## Abstract

The famous Rosenthal–Lacey theorem asserts that for each infinite compact space *K* the Banach space *C*(*K*) admits a quotient isomorphic to Banach spaces *c* or \(\ell _{2}\). The aim of the paper is to study a natural variant of this result for the space \(C_{p}(X)\) of continuous real-valued maps on a Tychonoff space *X* with the pointwise topology. Following Josefson–Nissenzweig theorem for infinite-dimensional Banach spaces we introduce a corresponding property (called Josefson–Nissenzweig property, briefly, the JNP) for \(C_{p}(X)\)-spaces. We prove: for a Tychonoff space *X* the space \(C_p(X)\) satisfies the JNP if and only if \(C_p(X)\) has a quotient isomorphic to \(c_{0}:=\{(x_n)_{n\in \mathbb N}\in \mathbb {R}^\mathbb {N}:x_n\rightarrow 0\}\) (with the product topology of \(\mathbb {R}^\mathbb {N}\)) if and only if \(C_{p}(X)\) contains a complemented subspace isomorphic to \(c_0\). The last statement provides a \(C_{p}\)-version of the Cembranos theorem stating that the Banach space *C*(*K*) is not a Grothendieck space if and only if *C*(*K*) contains a complemented copy of the Banach space \(c_{0}\) with the sup-norm topology. For a pseudocompact space *X* the space \(C_p(X)\) has the JNP if and only if \(C_p(X)\) has a complemented metrizable infinite-dimensional subspace. An example of a compact space *K* without infinite convergent sequences with \(C_{p}(K)\) containing a complemented subspace isomorphic to \(c_{0}\) is given.

## Introduction, motivations and two main problems

Let *X* be a Tychonoff space. By \(C_{p}(X)\) we denote the space of real-valued continuous functions on *X* endowed with the pointwise topology.

We will need the following fact stating that each metrizable (linear) quotient \(C_{p}(X)/Z\) of \(C_p(X)\) by a closed vector subspace *Z* of \(C_p(X)\) is *separable*. Indeed, this follows from the separability of metizable spaces of countable cellularity and the fact that \(C_p(X)\) has countable cellularity, being a dense subspace of \(\mathbb {R}^X\), see [2].

The classic Rosenthal–Lacey theorem, see [19, 23, 27], asserts that the Banach space *C*(*K*) of continuous real-valued maps on an infinite compact space *K* has a quotient isomorphic to Banach spaces *c* or \(\ell _{2},\) or equivalently, there exists a continuous linear (and open; by the open mapping Banach theorem) map from *C*(*K*) onto *c* or \(\ell _{2}\), see also a survey paper [14].

This theorem motivates the following natural question for spaces \(C_{p}(X)\).

### Problem 1

For which compact spaces *K* any of the following equivalent conditions holds:

- (1)
The space \(C_{p}(K)\) has an infinite dimensional metrizable quotient.

- (2)
The space \(C_{p}(K)\) has an infinite dimensional metrizable separable quotient.

- (3)
The space \(C_p(K)\) has a quotient isomorphic to a dense subspace of \(\mathbb {R}^{\mathbb {N}}\).

In [21] it was shown that \(C_{p}(K)\) has an *infinite-dimensional separable quotient algebra* if and only if *K* contains an infinite countable closed subset. Hence \(C_{p}(\beta \mathbb {N})\) lacks infinite-dimensional separable quotient algebras. Nevertheless, as proved in [22, Theorem 4], the space \(C_{p}(K)\) has infinite-dimensional separable quotient for any compact space *K* containing a copy of \(\beta \mathbb {N}\).

Problem 1 has been already partially studied in [3], where we proved that for a Tychonoff space *X* the space \(C_p(X)\) has an infinite-dimensional metrizable quotient if *X* either contains an infinite discrete \(C^{*}\)-embedded subspace or else *X* has a sequence \((K_n)_{n\in \mathbb {N}}\) of compact subsets such that for every *n* the space \(K_n\) contains two disjoint topological copies of \(K_{n+1}\). If fact, the first case (for example if compact *X* contains a copy of \(\beta \mathbb {N}\)) asserts that \(C_{p}(X)\) has a quotient isomorphic to the subspace \(\ell _\infty =\{(x_n)\in \mathbb {R}^{\mathbb {N}}:\sup _n |x_n|<\infty \}\) of \(\mathbb {R}^{\mathbb {N}}\) or to the product \(\mathbb {R}^{\mathbb {N}}\).

Consequently, this theorem reduces Problem 1 to the case when *K* is an *Efimov space* (i.e. *K* is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of \(\beta \mathbb {N}\)). Although, it is unknown if Efimov spaces exist in ZFC (see [7,8,9,10, 12, 13, 16, 18]) we showed in [22] that under \(\lozenge \) for some Efimov spaces *K* the function space \(C_{p}(K)\) has an infinite dimensional metrizable quotient.

