{i}\cap S_{j}\text{ for }i\neq j\in I\big{\\}}.$

Let us show that there always exists a nonempty set
$\mathcal{H}_{\alpha}\in\Omega_{\alpha}$. Consider any $p\notin\bigcup_{S\in
G_{\alpha}}S$, and write $P_{0}=\sup_{S\in G_{\alpha}}\\{S\cap[0,p]\\}$ (here
we are taking the maximum with respect to the natural order in $T$ given by:
For any $p,q\in T$, we say that $p\leq q$ if $[0,p]\subset[0,q]$). We will
prove that there exists an $\mathbb{R}$-subtree $S_{0}\in G_{\alpha}$ such
that $S_{0}\cap[0,p]=[0,P_{0}]$.

Consider a sequence $\\{p_{n}\\}_{n\in\mathbb{N}}$ in $\bigcup_{S\in
G_{\alpha}}S$ such that $p_{n}\leq p_{n+1}$ and $p_{n}\rightarrow P_{0}$. The
set $H(p_{n})\in G_{\text{height}(p_{n})}$ is in $G_{\alpha}$ for every
$n\in\mathbb{N}$. Note that $p_{n}$ belongs to $H(p_{m})$ for all $m\geq n$.
If there exists $n\in\mathbb{N}$ such that $H(p_{n})$ contains every $p_{m}$
for all $m\geq n$, then the limit point $P_{0}$ belongs to the closed
$\mathbb{R}$-subtree $H(p_{n})$, and we obtain that
$H(p_{n})\cap[0,p]=[0,P_{0}]$ as desired.

If not, we can assume by passing to a subsequence that $H(p_{m})$ is not a
subset of $H(p_{n})$ for every pair $n,m\in\mathbb{N}$ with $n<m$. This
implies that $H(p_{n})$ is a subset of $H(p_{m})$ for $n,m\in\mathbb{N}$ with
$n<m$, since otherwise the second part of condition $(b)$ shows that
$H(p_{n})\cap H(p_{m})$ belongs to $G_{\beta}$ for some
$\beta<\text{height}(p_{n})$, which contradicts the minimality of
$\text{height}(p_{n})$. Hence, applying condition $(c)$ to the increasing
sequence $\\{H(p_{n})\\}_{n\in\mathbb{N}}$ yields the sought
$\mathbb{R}$-subtree in $G_{\alpha}$.

Once we have an $\mathbb{R}$-subtree $S_{0}\in G_{\alpha}$ with
$S_{0}\cap[0,p]=[0,p_{0}]$, it is straightforward to see that the singleton
$\\{S_{0}\cup[0,p]\\}$ belongs to $\Omega_{\alpha}$. Choose any nonempty
$\mathcal{H}_{\alpha}\in\Omega_{\alpha}$ and put
$G_{\alpha+1}=G_{\alpha}\cup\mathcal{H}_{\alpha}$.

It is straightforward to check that $G_{\alpha+1}$ satisfies the inductive
hypothesis. Since for every ordinal $\alpha$ the set $\bigcup_{S\in
G_{\alpha}}S$ is strictly contained in $\bigcup_{S\in G_{\alpha+1}}S$, there
exists an ordinal $\kappa$ such that $T=\bigcup_{S\in G_{\kappa}}S$. This
finishes the inductive construction.

Once the induction is finished, for every $p\in T$ we keep the definition of
$\text{height}(p)$ and $H(p)\in G_{\text{height}(p)}$. Now, for every $p\in T$
such that $\text{height}(p)$ is a successor ordinal $\alpha+1$, we wish to
define $S(p)\in G_{\alpha}$ as the unique set in $G_{\kappa}$ that contains
every set $S$ in $G_{\kappa}$ strictly contained in $H(p)$. In order to show
that such a set exists and is unique, consider the family $\\{S\in
G_{\alpha}\colon S\subset H(p)\\}$, which is well ordered by $(a)$. Since
$\overline{\bigcup\\{S\in G_{\alpha}\colon S\subset H(p)\\}}$ is contained in
the separable set $H(p)$, there exists a sequence
$\\{S_{n}\\}_{n\in\mathbb{N}}$ such that

$\overline{\bigcup_{n\in\mathbb{N}}S_{n}}=\overline{\bigcup\\{S\in
G_{\alpha}\colon S\subset H(p)\\}}.$

Condition $(c)$ now implies that the set
$S(p)=\overline{\bigcup_{n\in\mathbb{N}}S_{n}}$ belongs to $G_{\alpha}$, which
clearly has the desired properties.

