\begin{document}

\theoremstyle{plain}\newtheorem{teo}{Theorem}[section]
\theoremstyle{plain}\newtheorem{prop}[teo]{Proposition}
\theoremstyle{plain}\newtheorem{lem}[teo]{Lemma}
\theoremstyle{plain}\newtheorem{cor}[teo]{Corollary}
\theoremstyle{definition}\newtheorem{defin}[teo]{Definition}
\theoremstyle{remark}\newtheorem{rem}[teo]{Remark}
\theoremstyle{plain} \newtheorem{assum}[teo]{Assumption}
\theoremstyle{definition}\newtheorem{example}[teo]{Example}

\begin{abstract}
In this work we investigate the $c_0$-extension property. This property generalizes Sobczyk's theorem in the context of nonseparable Banach spaces. We prove that a sufficient condition for a Banach space to have this property is that its closed dual unit ball is weak-star monolithic. We also present several results about the $c_0$-extension property in the context of $C(K)$ Banach spaces. An interesting result in the realm of $C(K)$ spaces is that the existence of a Corson compactum $K$ such that $C(K)$ does not have the $c_0$-extension property is independent from the axioms of $ZFC$.
\end{abstract}

\maketitle

\begin{section}{Introduction}

Sobczyk's theorem \cite{Sobczyk} is a classical result about the structure of separable Banach spaces. It states that if $X$ is a separable Banach space, then every $c_0$-valued bounded operator defined on a closed subspace of $X$ admits a $c_0$-valued bounded extension defined on $X$. The search for generalizations of Sobczyk's theorem in the context of nonseparable Banach spaces has attracted a lot of attention in the last decades \cite{SobczykLine, c0EP, c0EPline, DrygierPlebanek, Eloi, Molto, Patterson}. In \cite{SobczykLine}, the $c_0$-extension property was introduced. We say that a Banach space $X$ has the {\it $c_0$-extension property} ($c_0$EP) if every $c_0$-valued bounded operator defined on a closed subspace of $X$ admits a $c_0$-valued bounded extension defined on $X$. Clearly Sobczyk's theorem implies that every separable Banach space has the $c_0$EP. Let us recall the main results known about the $c_0$EP. An adaptation of Veech's proof of Sobczyk's theorem \cite{Veech} shows that every weakly compactly generated Banach space has the $c_0$EP\ \cite[Proposition~2.2]{c0EP}. Recall that a Banach space is said to be {\it weakly compactly generated} (WCG) if it contains a weakly compact subset that is linearly dense. Clearly every separable and every reflexive Banach space is WCG and the canonical example of a neither separable nor reflexive WCG space is given by $c_0(I)$, for any uncountable set $I$ \cite[page 575]{Fabian}.
In \cite[Theorem~2.2]{SobczykLine}, it was shown that if $K$ is compact line, then $C(K)$ has the $c_0$EP\ if and only if $K$ is monolithic. Here, as usual, $C(K)$ denotes the Banach space of real-valued continuous functions defined on a compact Hausdorff space $K$, endowed with the supremum norm. Recall that a {\it compact line} is a totally ordered set that is compact when endowed with the order topology. The notion of monolithicity plays a central role in this work. We say that a compact Hausdorff space $K$ is {\it monolithic} if every separable subspace of $K$ is second countable.

One of the main results of this work is Theorem \ref{monoc0EP}, where we show that if $X$ is a Banach space with closed dual unit ball weak-star monolithic, then $X$ has the $c_0$EP.
We denote the closed dual unit ball of a Banach space $X$ by $B_{X^*}$ and the weak-star topology by $w^*$-topology. Observe that all the spaces with the $c_0$EP\ mentioned above have closed dual unit ball $w^*$-monolithic. Indeed, it is well-known that if $X$ is a separable Banach space, then $B_{X^*}$ is $w^*$-metrizable \cite[Proposition 3.103]{Fabian} and therefore $B_{X^*}$ is $w^*$-monolithic. Moreover, if $X$ is a WCG Banach space, then $(B_{X^*}, w^*)$ is an Eberlein compactum \cite[Theorem~13.20]{Fabian} and therefore $B_{X^*}$ is $w^*$-monolithic. Recall that a compact space is an {\it Eberlein compactum} if it is homeomorphic to a weakly compact subset of a Banach space, endowed with the weak topology. Finally it follows from Lemma \ref{MonoMBolaMono} that if $K$ is a monolithic compact line, then $B_{C(K)^*}$ is $w^*$-monolithic.
A really interesting class of monolithic compact spaces is the class of Corson compacta. We say that a compact space $K$ is a {\it Corson compactum} if there exists a set $I$ such that $K$ is homeomorphic to a closed subset of $\Sigma(I)$, endowed with the product topology. By $\Sigma(I)$ we denote the subset of $\mathds R^I$ formed by functions with countable support.
Since every Corson compactum is monolithic, it follows from Theorem \ref{monoc0EP} that a Banach space $X$ has the $c_0$EP\ whenever $(B_{X^*}, w^*)$ is a Corson compactum. Those are precisely the {\it weakly Lindelöf determined} Banach spaces (WLD).
Therefore the class of Banach spaces with the $c_0$EP\ contains the class of WLD spaces. Note that there exist Banach spaces with the $c_0$EP\ that are not WLD. For instance, the space of continuous functions $C[0,\omega_1]$ defined on the ordinal segment $[0,\omega_1]$ has the $c_0$EP, since $[0,\omega_1]$ is a monolithic compact line, but $C[0,\omega_1]$ is not WLD. To see that $C[0,\omega_1]$ is not WLD, note that a necessary condition for a $C(K)$ space to be WLD is that $K$ is a Corson compactum and $[0,\omega_1]$ is not a Corson compactum. A classical class of Banach spaces that contains properly the WLD spaces is the class of $1$-Plichko spaces. We say that a Banach space $X$ is {\it $1$-Plichko} if $X^*$ contains a $1$-norming $\Sigma$-subspace (see \cite{Biorthogonal} for more information on those spaces). Recall that a Banach space $X$ is WLD if and only if $X^*$ is a $\Sigma$-subspace of itself \cite[Theorem~5.37]{Biorthogonal} and therefore every WLD space is $1$-Plichko. An example of a non WLD space that is $1$-Plichko is given by the space $\ell_1(I)$, for any uncountable set $I$ \cite[Example~6.9]{Kalenda}. A natural question at this point is whether every $1$-Plichko space has the $c_0$EP. In Proposition \ref{l1SemEP}, we answer this question negatively by showing that if $I$ is an uncountable set, then the space $\ell_1(I)$ does not have the $c_0$EP.