By \(c_{0}\) we mean the subspace \(\{(x_n)_{n\in \mathbb N}\in \mathbb {R}^{\mathbb {N}}:x_n\rightarrow 0\}\) of \(\mathbb {R}^{\mathbb {N}}\) endowed with the product topology. The term “the Banach space \({c}_{0}\)” means the classic Banach space of null-sequences with the sup-norm topology.

It is known that the Banach space *C*(*K*) over an infinite compact *K* contains a copy of the Banach space \(c_{0}\), see for example [6]. By a result of Cembranos, see [4, Theorem, page 74], the space *C*(*K*) is not a Grothendieck space if and only if *C*(*K*) contains a complemented copy of the Banach space \(c_{0}\). Recall a Banach space *E* is a *Grothendieck space* if every weak\(^{*}\) converging sequence in the dual \(E^{*}\) weakly converges in \(E^{*}\). It is well-known that if a compact space *K* contains a non-trivial converging sequence, *C*(*K*) is not a Grothendieck space; hence *C*(*K*) contains a complemented copy of the Banach space \(c_{0}\). It is also easy to see that for every infinite compact space *K* the space \(C_{p}(K)\) contains a closed copy of the space \(c_{0}\) endowed with the product topology of \(\mathbb {R}^{\mathbb {N}}\).

Cembranos theorem motivates the following next problem (connected with Problem 1).

### Problem 2

Characterize those spaces \(C_{p}(K)\) which contain a complemented copy of \(c_{0}\) with the product topology of \(\mathbb {R}^{\mathbb {N}}\).

## The main results

For a Tychonoff space *X* and a point \(x\in X\) let \(\delta _x:C_p(X)\rightarrow \mathbb {R},\,\,\, \delta _x:f\mapsto f(x),\) be the Dirac measure concentrated at *x*. The linear hull \(L_p(X)\) of the set \(\{\delta _x:x\in X\}\) in \(\mathbb {R}^{C_p(X)}\) can be identified with the dual space of \(C_p(X)\). We refer also the reader to [15] for more information about the dual \(L_p(X)\).

Elements of the space \(L_p(X)\) will be called *finitely supported sign-measures* (or simply *sign-measures*) on *X*.

Each \(\mu \in L_p(X)\) can be uniquely written as a linear combination of Dirac measures \(\mu =\sum _{x\in F}\alpha _x\delta _x\) for some finite set \(F\subset X\) and some non-zero real numbers \(\alpha _x\). The set *F* is called the *support* of the sign-measure \(\mu \) and is denoted by \(\mathrm {supp}(\mu )\). The measure \(\sum _{x\in F}|\alpha _x|\delta _x\) will be denoted by \(|\mu |\) and the real number \(\Vert \mu \Vert =\sum _{x\in F}|\alpha _x|\) coincides with the *norm* of \(\mu \) (in the dual Banach space \(C(\beta X)^{*}\)).

The sign-measure \(\mu =\sum _{x\in F}\alpha _x\delta _x\) determines the function \(\mu :2^X\rightarrow \mathbb {R}\) defined on the power-set of *X* and assigning to each subset \(A\subset X\) the real number \(\sum _{x\in A\cap F}\alpha _x\). So, a finitely supported sign-measure will be considered both as a linear functional on \(C_p(X)\) and an additive function on the power-set \(2^X\).

The famous *Josefson–Nissenzweig theorem* asserts that for each infinite-dimensional Banach space *E* there exists a null sequence in the weak\(^{*}\)-topology of the topological dual \(E^{*}\) of *E* and which is of norm one in the dual norm, see for example [6].

We propose the following corresponding property for spaces \(C_{p}(X)\).

### Definition 1

For a Tychonoff space *X* the space \(C_{p}(X)\) satisfies the *Josefson–Nissenzweig property* (JNP in short) if there exists a sequence \((\mu _n)\) of finitely supported sign-measures on *X* such that \(\Vert \mu _n\Vert =1\) for all \(n\in \mathbb {N}\), and \(\mu _n(f)\rightarrow _n 0\) for each \(f\in C_p(X)\).

Concerning the JNP of function spaces \(C_p(X)\) on compacta we have the following:

- (1)
*If a compact space**K**contains a non-trivial convergent sequence, say*\(x_{n}\rightarrow x\),*then*\(C_{p}(K)\)*satisfies the JNP.*This is witnessed by the weak\(^{*}\) null sequence \((\mu _n)\) of sign-measures \(\mu _{n}=\frac{1}{2}(\delta _{x_{n}}-\delta _{x})\), \(n\in \mathbb {N}\). - (2)
*The space*\(C_{p}(\beta \mathbb {N})\)*does not satisfy the JNP.*This follows directly from the Grothendieck theorem, see [5, Corollary 4.5.8]. - (3)
*There exists a compact space**K**containing a copy of*\(\beta \mathbb {N}\)*but without non-trivial convergent sequences such that*\(C_{p}(K)\)*satisfies the JNP,*see Example 1 below.