Denoting by $P_{S}\colon T\rightarrow S$ the metric projection onto any
$\mathbb{R}$-subtree $S$, we consider now the family of molecules given by:

$\displaystyle\Delta=\big{\\{}m_{p,P_{S(p)}(p)}\colon$ $\displaystyle p\in
D\text{ with }H(p)\in G_{\alpha+1}\text{ for some }\alpha<\kappa\big{\\}}.$

We start by showing that $\Delta$ is linearly dense in $\mathcal{F}(T)$. It is
enough to prove that $\delta(p)$ belongs to the closed linear span of $\Delta$
for every $p\in T$. We work by induction on the height of $p$. If
$\text{height}(p)=0$, then $p=0$ and the conclusion follows trivially.

Assume that $\text{height}(p)$ is a limit ordinal $\alpha$, and that
$\delta(q)\in\overline{\text{span}}(\Delta)$ for all $q\in M$ with
$\text{height}(q)<\alpha$. By definition of $G_{\alpha}$ there exists a
sequence of ordinals $\\{\beta_{n}\\}_{n\in\mathbb{N}}$ with
$\beta_{n}<\alpha$ for all $n\in\mathbb{N}$, and a sequence of sets
$\\{S_{\beta_{n}}\\}_{n\in\mathbb{N}}$ with $S_{\beta_{n}}\in G_{\beta_{n}}$
such that $H(p)=\overline{\bigcup_{n\in\mathbb{N}}S_{\beta_{n}}}$. Since $p\in
H(p)$, it follows by inductive hypothesis that
$p\in\overline{\text{span}}(\Delta)$.

Assume now that $\text{height}(p)=\alpha+1$ for some ordinal $\alpha$, and
that $\delta(q)\in\overline{\text{span}}(\Delta)$ for all $q\in M$ with
$\text{height}(q)=\alpha$. Suppose first that $p\in D$. Since $P_{S(p)}(p)$
belongs to $P_{S(p)}(M)$ and $\text{height}(q)=\alpha$ for all $q\in S(p)$, we
have by inductive hypothesis that
$\delta(P_{S(p)})\in\overline{\text{span}}(\Delta)$. Therefore, we obtain that

$\delta(p)=d\left(p,P_{S(p)}(p)\right)m_{p,P_{S(p)}(p)}+\delta(P_{S{p}})\in\overline{\text{span}}(\Delta).$

Having proven the claim for all $p\in D$, it follows by density of $D$ that
$\delta(p)\in\overline{\text{span}}(\Delta)$ for all $p\in M$ with
$\text{height}(p)=\alpha+1$. Hence, we conclude that
$M=\overline{\text{span}}(\Delta)$.

Finally, let $N=\\{f\in\text{Lip}_{0}\colon f\text{ is countably supported in
}\Delta\\}$. We will show that if $C$ is a separable subset of $T$ containing
the distinguished point $0$, and $\varphi$ is a $1$-Lipschitz function in
$\text{Lip}_{0}(C)$, there exists a $1$-Lipschitz function $f\in N$ that
extends $\varphi$. This implies, by Kalton’s Lemma, that $N$ is $1$-norming.

Consider such $C$ and $\varphi$. Since $C$ is separable, there exists a
countable subfamily $\\{S_{n}\\}_{n\in\mathbb{N}}$ of $G_{\kappa}$ such that
$C\subset S^{*}=\bigcup_{n\in\mathbb{N}}S_{n}$. Using McShane’s theorem, we
can extend $\varphi$ to a function $F\in\text{Lip}_{0}(S^{*})$ which is
$1$-Lipschitz. Now, define $f(p)=F(P_{S^{*}}(p))$ for all $p\in T$, where
$P_{S^{*}}$ is the metric projection onto the separable $\mathbb{R}$-subtree
$S^{*}$. We will prove that $f$ is finitely supported in $\Delta$ and thus
belongs to $N$.