In Section \ref{C(K)spaces}, we investigate the $c_0$EP\ in the context of $C(K)$ spaces. It is well-known that if $K$ is a compact Hausdorff space, then $C(K)$ is separable if and only if $K$ is metrizable \cite[Lemma~3.102]{Fabian}. Therefore, it follows from Sobczyk's theorem that $C(K)$ has the $c_0$EP, for every metrizable compact space $K$. This result can be extended to the class of Eberlein compacta that generalizes the class of metrizable compact spaces. Recall that if $K$ is a compact Hausdorff space, then $K$ is an Eberlein compactum if and only if $C(K)$ is WCG \cite[Theorem~14.9]{Fabian}. Thus $C(K)$ has the $c_0$EP\ whenever $K$ is an Eberlein compactum.
Note that the class of Corson compacta generalizes properly the class of Eberlein compacta. In this work we address the problem of determining whether $C(K)$ has the $c_0$EP, for every Corson compactum. In Theorem \ref{CorsonIndependent}, we establish that the existence of a Corson compactum $K$ such that $C(K)$ does not have the $c_0$EP\ is independent from the axioms of ZFC. Another interesting result presented here is the characterization of the $c_0$EP\ for spaces of continuous functions on scattered compact spaces with small height. More precisely, in Theorem \ref{Scatteredc0EPiiMono} we show that if $K$ is a scattered compact space with height at most $\omega+1$, then $C(K)$ has the $c_0$EP\ if and only if $K$ is monolithic. It is worthwhile to compare this result with a similar characterization of the $c_0$EP\ in the context of compact lines obtained in \cite[Theorem~2.2]{SobczykLine}.
\end{section}

\begin{section}{General Results}
\label{sec:GeneralResults}

In this work we consider only real Banach spaces. An apparently stronger property than the $c_0$EP\ is the $c_0$-extension property with some constant.

\begin{defin}
Let $X$ be a Banach space and $\lambda \ge 1$ be a real number. We say that $X$ has the {\it $c_0$-extension property with constant $\lambda$} ($\lambda$-$c_0$EP) if any bounded operator $T:Y \to c_0$, defined on a closed subspace $Y$ of $X$, admits a bounded extension $\tilde{T}: X \to c_0$ with $\Vert \tilde{T} \Vert \le \lambda \Vert T \Vert$.
\end{defin}

We observe that all known proofs of $c_0$EP\ are actually proofs of $2$-$c_0$EP. It was shown in \cite[Proposition~2.2]{c0EP} that every WCG Banach space has the $2$-$c_0$EP. Moreover, the proof of \cite[Theorem~2.2]{SobczykLine} establishes that if $K$ is a monolithic compact line, then $C(K)$ has the $2$-$c_0$EP.

In what follows we devote ourselves to the proof of Theorem \ref{monoc0EP} which states that if $X$ is a Banach space such that $B_{X^*}$ is $w^*$-monolithic, then $X$ has the $2$-$c_0$EP. This proof relies essentially on Proposition \ref{monoBallequiv} that establishes an interesting characterization of the monolithicity of the dual unit ball of a Banach space in terms of $\ell_\infty$-valued bounded operators. The key ingredient in the proof of Proposition \ref{monoBallequiv} is a translation of monolithicity in terms of absolute bipolar sets presented in Lemma \ref{monoBipolar}. Let us recall some terminology and facts.
Given a real normed space $X$, there is a natural bilinear pairing of $X$ and $X^*$ given by:
\[X \times X^* \ni (x, \alpha) \to \alpha(x) \in \mathds R.\]
Given a subset $A$ of $X^*$ the {\it absolute polar} $A^0$ of $A$, with respect to this bilinear pairing, is defined as:
\[A^0=\{x \in X: \vert \alpha(x) \vert \le 1, \ \forall \alpha \in A\}\]
and given a subset $B$ of $X$ the {\it absolute polar} $B^0$ of $B$, with respect to this bilinear pairing, is defined as:
\[B^0=\{\alpha \in X^*: \vert \alpha(x) \vert \le 1, \ \forall x \in B\}.\]
If $A$ is a subset of $X^*$, then the {\it absolute bipolar} $A^{00}$ of $A$ is defined as $A^{00}=(A^0)^0$.
Here we are considering only this natural bilinear pairing.
The proof of Lemma \ref{monoBipolar} relies on the deep relationship between the absolutely convex hull of subsets of $X^*$ and their absolute bipolars that is given by the {\em Absolute bipolar theorem} \cite[Theorem~3.1.1]{Bogachev}. For this particular bilinear pairing, the Absolute bipolar theorem ensures that $A^{00}$ coincides with the $w^*$-closure of the absolutely convex hull of $A$, for every subset $A$ of $X^*$.
Recall that if $A$ is a subset of $X^*$, then the {\it absolutely convex hull of $A$}, denoted here by $\mathrm{absconv}(A)$, is the smallest absolutely convex subset of $X^*$ containing $A$. It is easy to see that:
\[\mathrm{absconv}(A)=\big\{\sum_{i=1}^{n} t_i a_i: n \ge 1, a_i \in A, t_i \in \mathds R \ \text{and $\sum_{i=1}^{n} \vert t_i \vert \le 1$}\big\}.\]

\begin{lem}\label{monoBipolar}
Let $X$ be a Banach space. The following conditions are equivalent:
\begin{enumerate}
\item[(i)] $B_{X^*}$ is $w^*$-monolithic.