If a compact space *K* contains a closed subset *Z* that is metrizable, then \(C_p(K)\) has a complemented subspace isomorphic to \(C_p(Z)\). Consequently, if compact *K* contains a non-trivial convergent sequence, then \(C_{p}(K)\) has a complemented subspace isomorphic to \(c_{0}\). However for every infinite compact *K* the space \(C_{p}(K)\) contains a subspace isomorphic to \(c_{0}\) but not necessary complemented in \(C_{p}(K)\). Nevertheless, there exists a compact space *K* without infinite convergent sequences and such that \(C_{p}(K)\) enjoys the JNP and so contains a complemented subspace isomorphic to \(c_{0}\), as follows from Theorem 1 below.

It turns out that the Josefson–Nissenzweig property characterizes an interesting case related with Problem 1 and provides a complete solution to Problem 2.

### Theorem 1

For a Tychonoff space *X* the following conditions are equivalent:

- (1)
\(C_{p}(X)\) satisfies the JNP;

- (2)
\(C_p(X)\) contains a complemented subspace isomorphic to \(c_0\);

- (3)
\(C_p(X)\) has a quotient isomorphic to \(c_0\);

- (4)
\(C_p(X)\) admits a linear continuous map onto \(c_0.\) If the space

*X*is pseudocompact, then the conditions (1)–(4) are equivalent to - (5)
\(C_{p}(X)\) contains a complemented infinite-dimensional metrizable subspace;

- (6)
\(C_{p}(X)\) contains a complemented infinite-dimensional separable subspace;

- (7)
\(C_p(X)\) has an infinite-dimensional Polishable quotient.

We recall that a locally convex space *X* is *Polishable* if *X* admits a stronger Polish locally convex topology. Equivalently, Polishable locally convex spaces can be defined as images of separable Fréchet spaces under continuous linear maps. Clearly, the subspace \(c_0\) of \(\mathbb {R}^{\mathbb {N}}\) is Polishable.

A topological space *X* is *pseudocompact* if it is Tychonoff and each continuous real-valued function on *X* is bounded. It is known (see [3]) that a Tychonoff space *X* is not pseudocompact if and only if \(C_{p}(X)\) contains a complemented copy of \(\mathbb {R}^{\mathbb {N}}\). Combining this characterization with Theorem 1, we obtain another characterization related to Problem 1.

### Corollary 1

For a Tychonoff space *X* the following conditions are equivalent:

- (1)
\(C_p(X)\) has an infinite-dimensional Polishable quotient;

- (2)
\(C_p(X)\) contains a complemeneted infinite-dimensional Polishable subspace;

- (3)
\(C_p(X)\) contains a complemented subspace isomorphic to \(\mathbb {R}^{\mathbb {N}}\) or \(c_0\);

### Corollary 2

The space \(C_{p}(\beta \mathbb {N})\)

- (1)
has a quotient isomorphic to \(\ell _{\infty }\);

- (2)
contains a subspace isomorphic to \(c_{0}\);

- (3)
does not admit a continuous linear map onto \(c_0\);

- (4)
has no Polishable infinite-dimensional quotients;

- (4)
contains no complemented separable infinite-dimensional subspaces.

Indeed, the first claim follows from [3, Proposition], the others follow from Theorem 1 and the statement (2) after Definition 1.

In the final Sect. 5 we shall characterize Tychonoff spaces *X* whose function space \(C_p(X)\) is Polishable and prove the following theorem.

### Theorem 2

For a Tychonoff space *X* the following conditions are equivalent:

- (1)
\(C_p(X)\) is Polishable;

- (2)
\(C_k(X)\) is Polishable;

- (3)
\(C_k(X)\) is Polish;

- (4)
*X*is a submetrizable hemicompact*k*-space.

In this theorem \(C_k(X)\) denotes the space of continuous real-valued functions on *X*, endowed with the compact-open topology. It should be mentioned that a locally convex space is Polish if and only if it is a separable Fréchet space, by using, for example, the Birkhoff–Kakutani theorem [20, Theorem 9.1].

## Proof of Theorem 1

We start with the following

### Lemma 1

Let a Tychonoff space *X* be continuously embedded into a compact Hausdorff space *K* (i.e. there exists a continuous injection from *X* into *K*.) Let \((\mu _n)\) be a sequence of finitely supported sign-measures on *X* (and so, on *K*) such that

- (1)
\(\Vert \mu _n\Vert =1\) for all \(n\in \mathbb {N}\), and

- (2)
\(\mu _n(f)\rightarrow _n 0\) for all \(f\in C(K)\).