Let $p\in D$, with $H(p)\in G_{\alpha+1}$ for some ordinal $\alpha<\kappa$.
Suppose first that $H(p)\nsubseteq S^{*}$. Then $P_{S^{*}}(p)\neq p$. By
commutativity of the metric projections, we have that $P_{S^{*}}(p)\in
H(P_{S^{*}}(p))\cap H(p)$. This implies that $H(P_{S^{*}}(p))$ is contained in
$H(p)$, and the inclusion is strict since $H(P_{S^{*}}(p))\subset S^{*}$. We
conclude that $H(P_{S^{*}}(p))\subset S(p)$. Therefore, again by commutativity
of the metric projections, we obtain that
$P_{S^{*}}(P_{S(p)}(p))=P_{S^{*}}(p)$, which implies that $\langle
f,m_{p,P_{S(p)}(p)}\rangle=0$.

Now, if $H(p)\subset S^{*}$, then $p\in S^{*}$. Since $S^{*}$ is separable and
$D\cap S_{n}$ is countable for all $n\in\mathbb{N}$, the set $\\{p\in D\colon
H(p)\subseteq S^{*}\\}$ is countable. We conclude that $f$ is countably
supported in $\Delta$. Hence, $f\in N$ and we obtain that $N$ is $1$-norming.
∎

##  4\. Lipschitz-retractional trees

We will generalize the previous ideas in the proof of Theorem 3.1 to obtain
the same result for a bigger class of metric spaces. To define this concept,
we will use the notion of _tree_ in the partially ordered sense: A partially
ordered set $(\Gamma,\leq)$ is called a _tree_ if for every element
$s\in\Gamma$, the set $\\{t\in\Gamma\colon t<s\\}$ is a well-ordered set. A
tree is said to be _rooted_ if there is a minimum element in $\Gamma$, called
the _root_ and usually denoted by $0$, and is called _$\sigma$ -complete_ if
every totally ordered countable subset of $\Gamma$ has a supremum in $\Gamma$.

For any element $s$ in a tree $\Gamma$, we call $I_{s}=\\{t\in\Gamma\colon
t<s\\}$ the initial segment generated by $s$. Notice that $I_{s}$ is well
ordered for every $s\in\Gamma$ and is, therefore, order isomorphic to an
ordinal number, which we will refer to as the height of $s$. The height of a
tree $\Gamma$ is the supremum over the height of all elements of $\Gamma$.

In a $\sigma$-complete tree, given two points $s$ and $t$ of countable height,
it is direct to show that there exists a unique $s\wedge t\in\Gamma$, defined
as the supremum in $\Gamma$ over all points which are lower bounds of both $s$
and $t$.

Let $M$ be a complete metric space. A family of $\lambda$-Lipschitz
retractions $\\{R_{s}\\}_{s\in\Gamma}$ on $M$ indexed by a rooted,
$\sigma$-complete tree $\Gamma$ such that $\text{height}(s)<\omega_{1}$ for
all $s\in\Gamma$ is called a $\lambda$-_Lipschitz pre-retractional tree_ if
the following properties are satisfied:

  * (i)

$R_{0}(M)=\\{0\\}$ and $R_{s}(M)$ is a separable subset of $M$ for every
$s\in\Gamma$.

  * (ii)

For every $s,t\in\Gamma$ we have that $R_{s}\circ R_{t}=R_{t}\circ
R_{s}=R_{s\wedge t}$.

  * (iii)

If $(s_{n})_{n\in\mathbb{N}}$ is a totally ordered sequence in $\Gamma$ with
$s=\sup_{n\in\mathbb{N}}s_{n}$, then
$R_{s}(M)=\overline{\bigcup_{n\in\mathbb{N}}R_{s_{n}}(M)}$.

  * (iv)

$M=\bigcup_{n\in\mathbb{N}}R_{s}(M)$.

Notice that if $M$ admits a Lipschitz pre-retractional tree
$\\{R_{s}\\}_{s\in\Gamma}$ such that $\text{height}(s)<\omega_{1}$ for all
$s\in\Gamma$, given a point $p\in M$ we can find an element $s(p)\in\Gamma$
such that $p\in R_{s(p)}(M)$ and such that $s(p)$ is minimal in $\Gamma$ for
this property. Moreover, condition $(ii)$ ensures that $s(p)$ is unique. This
allows us to define the height of a point $p$ in a Lipschitz pre-retractional
tree $\\{R_{s}\\}_{s\in\Gamma}$ as the height of $s(p)$ in the tree $\Gamma$.