\item[(ii)] $\overline{\mathrm{absconv}(A)}^{w^*}$ is $w^*$-metrizable, for every countable subset $A$ of $B_{X^*}$.

\item[(iii)] $A^{00}$ is $w^*$-metrizable, for every countable subset $A$ of $B_{X^*}$.

\end{enumerate}
\end{lem}
\begin{proof}
The equivalence between (ii) and (iii) follows directly from the Absolute bipolar theorem.
Observe that Uryshon's metrization theorem implies that a compact space $K$ is monolithic if and only if the closure of any countable subset of $K$ is metrizable.
Clearly (ii) implies (i). To see that (i) implies (ii), note that if $A$ is a countable subset of $X^*$, then $\overline{\mathrm{absconv}(A)}^{w^*}$ is $w^*$-separable. This separability follows from the fact that the countable set
\[\big\{\sum_{i=1}^{n} q_i a_i: n \ge 1, a_i \in A, q_i \in \mathbb Q \ \text{and $\sum_{i=1}^{n} \vert q_i \vert \le 1$}\big\}\]
is $w^*$-dense in $\overline{\mathrm{absconv}(A)}^{w^*}$.
\end{proof}

Let $X$ be a real normed space and $A$ be a bounded subset of $X^*$. We denote by $\rho_A: X \to [0,+\infty[$ the seminorn defined as:
\[\rho_A(x)=\sup_{\alpha \in A}\vert \alpha (x) \vert, \ \forall x \in X.\]

\begin{lem}\label{seminormSep}
Let $X$ be a Banach space. The following conditions are equivalent:
\begin{enumerate}
\item[(a)] $B_{X^*}$ is $w^*$-monolithic.

\item[(b)] $(X, \rho_{A})$ is separable, for every countable subset $A$ of $B_{X^*}$.
\end{enumerate}
\end{lem}
\begin{proof}
To prove that $(a)$ and $(b)$ are equivalent it suffices to show the equivalence between $(b)$ and condition $(iii)$ of Lemma \ref{monoBipolar}. Towards this goal, we will show that $A^{00}$ coincides with the closed unit ball of the dual space $(X,\rho_A)^*$, for any subset $A$ of $B_{X^*}$. The result will then follow from the fact that $(X, \rho_A)$ is separable if and only if the closed unit ball of its dual space is $w^*$-metrizable \cite[Proposition~3.103]{Fabian}. Let $A$ be a subset of $B_{X^*}$ and note that:
\[A^{00}=\{\alpha \in X^*: \vert \alpha(x) \vert \le 1, \ \forall x \in B_{(X, \rho_A)}\},\]
where $B_{(X, \rho_A)}$ denotes the closed unit ball of $(X, \rho_A)$.
Finally, to conclude the result just observe that $A^{00} \subset (X, \rho_A)^*$ and that $(X, \rho_A)^* \subset X^*$.
\end{proof}

Recall that if $X$ is a real normed space, then there exists a bijective correspondence between bounded operators $X \to \ell_\infty$ and bounded sequences in $X^*$.
More precisely, if $S:X \to \ell_\infty$ is a bounded operator, then the {\it sequence associated to $S$} is $(\pi_n \circ S)_{n \ge 1}$, where $\pi_n: \ell_\infty \to \mathds R$ denotes the $n^{th}$ projection. Conversely if $(\alpha_n)_{n \ge 1}$ is a bounded sequence in $X^*$, then the {\it operator associated} to this sequence is defined as $S(x)=\big(\alpha_n(x)\big)_{n \ge 1}$, for every $x \in X$. Note that if $(\alpha_n)_{n \ge 1}$ is the sequence associated to a bounded operator $S:X \to \ell_\infty$, then $\Vert S \Vert=\sup_{n \ge 1} \Vert \alpha_n \Vert$. It follows from this correspondence and Hahn-Banach's theorem that every bounded $\ell_\infty$-valued operator defined on a subspace of $X$ admits a bounded $\ell_\infty$-valued extension defined on $X$ with the same norm. Any of those extensions is called a {\it Hahn-Banach extension} of the original operator. Given an operator $S$, we denote its image by $\Img S$.

\begin{prop}\label{monoBallequiv}
Let $X$ be a Banach space. The following conditions are equivalent:
\begin{enumerate}
\item $B_{X^*}$ is $w^*$-monolithic.