Then there exists an infinite subset \(\Omega \) of \(\mathbb {N}\) such that

- (a)
the closed subspace \(Z=\bigcap _{k\in \Omega }\{f\in C_p(X):\mu _k(f)=0\}\) of \(C_p(X)\) is complemented in the subspace \(L=\big \{f\in C_p(X):\lim _{k\in \Omega }\mu _k(f)=0\big \}\) of \(C_p(X)\);

- (b)
the quotient space

*L*/*Z*is isomorphic to the subspace \(c_0\) of \(\mathbb {R}^\mathbb {N}\); - (c)
*L*contains a complemented subspace isomorphic to \(c_0\); - (d)
the quotient space \(C_p(X)/Z\) is infinite-dimensional and metrizable (and so, separable).

### Proof

- (I)
First we show that the set \(M=\{\mu _n: n\in \mathbb {N}\}\) is not relatively weakly compact in the dual of the Banach space

*C*(*K*). Indeed, assume on the contrary that the closure \(\overline{M}\) of*M*in the weak topology of \(C(K)^{*}\) is weakly compact. Applying the Eberlein–Šmulian theorem [1, Theorem 1.6.3], we conclude that \(\overline{M}\) is weakly sequentially compact. Thus \((\mu _n)\) has a subsequence \((\mu _{k_n})\) that weakly converges to some element \(\mu _0\in C(K)^{*}\). Taking into account that the sequence \((\mu _n)\) converges to zero in the weak\(^{*}\) topology of \(C(K)^{*}\), we conclude that \(\mu _0=0\) and hence \((\mu _{k_n})\) is weakly convergent to zero in \(C(K)^{*}\). Denote by*W*the countable set \(\bigcup _{n\in \mathbb {N}}\mathrm {supp}(\mu _n)\). The measures \(\mu _n, n\in \mathbb {N},\) can be considered as elements of the unit sphere of the Banach space \(\ell _1(W)\subset C(K)^{*}\). By the Schur theorem [1, Theorem 2.3.6], the weakly convergent sequence \((\mu _{k_n})\) is convergent to zero in the norm topology of \(\ell _1(W)\), which is not possible as \(\Vert \mu _n\Vert =1\) for all \(n\in \mathbb {N}\). Thus the set*M*is not relatively weakly compact in \(C(K)^{*}\). - (II)
By the Grothendieck theorem [1, Theorem 5.3.2] there exist a number \(\epsilon >0\), a sequence \((m_n) \subset \mathbb {N}\) and a sequence \((U_n)\) of pairwise disjoint open sets in

*K*such that \(|\mu _{m_n}(U_n)|>\epsilon \) for any \(n\in \mathbb {N}\). Clearly, \(\lim _{n\rightarrow \infty }\mu _k(U_n)=0\) for any \(k\in \mathbb {N}\), since$$\begin{aligned} \sum _{n\in \mathbb {N}} |\mu _k|(U_n) = |\mu _k|\left( \bigcup _{n\in \mathbb {N}} U_n\right) \le |\mu _k|(K)=1. \end{aligned}$$Thus we can assume that the sequence \((m_n)\) is strictly increasing.

For some strictly increasing sequence \((n_k)\subset \mathbb {N}\) we have \(U_{n_k}\cap \mathrm {supp}(\mu _{m_{n_i}})=\emptyset \) for all \(k,i\in \mathbb {N}\) with \(k>i\).

Put \(\nu _k=\mu _{m_{n_k}}\) and \(W_k=U_{n_k}\) for all \(k\in \mathbb {N}.\) Then

- (A1)
\(\nu _k(f)\rightarrow _k 0\) for every \(f\in C(K);\)

- (A2)
\(|\nu _k(W_k)|>\epsilon \) for every \(k\in \mathbb {N};\)

- (A3)
\(|\nu _k|(W_n)=0\) for all \(k,n\in \mathbb {N}\) with \(k<n.\)

- (III)
By induction we shall construct a decreasing sequence \((N_k)\) of infinite subsets of \(\mathbb {N}\) with \(\min N_k< \min N_{k+1}\) for \(k\in \mathbb {N}\) such that \(|\nu _n|(W_m)\le \epsilon /3^k\) for every \(k\in \mathbb {N}, m=\min N_k, n\in N_k\) and \(n>m\). Let \(N_0=\mathbb {N}.\) Assume that for some \(k\in \mathbb {N}\) an infinite subset \(N_{k-1}\) of \(\mathbb {N}\) has been constructed. Let