Now, given a $\sigma$-complete subset $A\subset\Gamma$, we may define a map
$R_{A}\colon M\rightarrow\bigcup_{s\in A}R_{s}(M)$ given by
$R_{A}(p)=R_{s(p)\wedge A}(p)$. Condition $(ii)$ ensures that this map is well
defined, and it clearly is a retraction. We can finally define the structure
we will be studying in this section:

######  Definition 4.1.

Let $M$ be a complete metric space. A family of $\lambda$-Lipschitz
retractions $\\{R_{s}\\}_{s\in\Gamma}$ on $M$ indexed by a rooted,
$\sigma$-complete tree $\Gamma$ such that $\text{height}(s)<\omega_{1}$ for
all $s\in\Gamma$ is called a $\lambda$-_Lipschitz retractional tree_ if
$\\{R_{s}\\}_{s\in\Gamma}$ is a $\lambda$-Lipschitz pre-retractional tree
which additionally satisfies the following condition:

  * (v)

For every $\sigma$-complete countable set $A$ of $\Gamma$, the set $R_{A}(M)$
is closed and the retraction $R_{A}$ is $\lambda$-Lipschitz.

In the proof of Theorem 3.1 we construct a similar structure to a
$1$-Lipschitz retractional tree in every $\mathbb{R}$-tree. We are going to
show that this construction is enough to obtain the Plichko property in the
Lipschitz-free space witnessed by molecules and that it can be applied to
metric spaces which are not isometric to an $\mathbb{R}$-tree.

Let us prove first one technical lemma:

######  Lemma 4.2.

Let $M$ be a metric space. Suppose $M$ admits a Lipschitz retractional tree
$\\{R_{s}\\}_{s\in\Gamma}$. Then there exists a dense set $D$ in $M$ such that
$D\cap R_{s}(M)$ is countable for every $s\in\Gamma$.

###### Proof.

Let $\kappa$ be the height of the tree $\Gamma$. We are going to define by
transfinite induction a family of sets $\\{D_{\alpha}\\}_{\alpha\leq\kappa}$
with the following properties for every $\alpha<\kappa$:

  * (1).

$D_{\alpha}$ contains $D_{\beta}$ for all $\beta<\alpha$.

  * (2).

$D_{\alpha}\cap R_{s}(M)$ is a countable dense subset of $R_{s}(M)$ for every
$s\in\Gamma$ such that $\text{height}(s)\leq\alpha$.

It is clear that once we construct such a family, the set $D=D_{\kappa}$ will
satisfy the conditions of the lemma.

Put $D_{0}=\\{0\\}$. Suppose we have constructed
$\\{D_{\beta}\\}_{\beta<\alpha}$ for some ordinal $\alpha<\kappa$. If $\alpha$
is a limit ordinal, simply define
$D_{\alpha}=\bigcup_{\beta<\alpha}D_{\beta}$. It follows from property $(iii)$
of the Lipschitz retractional tree that $D_{\alpha}$ satisfies condition $(2)$
of the inductive process.

If $\alpha$ is not a limit ordinal, then $\alpha=\eta+1$ for some ordinal
$\eta<\kappa$. For each $s\in\Gamma$ such that $\text{height}(s)=\eta+1$,
consider a countable dense subset $D_{s}$ of
$R_{s}(M)\setminus\overline{D_{\eta}}$. Finally, Put

$D_{\eta+1}=D_{\eta}\cup\bigg{(}\bigcup\\{D_{s}\colon s\in\Gamma\text{ and
}\text{height}(s)=\eta+1\\}\bigg{)}.$

Clearly, condition $(1)$ is satisfied, and condition $(2)$ follows from the
definition of each $D_{s}$ and condition $(ii)$ in the definition of Lipschitz
pre-retractional trees. ∎

We can now prove the main theorem of the section.

######  Theorem 4.3.

Let $M$ be a complete metric space. If $M$ admits a $\lambda$-Lipschitz
retractional tree, then $\mathcal{F}(M)$ is $\lambda$-Plichko witnessed by a
pair $(\Delta,N)$, where $\Delta$ is a subset of molecules of $M$.

###### Proof.

Let $\\{R_{s}\\}_{s\in\Gamma}$ be a $\lambda$-Lipschitz retractional tree.
Using Lemma 4.2, choose $D$ to be a dense subset of $M$ such that $D\cap
R_{s}(M)$ is countable for every $s\in\Gamma$.