\item $\Img S$ is separable, for every bounded operator $S:X \to \ell_\infty$.
\end{enumerate}
\end{prop}
\begin{proof}
Assume $(1)$ and let $S:X \to \ell_\infty$ be a bounded operator. Without loss of generality, we may assume that $\Vert S \Vert=1$. Denote by $(\alpha_n)_{n \ge 1}$ the sequence associated to $S$ and set $A=\{\alpha_n: n \ge 1\}$. Since $A \subset B_{X^*}$, Lemma \ref{seminormSep} ensures that $(X,\rho_A)$ is separable. The separability of $\Img S$ follows from the fact that the onto operator $S:(X, \rho_A) \to \Img S$ is bounded and therefore continuous. Now let us prove that $(2)$ implies condition (b) of Lemma \ref{seminormSep}. Let $A$ be a countable subset of $B_{X^*}$ and enumerate $A=\{\alpha_n: n \ge1\}$. Denote by $S:X \to \ell_\infty$ the bounded operator associated to the sequence $(\alpha_n)_{n \ge 1}$. Let $E$ be a countable and dense subset of $\Img S$ given by $(2)$ and denote by $D$ a countable subset of $X$ such that $S[D]=E$. It is easy to see that $D$ is dense in $(X, \rho_A)$. Indeed, fixed $x \in X$ and $\epsilon>0$, the density of $E$ in $\Img S$ implies that there exists $e \in E$ with $\Vert S(x)-e \Vert< \epsilon$. If $d \in D$ satisfies $S(d)=e$, then:
\[\rho_A(x-d)=\sup_{n \ge 1} \vert \alpha_n(x-d) \vert=\Vert S(x-d) \Vert=\Vert S(x)-e \Vert< \epsilon.\]

\end{proof}

\begin{teo}\label{monoc0EP}
Let $X$ be a Banach space. If $B_{X^*}$ is $w^*$-monolithic, then $X$ has the $2$-$c_0$EP.
\end{teo}
\begin{proof}
Let $Y$ be a closed subspace of $X$ and $T:Y \to c_0$ be a bounded operator. If $S: X \to \ell_\infty$ denotes a Hahn-Banach extension of $T$, then Proposition \ref{monoBallequiv} ensures that $\Img S$ is a separable subspace of $\ell_\infty$. Therefore $\overline{\Img T}$ is a closed subspace of the separable Banach space $\overline{\Img S}$. It follows from Sobczyk's theorem that the inclusion map $i: \overline{\Img T} \to c_0$ admits a bounded extension $L: \overline{\Img S} \to c_0$ with $\Vert L \Vert \le 2 \Vert i \Vert \le 2$. The map $L \circ S: X \to c_0$ is a bounded extension of $T$ and $\Vert L \circ S \Vert \le \Vert S \Vert \Vert L \Vert\le 2 \Vert T\Vert$.
\end{proof}

We do not know if the converse of Theorem \ref{monoc0EP} holds.

\begin{cor}
If $X$ is a WLD Banach space, then $X$ has the $2$-$c_0$EP.
\end{cor}
\begin{proof}
It follows from Theorem \ref{monoc0EP} and the fact that every Corson compactum is monolithic.
\end{proof}

Now let us see some stability properties of the class of Banach spaces with the $c_0$EP.

\begin{prop}\label{fechamentosEP}
Let $X$ and $Z$ be Banach spaces and assume that $X$ has the $c_0$EP.
\begin{enumerate}
\item[(a)] If $Z$ is a subspace of $X$, then $Z$ has the $c_0$EP.

\item[(b)] If $Q:X \to Z$ is a bounded and onto operator, then $Z$ has the $c_0$EP.
\end{enumerate}
\end{prop}
\begin{proof}
The proof of (a) is straightforward. To prove (b), let $Y$ be a closed subspace of $Z$ and let $T:Y \to c_0$ be a bounded operator. Since $X$ has the $c_0$EP, the bounded operator $T \circ Q|_{Q^{-1}[Y]}$ admits a bounded extension $S: X \to c_0$. Define $\tilde{T}: Z \to c_0$ as $\tilde{T}(z)=S(x)$, where $x$ is any element of $X$ satisfying $Q(x)=z$. The fact that $\tilde{T}$ is well-defined follows from the fact that $\Ker Q$ is contained in $Q^{-1}[Y]$. Indeed, if $x_1$ and $x_2$ are elements of $X$ with $Q(x_1)=Q(x_2)$, then
$x_1-x_2 \in \Ker Q \subset Q^{-1}[Y]$. Therefore, we have:
\[S(x_1-x_2)=T(Q(x_1-x_2))=0.\]
It is easy to see that the map $\tilde{T}$ is a bounded extension of $T$.
\end{proof}

Next proposition states that the $1$-Plichko space $\ell_1(I)$ does not have the $c_0$EP, for any uncountable set $I$. We present its proof after Remark \ref{valdiviaSemEP},
since it depends on results developed in Section \ref{C(K)spaces}.

\begin{prop}\label{l1SemEP}
If $I$ is an uncountable set, then $\ell_1(I)$ does not have the $c_0$EP.
\end{prop}

\begin{cor}\label{copial1SemEP}
Let $X$ be a Banach space and $I$ be an uncountable set. If $X$ contains an isomorphic copy of $\ell_1(I)$, then $X$ does not have the $c_0$EP.
\end{cor}
\begin{proof}
It follows from Propositions \ref{fechamentosEP}(a) and \ref{l1SemEP}.
\end{proof}

It is interesting to observe that even though not every $1$-Plichko space has the $c_0$EP, these spaces have a weaker property that was also introduced in \cite{SobczykLine}; namely the separable $c_0$-extension property. We say that a Banach space $X$ has the {\it separable $c_0$-extension property} (separable $c_0$EP) if every $c_0$-valued bounded operator defined on a closed and separable subspace of $X$ admits a $c_0$-valued bounded extension defined on $X$. The fact that every $1$-Plichko space has the separable $c_0$EP\ follows from the fact that such spaces have the separable complementation property (SCP) \cite[pg. 105]{Biorthogonal} and clearly the SCP implies the separable $c_0$EP. Note that Proposition \ref{l1SemEP} provides an example of a space with the SCP but not the $c_0$EP. A stronger property than the SCP is the controlled separable complementation property (CSCP) that was introduced in \cite{Woj} and studied in a series of papers \cite{Ferrer1, Ferrer2, Ferrer3}. A surprising consequence of Theorem \ref{monoc0EP} is that the CSCP implies the $c_0$EP.