*F*be a finite subset of \(N_{k-1}\) with \(|F|>3^k/\epsilon \) and \(\min F> \min N_{k-1}.\) For every \(i\in F\) consider the set$$\begin{aligned} \Lambda _i=\{n\in N_{k-1}: |\nu _n|(W_i) \le \epsilon /3^k\}. \end{aligned}$$For every \(n\in N_{k-1}\) we get \(|\nu _n|(X)\ge \sum _{i\in F} |\nu _n|(W_i).\) Hence there exists \(i\in F\) such that

$$\begin{aligned} |\nu _n|(W_i)\le 1/|F| \le \epsilon /3^k. \end{aligned}$$Thus \(N_{k-1}=\bigcup _{i\in F} \Lambda _i,\) so for some \(m\in F\) the set \(\Lambda _m\) is infinite. Put

$$\begin{aligned} N_k=\{n\in \Lambda _m: n>m\}\cup \{m\}. \end{aligned}$$Then \(\min N_{k-1}< \min F \le m=\min N_k\) and \(|\nu _n|(W_m)\le \epsilon /3^k\) for \(n\in N_k\) with \(n>m.\)

- (IV)
Let \(i_k= \min N_k, \lambda _k=\nu _{i_k}\) and \(V_k=W_{i_k}\) for \(k\in \mathbb {N}.\) Then

- (B1)
\(\lambda _k(f)\rightarrow _k 0\) for every \(f\in C(K);\)

- (B2)
\(|\lambda _k(V_k)|> \epsilon \) for every \(k\in \mathbb {N}\);

- (B3)
\(|\lambda _k|(V_l)=0\) and \(|\lambda _l|(V_k) \le \epsilon /3^k\) for all \(k,l\in \mathbb {N}\) with \(k<l.\)

- (B1)

Clearly, the set

is infinite. Put

and \(L=\{f\in C_p(X): \lambda _n(f)\rightarrow _n 0\}.\) Clearly, *Z* and *L* are subspaces of \(C_p(X)\) and *Z* is closed in *L* and in \(C_p(X)\). The linear operator

is continuous and \(\ker S=Z.\)

We shall construct a linear continuous map \(P:c_0 \rightarrow L\) such that \(S\circ P\) is the identity map on \(c_0\). For every \(k\in \mathbb {N}\) there exists a continuous function \(\varphi _k: K\rightarrow [-1,1]\) such that

for \(s\in V_k \cap \mathrm {supp}(\lambda _k)\) and \(\varphi _k(s)=0\) for \(s\in (K{\setminus } V_k).\) Then

\(\lambda _n(\varphi _k)=0\) for all \(n,k \in \mathbb {N}\) with \(n<k\) and

for all \(n,k \in \mathbb {N}\) with \(n>k.\)

- (V)
Let \((x_n)\in c_0\). Define a sequence \((x'_n)\in \mathbb {R}^{\mathbb {N}}\) by the recursive formula

$$\begin{aligned} x'_n:=\left[ x_n-\sum _{1\le k<n}x'_k \lambda _n(\varphi _k)\right] /\lambda _n(\varphi _n) \ \text{ for } \ n\in \mathbb {N}. \end{aligned}$$

We shall prove that \((x'_n)\in c_0\).

First we show that \(\sup _n |x'_n|<\infty \). Since \((x_n)\in c_0\), there exists \(m\in \mathbb {N}\) such that \(\sup _{n\ge m}|x_n|<\epsilon .\) Put

for \(n\ge 2.\) For every \(n>m\) we get

Hence \(M_{n+1}=\max \{M_n, |x'_n|\}\le M_n\) for \(n>m.\) Thus \(d:=\sup _n |x'_n|\le M_{m+1}<\infty .\)

Now we show that \(x'_n\rightarrow _n 0\). Given any \(\delta >0\), find \(v\in \mathbb {N}\) such that \(d<3^v\delta \). Since \((x_n)\in c_0\) and \(\lambda _n(f)\rightarrow _n 0\) for any \(f\in C(K)\), there exists \(m>v\) such that for every \(n\ge m\)

Then for \(n\ge m\) we obtain

Thus \((x'_n)\in c_0.\)

Clearly, the operator

is linear and continuous. We prove that \(\Theta \) is surjective. Let \((y_n)\in c_0\). Set \(t=\sup _n |y_n|.\) Let

First we show that \((x_n)\in c_0.\) Given any \(\delta >0\), find \(v\in \mathbb {N}\) with \(\epsilon t<\delta 3^v\). Clearly, there exists \(m>v+2\) such that \(|y_n|<\delta \) and \(\sum _{k=1}^v t|\lambda _n(\varphi _k)|<\delta \) for \(n\ge m.\) Then for every \(n\ge m\) we obtain

Thus \((x_n)\in c_0.\) Clearly, \(\Theta ((x_n)_{n\in \mathbb N})=(y_n)_{n\in \mathbb N};\) so \(\Theta \) is surjective.