Let $\kappa$ be the height of the tree $\Gamma$. Fix a point $p\in D$ such
that $\text{height}(p)$ is $\alpha+1$ for some ordinal number $\alpha<\kappa$.
Since the initial segment $I_{s(p)}$ is order isomorphic to the ordinal
$\alpha+1$, there exists a unique element $t(p)$ in $I_{s(p)}$ with
$\text{height}(t(p))=\alpha$. Define now

$\Delta=\bigcup\\{m_{p,R_{t(p)}(p)}\colon p\in
D,~{}\text{height}(p)=\alpha+1\text{ for some }\alpha<\kappa\\}.$

Notice that $\Delta$ is a subset of molecules of $M$.

With a similar inductive reasoning as in the proof of Theorem 3.1, we obtain
that $\Delta$ is linearly dense in $\mathcal{F}(M)$.

Finally, we show that the set $N=\\{f\in\text{Lip}_{0}(M)\colon f\text{ is
countably supported in }\Delta\\}$ is $\lambda$-norming. We will show that if
$S$ is a separable subset of $M$ containing the distinguished point $0$, and
$g$ is a $1$-Lipschitz function in $\text{Lip}_{0}(S)$, there exists a
$\lambda$-Lipschitz function $f\in N$ that extends $g$.

Consider $S$ and $g$ with such properties. Since $S$ is separable and
$M=\bigcup_{s\in\Gamma}R_{s}(M)$, there exists a $\sigma$-complete and
countable subset $A$ of $\Gamma$ such that $S\subset\bigcup_{s\in A}R_{s}(A)$.
Define $f\in\text{Lip}_{0}(M)$ as $f=g\circ R_{A}$. Since $R_{A}$ is
$\lambda$-Lipschitz, it follows that $\|f\|_{\text{Lip}}\leq\lambda$, and
clearly $f$ extends $g$ since $R_{A}$ is a retraction onto $\bigcup_{s\in
A}R_{s}(A)$. It remains to show that $f$ is countably supported in $\Delta$.

To this end, we will show that for any point $p\in D$ such that
$\text{height}(p)$ is a successor ordinal and such that $s(p)\notin A$, we
have that $\langle f,m_{p,R_{t(p)}(p)}\rangle=0$. In this situation, since
$s(p)\notin A$, we have that $t(p)$ is greater or equal than $s(p)\wedge A$,
and thus it follows that $t(p)\wedge A=s(p)\wedge A$ and
$R_{A}(p)=R_{A}(R_{t(p)}(p))$. This implies that $f(p)=f(R_{t(p)}(p))$, which
finishes the proof. ∎

We finish with an example of a non-separable metric space that is not
isometric to an $\mathbb{R}$-tree though it admits a $1$-Lipschitz
retractional tree.

######  Example 4.4.

Let $\kappa$ be an uncountable set. Let $M=\\{0,1\\}\cup\\{(t,\alpha)\colon
t\in(0,1),~{}\alpha\in\kappa\\}$, endowed with the unique distance $d\colon
M\times M\rightarrow[0,\infty)$ satisfying

$d(p,q)=\begin{cases}1,&\text{ if }p=0,~{}q=1\\\ t,&\text{ if
}p=0,~{}q=(t,\alpha),\text{ for some }\alpha\in\kappa\\\ 1-t,&\text{ if
}p=1,~{}q=(t,\alpha),\text{ for some }\alpha\in\kappa\\\ |t-s|,&\text{ if
}p=(t,\alpha),~{}q=(s,\alpha),\text{ for some }\alpha\in\kappa\\\
\min\\{d(p,0)+d(0,q),d(p,1)+d(1,q)\\},&\text{ otherwise}.\end{cases}$

It is clear that $M$ is non-separable and is not isometric to an
$\mathbb{R}$-tree. For every $\alpha\in\kappa$, write
$M_{\alpha}=\\{0,1\\}\cup\\{(t,\alpha)\colon t\in(0,1),~{}\alpha\in\kappa\\}$.