\begin{prop}
Let $X$ be a Banach space. If $X$ has the CSCP, then $X$ has the $2$-$c_0$EP.
\end{prop}
\begin{proof}
It follows from Theorem \ref{monoc0EP} and the fact that if $X$ has the CSCP, then $B_{X^*}$ is $w^*$-monolithic \cite[Proposition~1.5]{KK}.
\end{proof}

\end{section}

\begin{section}{The $c_0$EP\ for $C(K)$ spaces}
\label{C(K)spaces}

In this section, we develop the theory of the $c_0$EP\ in the context of Banach spaces of the form $C(K)$. Throughout this section we will refer to a compact and Hausdorff space just as a compact space. For a compact space $K$, we identify the dual space $C(K)^*$ with the space $M(K)$ of regular signed finite Borel measures on $K$, endowed with the total variation norm. In Lemma \ref{MonoMBolaMono}, we establish conditions on a compact space $K$ that ensure that $B_{M(K)}$ is $w^*$-monolithic. Clearly if $B_{M(K)}$ is $w^*$-monolithic, then $K$ is monolithic, since $K$ is homeomorphic to a closed subspace of $\big(B_{M(K)}, w^*\big)$. The additional condition on monolithic compacta used in Lemma \ref{MonoMBolaMono} is Property (M). We say that a compact space $K$ has {\it Property (M)} if the support of every measure belonging to $M(K)$ is separable. Recall that if $\mu$ is an element of $M(K)$, then its {\it support} is defined as:
\[\supp \mu=\{p \in K: \vert \mu \vert(U)>0, \ \text{for every open neighborhood $U$ of $p$}\},\]
where $\vert \mu \vert$ denotes the total variation of $\mu$.

\begin{lem}\label{MonoMBolaMono}
If $K$ is monolithic compact space with Property (M), then $B_{M(K)}$ is $w^*$-monolithic.
\end{lem}
\begin{proof}
Let $\{\mu_n: n \ge 1\}$ be a countable subset of $B_{M(K)}$ and denote by $L$ the closed subset of $K$ defined as $L=\overline{\bigcup_{n \ge 1} \supp \mu_n}$. Since $K$ has Property (M) and is monolithic, we have that $L$ is a compact metric space and therefore $B_{M(L)}$ is $w^*$-metrizable. To show that $\overline{\{\mu_n: n \ge 1\}}^{w^*}$ is $w^*$-metrizable, we will prove that this space is a subspace of an homeomorphic image of $(B_{M(L)}, w^*)$. Denote by $i_{*}: M(L) \to M(K)$ the operator defined as $i_*(\nu)(B)=\nu(B \cap L)$, for every Borel subset $B$ of $K$ and every $\nu \in M(L)$. Note that $i_*$ is injective and $w^*$-continuous. Therefore the $w^*$-compactness of $B_{M(L)}$ ensures that $\big(i_*[B_{M(L)}], w^*\big)$ is homeomorphic to $(B_{M(L)}, w^*)$.
Finally, observe that if $\mu \in B_{M(K)}$ and $\supp \mu \subset L$, then $\mu \in i_*[B_{M(L)}]$. This implies that $\overline{\{\mu_n: n \ge 1\}}^{w^*} \subset i_*[B_{M(L)}]$.
\end{proof}

\begin{teo}\label{monoMEP}
If $K$ is a monolithic compact space with Property (M), then $C(K)$ has the $2$-$c_0$EP.
\end{teo}
\begin{proof}
It follows from Theorem \ref{monoc0EP} and Lemma \ref{MonoMBolaMono}.
\end{proof}

Clearly every metrizable compact space has Property (M) and it is easy to see that every scattered compact space has Property (M). We say that a topological space $X$ is {\it scattered} if for every closed subspace $Y$ of $X$, the set of isolated points of $Y$ is dense in $Y$. Other examples of spaces with Property (M) are compact lines \cite[Lemma~2.1]{KK}, Eberlein compacta and Rosenthal compacta \cite[Remark~3.2]{ArgyrosCorson}. Recall that a {\it Rosenthal compactum} is a pointwise compact subset of the space of Baire class one real-valued functions on a Polish space.

\begin{cor}\label{ghat}
If $K$ is a monolithic compact space belonging to any of the following classes, then $C(K)$ has the $2$-$c_0$EP:
\begin{enumerate}
\item[(a)] Compact lines.

\item[(b)] Scattered spaces.

\item[(c)] Rosenthal compacta.
\end{enumerate}
\end{cor}
\begin{proof}
It follows from Theorem \ref{monoMEP} and the fact that those spaces have Property (M).
\end{proof}

It is well-known that the existence of a Corson compactum without Property (M) is independent from the axioms of $ZFC$. It was shown in \cite[Remark~3.2.3]{ArgyrosCorson} that if we assume Martin's axiom and the negation of the Continuum hypothesis ($MA+\neg CH$), then every Corson compactum has Property (M). Therefore Theorem \ref{monoMEP} implies that, under $MA+\neg CH$, $C(K)$ has the $2$-$c_0$EP, for every Corson compactum $K$. On the other hand, if we assume $CH$, then there exists a Corson compactum without Property (M) \cite[Theorem~3.12]{ArgyrosCorson}. It turns out that the space of continuous functions on the Corson compactum constructed in \cite[Theorem~3.12]{ArgyrosCorson} does not have the $c_0$EP.