- (VI)
The operator

$$\begin{aligned} T: c_0 \rightarrow C_p(X),\;\; T:(x_n)\mapsto \sum _{n=1}^{\infty } x_n{\cdot }\varphi _n|X, \end{aligned}$$

is well-defined, linear and continuous, since the functions \(\varphi _n, n\in \mathbb {N},\) have pairwise disjoint supports and \(\varphi _n(X)\subset [-1,1], n\in \mathbb {N}.\) Thus the linear operator

is continuous.

Let \(x=(x_k) \in c_0\) and \(x'=(x'_k)=\Theta (x).\) Then

Using (B3) and the definition of \(\Theta \), we get for every \(n\in \mathbb {N}\)

so \(\lambda _n(\Phi (x))\rightarrow _n 0.\) This implies that \(\Phi (x)\in L\) and \(S\circ \Phi (x)=x\) for every \(x\in c_0.\) Therefore the operator \(P:=\Phi \circ S{:}L\rightarrow L\) is a continuous linear projection with \(\ker P=\ker S=Z.\) Thus the subspace *Z* is complemented in *L*. Since \(S\circ \Phi \) is the identity map on \(c_0\), the map \(S: L\rightarrow c_0\) is open. Indeed, let *U* be a neighbourhood of zero in *L*; then \(V=\Phi ^{-1}(U)\) is a neighbourhood of zero in \(c_0\) and

Thus the quotient space *L* / *Z* is topologically isomorphic to \(c_0\) and \(\Phi (c_0)\) is a complemented subspace of *L*, isomorphic to \(c_0\). In particular, *Z* has infinite codimension in *L* and in \(C_p(X).\)

- (VII)
Finally we prove that the quotient space \(C_p(X)/Z\) is first countable and hence metrizable. Let

$$\begin{aligned} U_n=\{f\in C_p(X): |f(x)|<1/n\;\text{ for } \text{ every }\;x\in \bigcup _{k=1}^n \mathrm {supp}(\lambda _k)\}, n\in \mathbb {N}. \end{aligned}$$The first countability of the quotient space \(C_p(X)/Z\) will follow as soon as for every neighbourhood

*U*of zero in \(C_p(X)\) we find \(n\in \mathbb {N}\) with \(Z+U_n \subset Z+U.\) Clearly we can assume that$$\begin{aligned} U=\bigcap _{x\in F}\{f\in C_p(X): |f(x)|<\delta \} \end{aligned}$$for some finite subset

*F*of*X*and some \(\delta >0.\)

By the continuity of the operator \(\Phi : c_0 \rightarrow C_p(X),\) there exists \(n\in \mathbb {N}\) such that for any \(y=(y_k)\in c_0\) with

we get \(\Phi (y)\in \frac{1}{2}U\). Replacing *n* by a larger number, we can assume that \(\frac{1}{n}<\frac{1}{2}\delta \) and

Let \(f\in U_n.\) Choose a function \(h\in C_p(K)\) such that \(h(x)=f(x)\) for every

and \(h(x)=0\) for every \(x\in \bigcup _{k=1}^n \mathrm {supp}(\lambda _k)\). Put \(g=h|X.\) Then \(g\in L\), since \(\lambda _k (g)=\lambda _k (h)\rightarrow _k 0\). Put \(y=S(g)\) and \(\xi =\Phi (y).\) Since \(g(x)=0\) for

we have \(|\lambda _k(g)|=0<\frac{1}{n}\) for \(1\le k \le n\), so \(\max _{1\le k \le n} |y_k|< \frac{1}{n}\). Hence \(\xi =\Phi (y) \in \frac{1}{2}U,\) so \(\max _{x\in F} |\xi (x)|<\frac{1}{2}\delta .\) For \(\varsigma =g-\xi \) we obtain

so \(\varsigma \in Z.\) Moreover \(f-\varsigma \in U.\) Indeed, we have

for \(x\in F{\setminus }\bigcup _{k=1}^{\infty }\mathrm {supp}(\lambda _k)\) and

for every \(x\in \bigcup _{k=1}^n \mathrm {supp}(\lambda _k)\). Thus \(f=\varsigma + (f-\varsigma ) \in Z+U,\) so \(U_n \subset Z+U.\) Hence \(Z+U_n \subset Z+U\). \(\square \)

### Lemma 2

Let *X* be a Tychonoff space. Each metrizable continuous image of \(C_p(X)\) is separable.