Fix any $\alpha_{0}\in\kappa$, and define $\Gamma=\\{0\\}\cup\kappa$, with the
order $\preceq$ given by

$\displaystyle 0\prec\alpha\qquad$ $\displaystyle\text{for every
}\alpha\in\kappa$ $\displaystyle\alpha\prec\beta\qquad$
$\displaystyle\text{for }\alpha,\beta\in\kappa\text{ if and only if
}\alpha=\alpha_{0}\text{ and }\beta\neq\alpha_{0}.$

Clearly $\left(\Gamma,\preceq\right)$ is a rooted, $\sigma$-complete tree with
finite height. Define $R_{0}\colon M\rightarrow\\{0\\}$ as the trivial
retraction, and $R_{\alpha_{0}}\colon M\rightarrow M_{\alpha_{0}}$ as the
identity map on $M_{\alpha_{0}}$ and $R_{\alpha_{0}}(t,\alpha)=(t,\alpha_{0})$
for every $t\in(0,1)$ and $\alpha\in\kappa\setminus{\alpha_{0}}$. Finally, for
every $\alpha\in\kappa\setminus\\{\alpha_{0}\\}$, define $R_{\alpha}\colon
M\rightarrow M_{\alpha_{0}}\cup M_{\alpha}$ as the identity on
$M_{\alpha_{0}}\cup M_{\alpha}$, and $R_{\alpha_{0}}(t,\alpha)=(t,\alpha_{0})$
for every $t\in(0,1)$ and $\alpha\in\kappa\setminus\\{\alpha_{0},\alpha\\}$.

It is now routine to check that $\\{R_{s}\\}_{s\in\Gamma}$ is indeed a
$1$-Lipschitz retractional tree.

##  5\. Linear structure of Lipschitz-free spaces associated to Banach spaces

As mentioned throughout this note, it is a standard application of the
linearization property of Lipschitz-free spaces that if $M$ is a complete
metric space and $R\colon M\rightarrow S$ is a Lipschitz retraction onto a
subset $S$ of $M$, there exists a bounded linear projection
$\widehat{R}\colon\mathcal{F}(M)\rightarrow\mathcal{F}(S)$ between the
associated Lipschitz-free spaces. It is also not difficult to show that not
all bounded and linear projections in Lipschitz-free spaces are obtained in
this way: a bounded and linear projection
$P\colon\mathcal{F}(M)\rightarrow\mathcal{F}(S)$ is of the form
$P=\widehat{R}$ for some Lipschitz retraction $R\colon M\rightarrow S$ if and
only if $P(\delta(p))\in\delta(S)$ for all $p\in M$.

However, if $X$ is a Banach space, we are able to define the _barycenter_ map
$\beta_{X}\colon\mathcal{F}(X)\rightarrow X$, which is a linear map with
$\|\beta_{X}\|\leq 1$ such that $\beta_{X}(\delta(x))=x$ for all $x\in X$ (see
e.g.: [GK03]). By composing with the isometry $\delta$, we obtain that every
bounded and linear projection $P\colon\mathcal{F}(X)\rightarrow\mathcal{F}(Y)$
where $Y$ is a linear subspace of $X$ induces a Lipschitz retraction
$R=\beta_{Y}\circ P\circ\delta\colon X\rightarrow Y$ with
$\|R\|_{\text{Lip}}=\|P\|$.

If $\mathcal{F}(X)$ is $r$-Plichko, then it admits in particular an
$r$-projectional skeleton $\\{P_{s}\\}_{s\in\Gamma}$. By Proposition 2.2 in
[HQ22], we may assume that $P_{s}(\mathcal{F}(X))=\mathcal{F}(Y_{s})$, where
$Y_{s}$ is a separable subspace of $X$ for every $s\in\Gamma$. Therefore,
using the definition of $r$-projectional skeleton and the previous discussion
we arrive at the following observation:

######  Remark 5.1.

If $\mathcal{F}(X)$ is $r$-Plichko for some $r\geq 1$, every separable
subspace of $X$ is contained in a separable subspace which is an $r$-Lipschitz
retract of $X$.

Recall that a Banach space has the _Separable Complementation Property (SCP)_
if every separable subspace of $X$ is contained in a separable subspace that
is linearly complemented in $X$. We say that $X$ has the $r$-SCP for $r\geq 1$
if the norm of the projections of the separable superspaces can be uniformly
bounded by $r$. Analogously, we say that a metric space $M$ has the _Lipschitz
Separable Retractional Property (Lipschitz SRP)_ if every separable subset of
$M$ is contained in a separable subset which is a Lipschitz retract of $M$.
For $r\geq 1$, we say that $M$ has the $r$-Lipschitz SRP if the norm of the
Lipschitz retractions onto the separable supersets can be uniformly bounded by
$r$.