\begin{prop}\label{CorsonSemEP}
Assume $CH$. There exists a Corson compactum $K$ such that $C(K)$ does not have the $c_0$EP.
\end{prop}
\begin{proof}
Let $K$ denote the Corson compactum without Property (M) constructed in \cite[Theorem~3.12]{ArgyrosCorson}. It follows from \cite[Theorem~3.13]{ArgyrosCorson} that $C(K)$ contains an isomorphic copy of $\ell_1(\omega_1)$. Therefore the result follows from Corollary \ref{copial1SemEP}.
\end{proof}

From what was discussed above we conclude the following result.

\begin{teo}\label{CorsonIndependent}
The existence of a Corson compactum $K$ such that $C(K)$ does not have the $c_0$EP\ is independent from the axioms of ZFC.
\end{teo}

\begin{rem}
The Corson compactum without Property (M) constructed in \cite[Theorem~3.12]{ArgyrosCorson} is defined by an adequate family of sets. In \cite[Theorem~3.13]{ArgyrosCorson}, it was shown that if $K$ is a Corson compactum defined by an adequate family of sets, then $K$ has Property (M) if and only if $\ell_1(\omega_1)$ does not embed isomorphically in $C(K)$. Thus Corollary \ref{copial1SemEP} and Theorem \ref{monoMEP} imply that if $K$ is a Corson compactum defined by an adequate family of sets, then $C(K)$ has the $c_0$EP\ if and only if $K$ has Property (M). We do not know if this equivalence holds for any Corson compactum. For instance it is unclear if $C(K)$ has the $c_0$EP\ for the Corson compactum $K$ without Property (M) constructed in \cite{Kunen}.
\end{rem}

Note that Proposition \ref{CorsonSemEP} also shows that if we assume $CH$, then there exists a monolithic compact space $K$ such that $C(K)$ does not have the $c_0$EP. However this is not possible under $MA+\neg CH$.

\begin{prop}\label{Martinmonotemc0EPsempre}
Assume $MA+\neg CH$. If $K$ is a monolithic compact space, then $C(K)$ has the $2$-$c_0$EP.
\end{prop}
\begin{proof}
It was shown in \cite[Theorem~3]{Arkangelski} that if we assume $MA+\neg CH$, then every monolithic compact space with the countable chain condition is separable. It is easy to see that this implies that, under $MA+\neg CH$, every monolithic compact space has Property (M). Thus the result follows from Theorem \ref{monoMEP}.
\end{proof}

From what was discussed above we conclude the following result.

\begin{teo}
The existence of a monolithic compact space $K$ such that $C(K)$ does not have the $c_0$EP\ is independent from the axioms of $ZFC$.
\end{teo}

Now we present some useful stability results for the $c_0$EP\ in the context of $C(K)$ spaces.

\begin{prop}\label{FecCompactEP}
Let $K$ and $L$ be compact spaces and assume that $C(K)$ has the $c_0$EP.
\begin{enumerate}
\item If $L$ is a subspace of $K$, then $C(L)$ has the $c_0$EP.

\item If $L$ is a quotient of $K$, then $C(L)$ has the $c_0$EP.
\end{enumerate}
\end{prop}
\begin{proof}
To prove (1), note that the restriction operator $C(K) \to C(L)$ is a bounded and onto operator. Therefore the result follows from Proposition \ref{fechamentosEP}(b).
Now let us prove (2). It is easy to see that if $\phi: K \to L$ is a continuous and onto map, then the composition operator $\phi^*: C(L) \to C(K)$ defined as $\phi^*(f)=f \circ \phi$, for all $f \in C(L)$, is an isometric embedding. Proposition \ref{fechamentosEP}(a) ensures that $\phi^*[C(L)]$ has the $c_0$EP\ and therefore $C(L)$ has the $c_0$EP.
\end{proof}

In what follows, we investigate the $c_0$EP\ in the class of scattered compact spaces. Let us recall some basic definitions and facts. Given a topological space $X$ and an ordinal number $\alpha$, we denote by $X^{(\alpha)}$ the $\alpha^{th}$ Cantor--Bendixson derivative of $X$.
It is well-known that a topological space $X$ is scattered if and only if there exists an ordinal $\alpha$ such that $X^{(\alpha)}=\emptyset$ \cite[Proposition~17.8]{Koppelberg}. The {\it height} of a scattered topological space $X$, denoted here by $\height(X)$, is defined as the least ordinal $\alpha$ such that $X^{(\alpha)}=\emptyset$. Finally if the height of $X$ is a natural number, then we say that $X$ has {\it finite height}.
In Corollary \ref{ghat}(b), we proved that if $K$ is a monolithic compact scattered space, then $C(K)$ has the $c_0$EP. In Theorem \ref{finiteheightEPMONO} and Corollary \ref{htw1}, we will show that if $K$ is a scattered compact space with height at most $\omega+1$ such that $C(K)$ has the $c_0$EP, then $K$ is monolithic.