### Proof

It is well-known [11, 2.3.18] that the Tychonoff product \(\mathbb {R}^{X}\) has *countable cellularity*, which means that \(\mathbb {R}^{X}\) contains no uncountable family of pairwise disjoint non-empty open sets. Then the dense subspace \(C_p(X)\) of \(\mathbb {R}^{X}\) also has countable cellularity and so does any continuous image *Y* of \(C_p(X)\). If *Y* is metrizable, then *Y* is separable according to Theorem 4.1.15 in [11]. \(\square \)

### Lemma 3

Let *X* be a pseudocompact space. A closed linear subspace *S* of \(C_p(X)\) is separable if and only if *S* is Polishable.

### Proof

If *S* is Polishable, then *S* is separable, being a continuous image of a separable Fréchet locally convex space. Now assume that *S* is separable. Fix a countable dense subset \(\{f_n\}_{n\in \mathbb N}\) in *S* and consider the continuous map

By the pseudocompactness of *X* and the metrizability of \(\mathbb {R}^{\mathbb {N}}\), the image \(M:=f(X)\) is a compact metrizable space. The continuous surjective map \(f:X\rightarrow M\) induces an isomorphic embedding

So, we can identify the space \(C_p(M)\) with its image \(C_pf(C_p(M))\) in \(C_p(X)\). We claim that \(C_p(M)\) is closed in \(C_p(X)\). Given any function \(\varphi \in C_p(X){\setminus }C_p(M)\), we should find a neighborhood \(O_\varphi \subset C_p(X)\) of \(\varphi \), which is disjoint with \(C_p(M)\).

We claim that there exist points \(x,y\in X\) such that \(f(x)=f(y)\) and \(\varphi (x)\ne \varphi (y)\). In the opposite case, \(\varphi =\psi \circ f\) for some bounded function \(\psi :M\rightarrow \mathbb {R}\). Let us show that the function \(\psi \) is continuous. Consider the continuous map

The pseudocompactness of *X* implies that the image \(h(X)\subset M\times \mathbb {R}\) is a compact closed subset of \(M\times \mathbb {R}\). Let \(\mathrm {pr}_M:h(X)\rightarrow M\) and \(\mathrm {pr}_\mathbb {R}:h(X)\rightarrow \mathbb {R}\) be the coordinate projections. It follows that

which implies that \(\mathrm {pr}_\mathbb {R}=\psi \circ \mathrm {pr}_M\). The map \(\mathrm {pr}_M:h(X)\rightarrow M\) between the compact metrizable spaces *h*(*X*) and *M* is closed and hence is quotient. Then the continuity of the map \(\mathrm {pr}_\mathbb {R}=\psi \circ \mathrm {pr}_M\) implies the continuity of \(\psi \). Now we see that the function \(\varphi =\psi \circ f\) belongs to the subspace \(C_p(M)\subset C_p(X)\), which contradicts the choice of \(\varphi \). This contradiction shows that \(\varphi (x)\ne \varphi (y)\) for some points \(x,y\in X\) with \(f(x)=f(y)\). Then

is a required neighborhood of \(\varphi \), disjoint with \(C_p(M)\).

Therefore the subspace \(C_p(M)\) of \(C_p(X)\) is closed and hence \(C_p(M)\) contains the closure *S* of the dense set \(\{f_n\}_{n\in \mathbb N}\) in *S*. Since the space \(C_p(M)\) is Polishable, so is its closed subspace *S*. \(\square \)

Now we are at the position to prove the main Theorem 1:

### Proof of Theorem 1

First, for a Tychonoff space *X* we prove the equivalence of the conditions:

- (1)
\(C_{p}(X)\) satisfies the JNP;

- (2)
\(C_p(X)\) contains a complemented subspace isomorphic to \(c_0\);

- (3)
\(C_{p}(X)\) has a quotient isomorphic to \(c_0\);

- (4)
\(C_p(X)\) admits a continuous linear map onto \(c_0.\)

The implication \((1)\Rightarrow (2)\) follows from Lemma 1, applied to the Stone-Čech compactification \(K=\beta X\) of *X*. The implications \((2)\Rightarrow (3)\Rightarrow (4)\) are trivial.

To prove the implication \((4)\Rightarrow (1)\) assume that there exists a continuous linear map *T* from \(C_p(X)\) onto \(c_0\). Let

be the sequence of coordinate functionals. By definition of \(c_0\), \(e^{*}_n(y)\rightarrow _n 0\) for every \(y\in c_0\). For every \(n\in \mathbb {N}\) consider the linear continuous functional

which can be thought as a finitely supported sign-measure on *X*. It follows that \(\lambda _n(f)=e^{*}_n(Tf)\rightarrow _n0\) for every \(f\in C_p(X)\).