Let us note that in the metric space setting, Remark 5.1 fails in a strong
sense, as there exists a metric space $M$ such that $\mathcal{F}(M)$ is
$1$-Plichko, but with two specific points $\\{0,1\\}\subset M$ such that every
separable set containing them is not a Lipschitz retract of $M$ (see Theorem
3.7 and Remark 3.8 in [HQ22]).

Remark 5.1 suggests that a strategy for finding a Lipschitz-free space without
the Plichko property could be to construct a Banach space failing the
Lipschitz SRP. This is a difficult problem, which implies the existence of a
separable Banach space that is not a Lipschitz retraction of its bidual; a
problem that goes back to the seminal paper [Lin64] by Lindenstrauss, and
whose non-separable version was solved by Kalton in [Kal11]. Indeed, suppose
there existed a Banach space $X$ and a separable subspace $Y$ which is not
contained in any separable Lipschitz retract of $X$. By a classic result of
Heinrich and Mankiewicz [HM82], there exists a separable subspace $Z$
containing $Y$ which is locally complemented, i.e.: $Z^{**}$ is linearly
complemented in $X^{**}$. Therefore, $Z$ is not a Lipschitz retract of
$Z^{**}$, since this would imply that $Z$ is Lipschitz retract of $X$.

We finish this note with a brief result on $\ell_{\infty}$. Although
$\ell_{\infty}$ fails the SCP, it enjoys the $2$-Lipschitz SRP (this holds in
general for all $C(K)$ spaces; see Proposition 2.11 in [HQ22]). Modifying the
classical argument that shows that $c_{0}$ is not an $r$-Lipschitz retract of
$\ell_{\infty}$ for $r<2$, we prove that the constant $2$ is also optimal for
the Lipschitz SRP in $\ell_{\infty}$.

######  Theorem 5.2.

Let $S$ be a separable subset of $\ell_{\infty}$ containing $c_{0}$, and
suppose there exists a Lipschitz retraction $R\colon\ell_{\infty}\rightarrow
S$. Then $\|R\|_{\text{Lip}}\geq 2$. Consequently,
$\mathcal{F}(\ell_{\infty})$ is not $r$-Plichko for $r<2$.

###### Proof.

Put $S=\overline{\\{x_{n}=(x_{n}(k))_{k\in\mathbb{N}}\in\ell_{\infty}\colon
n\in\mathbb{N}\\}}$. Consider the sequence $z\in\ell_{\infty}$ given by

$z(k)=-\text{sign}(x_{k}(k))\qquad\text{ for }k\in\mathbb{N}.$

For every $k\in\mathbb{N}$, let
$P_{k}\colon\ell_{\infty}\rightarrow\ell_{\infty}$ be the linear projection
given by $\left(P_{k}x\right)(i)=x(i)$ if $i\leq k$ and
$\left(P_{k}x\right)(i)=0$ if $i>k$ for every $x\in\ell_{\infty}$. With this
notation, define for every $k\in\mathbb{N}$ the sequence $y_{k}=2P_{k}(z)$,
which belongs to $S$ since $S$ contains $c_{0}$. It is straightforward to
check that $\|z-y_{k}\|_{\infty}=1$ for all $k\in\mathbb{N}$.

For every $n\in\mathbb{N}$ and every $k\geq n$, it holds that
$\|x_{n}-y_{k}\|_{\infty}\geq 2$, since $|y_{k}(n)|=2$ and the $n$-th
coordinate of $x_{n}$ has the opposite sign. It follows that
$\limsup_{k\rightarrow+\infty}\|x-y_{k}\|_{\infty}\geq 2$ for every $x\in S$.
Therefore:

$2\leq\limsup_{k\rightarrow+\infty}\|R(z)-y_{k}\|_{\infty}\leq\limsup_{k\rightarrow+\infty}\|R\|_{\text{Lip}}\|z-y_{k}\|_{\infty}=\|R\|_{\text{Lip}},$

and the first claim is proven. The second part of the statement follows from
Remark 5.1. ∎

##  6\. Acknowledgements

This research was supported by grant PID2021-122126NB-C33 funded by
MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. The
second author’s research has been supported by PAID-01-19, by GA23-04776S and
by the project SGS23/056/OHK3/1T/13.

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