\begin{lem}\label{MrowkaSemEP}
Let $K$ be a scattered compact space with height $3$. If $K$ is separable and nonmetrizable, then $C(K)$ does not have the $c_0$EP.
\end{lem}
\begin{proof}
Firstly let us prove that we can assume that $K^{(2)}$ has only one point. Write $K^{(2)}=\{p_1,\ldots,p_k\}$, where $k=\vert K^{(2)}\vert$. Clearly, there exist disjoint clopen subsets $C_1,\ldots,C_k$ of $K$ such that $K=\bigcup_{i=1}^{k} C_i$ and $p_i \in C_i$, for every $i=1,\ldots,k$. It is easy to see that $C_i$ is separable and $C_i^{(2)}=\{p_i\}$, for every $i=1,\ldots,k$ and that there exists an index $i_0$ such that $C_{i_0}$ is nonmetrizable.
Thus Proposition \ref{FecCompactEP}(1) ensures that the result follows if we show that $C(C_{i_0})$ does not have the $c_0$EP. Note that the restriction operator $R: C(K) \to C(K^{(1)})$ provides the following short exact sequence:
\[0 \longrightarrow \Ker(R) \longrightarrow  C(K) \overset{R}\longrightarrow C(K^{(1)}) \longrightarrow 0.\]
It is well-known that $\Ker(R)$ is isomorphic to $c_0$ and that $C(K^{(1)})$ is isomorphic to $c_0(K^{(1)})$. Therefore we have an exact sequence of the form:
\begin{equation}\label{exactSequence}
0 \longrightarrow c_0 \longrightarrow  C(K) \longrightarrow c_0(K^{(1)}) \longrightarrow 0.
\end{equation}
In order to conclude that $C(K)$ does not have the $c_0$EP, we will show that the isomorphic copy of $c_0$ inside $C(K)$ given by \eqref{exactSequence} is not complemented in $C(K)$.
Assume by contradiction that this copy of $c_0$ is complemented in $C(K)$. In this case $C(K)$ is isomorphic to $c_0 \bigoplus c_0(K^{(1)})$ and therefore $C(K)$ is a WCG space. This implies that $K$ is an Eberlein compactum. But this is not possible, since $K$ is not monolithic.
\end{proof}

An interesting consequence of Lemma \ref{MrowkaSemEP} is that the $c_0$EP\ is not a three-space property. We say that a property $\mathcal P$ is a {\it three-space property} if every twisted sum of Banach spaces with property $\mathcal P$ also has property $\mathcal P$. Classical examples of three-space properties for Banach spaces are separability and reflexivity \cite{CastilloGonzalez}.

\begin{cor}\label{Not3Propc0EP}
The $c_0$EP\ is not a three-space property.
\end{cor}
\begin{proof}
Consider the exact sequence \eqref{exactSequence} described in the proof Lemma \ref{MrowkaSemEP}. The spaces $c_0$ and $c_0(K^{(1)})$  have the $c_0$EP, since they are WCG spaces, but $C(K)$ does not have the $c_0$EP.
\end{proof}

The strategy to prove Theorem \ref{finiteheightEPMONO} is to proceed by induction on the height of $K$. In order to decrease the height and use the induction hypothesis we will use a quotient of $K$. Given a compact space $K$ and a closed subset $F$ of $K$, we define the following equivalence relation on K:
\[x \sim_F y \Leftrightarrow x=y \ \text{or} \ x,y \in F.\]
Clearly the quotient space $K/\sim_F$ is compact and it is easy to see that it is also Hausdorff.

\begin{lem}\label{QuotientF}
Given an ordinal $\alpha$, if $F=K^{(\alpha)}$, then $\big(K/\sim_F\big)^{(\alpha+1)}=\emptyset$.
\end{lem}
\begin{proof}
It follows from the fact that if $K$ and $L$ are compact spaces and $q:K \to L$ is a continuous and onto map, then $L^{(\alpha)} \subset q[K^{(\alpha)}]$, for every ordinal $\alpha$ \cite[Lemma~1.8 and Theorem~1.9]{QuotientScattered}.
\end{proof}

\begin{teo}\label{finiteheightEPMONO}
Let $K$ be a compact space with finite height. If $C(K)$ has the $c_0$EP, then $K$ is monolithic.
\end{teo}
\begin{proof}
It follows from Proposition \ref{FecCompactEP}(1) that it suffices to prove that if $K$ is a separable compact space with finite height such that $C(K)$ has the $c_0$EP, then $K$ is metrizable. Let us proceed by induction on the height of $K$. The case when $\height(K)\le 2$ is trivial and the case when $\height(K)=3$ is established in Lemma \ref{MrowkaSemEP}.
Now assume that $\height(K)=N+1$ with $N \ge 3$ and denote by $F$ the closed subset $K^{(N-1)}$ of $K$. Note that Lemma \ref{QuotientF} implies that $K/\sim_F$ is a scattered space with $\height\big(K/\sim_F\big) \le N$. Since Proposition \ref{FecCompactEP}(2) ensures that $C\big(K/\sim_F\big)$ also has the $c_0$EP, it follows from the induction hypothesis that $K/\sim_F$ is metrizable. Recall that if an infinite compact space is scattered, then its weight coincides with its cardinality \cite[Proposition~17.10]{Koppelberg}. Thus we have that $K/\sim_F$ is countable and consequently $K \setminus F$ is countable. This implies that $K^{(1)} \setminus K^{(2)}$ is countable and therefore $K^{(1)}$ is separable.
Note that $\height(K^{(1)})=N$ and that Proposition \ref{FecCompactEP}(1) ensures that $C(K^{(1)})$ has the $c_0$EP. Thus it follows from the induction hypothesis that $K^{(1)}$ is metrizable and therefore $K^{(1)}$ is countable, since it is scattered.
Finally the metrizability of $K$ follows from the fact that $K \setminus K^{(1)}$ is also countable, since $K$ is separable.
\end{proof}

\begin{cor}\label{htw1}
Let $K$ be a compact space with height $\omega+1$. If $C(K)$ has the $c_0$EP, then $K$ is monolithic.
\end{cor}
\begin{proof}
It follows from Proposition \ref{FecCompactEP}(1) that it suffices to prove that if $K$ is a separable compact space with height $\omega+1$ such that $C(K)$ has the $c_0$EP, then $K$ is metrizable. In order to do so, we will prove that $K \setminus K^{(N+1)}$ is countable, for every $N \in \omega$. This will imply that $K$ is countable and therefore metrizable, since $K=K^{(\omega)} \cup \bigcup_{N \in \omega} K \setminus K^{(N+1)}$ and $K^{(\omega)}$ is finite. Given $N \in \omega$, denote by $F$ the closed subset $K^{(N+1)}$ of $K$ and note that Lemma \ref{QuotientF} implies that $K/\sim_F$ is a scattered compact space with $\height\big(K/\sim_F\big) \le N+2$. Since Proposition \ref{FecCompactEP}(2) ensures that $C\big(K/\sim_F\big)$ also has the $c_0$EP, it follows from Theorem \ref{finiteheightEPMONO} that $K/\sim_F$ is metrizable. This implies that $K/\sim_F$ is countable and thus $K \setminus F$ is countable.
\end{proof}

\begin{teo}\label{Scatteredc0EPiiMono}
If $K$ is a scattered compact space with height at most $\omega+1$, then $C(K)$ has the $c_0$EP\ if and only if $K$ is monolithic.
\end{teo}
\begin{proof}
It follows from Corollary \ref{ghat}(b), Theorem \ref{finiteheightEPMONO} and Corollary \ref{htw1}.
\end{proof}

An interesting consequence of Lemma \ref{MrowkaSemEP} is the existence of Valdivia compacta whose spaces of continuous functions do not have the $c_0$EP. Recall that a compact space $K$ is said to be a {\it Valdivia compactum} if there exists a set $I$ and an homeomorphic embedding $\varphi: K \to \mathds R^I$ such that $\varphi^{-1}[\Sigma(I)]$ is dense in $K$. Clearly, every Corson compactum is Valdivia and it is easy to see that if $\kappa$ is an uncountable cardinal, then the Cantor cube $2^\kappa$ is a non-Corson Valdivia compactum \cite[Theorem~3.29]{Kalenda}.

\begin{prop}\label{2kappanoEP}
If $\kappa$ is an uncountable cardinal, then the space $C(2^\kappa)$ does not have the $c_0$EP.
\end{prop}
\begin{proof}
Proposition \ref{FecCompactEP}(1) ensures that it suffices to show that $C(2^{\omega_1})$ does not have the $c_0$EP, since $2^{\omega_1}$ embeds homeomorphically in $2^\kappa$, for any uncountable cardinal $\kappa$. Let $K$ be a separable scattered compact space with height $3$ and weight $\omega_1$.
Since $K$ is zero-dimensional, we have that $K$ embeds homeomorphically in $2^{\omega_1}$. Therefore the result follows from Proposition \ref{FecCompactEP}(1) and Lemma \ref{MrowkaSemEP}.
\end{proof}

\begin{rem}\label{valdiviaSemEP}
Recall that if $K$ is a Valdivia compactum, then $C(K)$ is a $1$-Plichko space \cite[Theorem~5.55]{Biorthogonal}. Thus it follows from what was discussed in the previous section that $C(K)$ has the separable $c_0$EP, for every Valdivia compactum $K$.
\end{rem}

\begin{proof}[Proof of Proposition \ref{l1SemEP}]
According to Proposition \ref{fechamentosEP}(b), it suffices to show that if $I$ is an uncountable set, then $\ell_1(I)$ has a Banach quotient without the $c_0$EP. It is well-known that every Banach space with density $\vert I \vert$ is a quotient of $\ell_1(I)$. The result follows from Proposition \ref{2kappanoEP}, since the density of $C(2^{I})$ is $\vert I \vert$.
\end{proof}

We can generalize Proposition \ref{2kappanoEP} to the class of dyadic compacta. A compact space is said to be a {\it dyadic compactum} if it is a continuous image of the Cantor cube $2^\kappa$, for some cardinal $\kappa$.

\begin{cor}
If $K$ is a nonmetrizable dyadic compactum, then $C(K)$ does not have the $c_0$EP.
\end{cor}
\begin{proof}
Gerlits and Efimov showed that every nonmetrizable dyadic compactum contains an homeomorphic copy of the Cantor cube $2^{\omega_1}$ (see \cite[3.12.12]{Engelking}). Therefore the result follows from Propositions \ref{FecCompactEP}(1) and \ref{2kappanoEP}.
\end{proof}

We conclude this section by showing that if $K$ is a compact space such that $C(K)$ has the $c_0$EP, then $K$ admits only measures with small Maharam type. Given a compact space $K$ and a cardinal $\kappa$, we say that a nonnegative measure $\mu \in M(K)$ has {\it Maharam type} $\kappa$ if $\kappa$ is the density of the Banach space $L_1(\mu)$.

\begin{prop}
Let $K$ be a compact space. If $C(K)$ has the $c_0$EP, then every nonnegative measure in $M(K)$ has Maharam type at most $\omega_1$.
\end{prop}
\begin{proof}
If there exists a nonnegative measure in $M(K)$ with Maharam type strictly greater than $\omega_1$, then $C(K)$ contains an isomorphic copy of $\ell_1(\omega_1)$ \cite{PlebanekMaharam}. Therefore Corollary \ref{copial1SemEP} implies that $C(K)$ does not have the $c_0$EP.
\end{proof}

Assuming $MA+\neg CH$, we obtain the following stronger result.

\begin{prop}
Assume $MA+\neg CH$. Let $K$ be a compact space. If $C(K)$ has the $c_0$EP, then every nonnegative measure in $M(K)$ has countable Maharam type.
\end{prop}
\begin{proof}
Under $MA+\neg CH$, if there exists a nonnegative measure in $M(K)$ with uncountable Maharam type, then $C(K)$ contains an isomorphic copy of $\ell_1(\omega_1)$ \cite{PlebanekMaharam}. Therefore, Corollary \ref{copial1SemEP} implies that $C(K)$ does not have the $c_0$EP.
\end{proof}

\end{section}

\end{document}