We shall show that the union \(S=\bigcup _{n=1}^{\infty } S_n\) of supports \(S_n=\mathrm {supp}(\lambda _n)\) of the sign-measures \(\lambda _n\) is bounded in *X* in the sense that for any \(\varphi \in C_p(X)\) the image \(\varphi (S)\) is bounded in \(\mathbb {R}\), since in the opposite case we get a function \(\psi \in C_p(X)\) with \(\lambda _n (\psi )\not \rightarrow 0.\) Indeed, suppose that for some \(\varphi \in C_p(X)\) the image \(\varphi (S)\) is unbounded in \(\mathbb {R}\); without loss of generality we can assume that \(\varphi \) is non-negative.

We can find inductively an increasing sequence \(\{n_k\}_{k\in \mathbb {N}} \subset \mathbb {N}\) such that

for all \(k,i \in \mathbb {N}\) with \(k>i\). For every \(k\in \mathbb {N}\) choose \(x_k\in S_{n_k}\) with \(\varphi (x_k)= \max \varphi (S_{n_k})\). Then \(\varphi (x_k) - \varphi (x_i)>3\) for all \(k,i \in \mathbb {N}\) with \(k>i\). Since the space *X* is Tychonoff, for every \(k\in \mathbb {N}\) we can find an open neighborhood \(U_k\subset \{x\in X:|\varphi (x)-\varphi (x_k)|<1\}\) of \(x_k\) such that \(U_k\cap \mathrm {supp}(\lambda _{n_k})=\{x_k\}\).

Observe that for any \(k>i\) and any points \(x\in U_k\) and \(y\in U_i\cup \mathrm {supp}(\lambda _{n_i})\) we get

which implies \(U_k\cap \mathrm {supp}(\lambda _{n_i})=\emptyset \) and that the family \(\big (\varphi (U_k)\big )_{k\in \mathbb {N}}\) is discrete in \(\mathbb {R}\) and consequently \((U_k)_{k\in \mathbb {N}}\) is discrete in *X*.

Taking into account that \(U_k\cap \mathrm {supp}(\lambda _{n_k})=\{x_k\}\), we can inductively construct a sequence \((\psi _k)_{k\in \mathbb {N}}\) of functions \(\psi _k:X\rightarrow \mathbb {R}\) such that \(\mathrm {supp}(\psi _k):=\{x\in X:\psi _i(x)\ne 0\}\subset U_k\) and

for all \(k\in \mathbb {N}\).

Since the family \((U_k)_{k\in \mathbb {N}}\) is discrete, the function

is well-defined and continuous. For every \(k\in \mathbb {N}\) and \(i>k\) we have \(U_i\cap \mathrm {supp}(\lambda _{n_k})=\emptyset \) and hence

But this contradicts \(\lambda _n(\psi )\rightarrow _n 0\). This contradiction shows that the set \(S=\bigcup _{k\in \mathbb {N}}\mathrm {supp}(\lambda _k)\) is bounded in *X*.

Next, we show that the set \(B=\{(x_n)\in c_0:\forall n\in \mathbb {N}\;\; |x_n|\le \Vert \lambda _n\Vert \}\) is absorbing. Given any sequence \((x_n) \in c_0\), find \(f\in C_p(X)\) with \(Tf=(x_n)\). Put \(\Vert f\Vert _S:= \sup \{|f(x)|: x\in S\}\) and observe that \(|x_n|=|e^{*}_n(Tf)|=|\lambda _n (f)| \le \Vert \lambda _n\Vert {\cdot }\Vert f\Vert _S\), so \((x_n) \in \Vert f\Vert _S\cdot B\). Thus *B* is absorbing. Clearly, *B* is absolutely convex and closed in \(c_0\), so it is a barrel in the Banach space \((c_0, \Vert \cdot \Vert _{\infty })\). Thus *B* is a neighbourhood of zero in \((c_0, \Vert \cdot \Vert _{\infty })\) and consequently, \(\inf _n \Vert \lambda _n\Vert >0\). For every \(n\in \mathbb {N}\) put

and observe that the sequence \((\mu _n)\) witnesses that the function space \(C_p(X)\) has the JNP.

Now assuming that the space *X* is pseudocompact, we shall prove that the conditions (1)–(4) are equivalent to

- (5)
\(C_p(X)\) contains a complemented infinite-dimensional metrizable subspace;

- (6)
\(C_{p}(X)\) contains a complemented infinite-dimensional separable subspace;

- (7)
\(C_p(X)\) has an infinite-dimensional Polishable quotient.

It suffices to prove the implications \((2)\Rightarrow (5)\Rightarrow (6)\Rightarrow (7)\Rightarrow (1)\). The implication \((2)\Rightarrow (5)\) is trivial and \((5)\Rightarrow (6)\Rightarrow (7)\) follow from Lemmas 2 and 3, respectively.

\((7) \Rightarrow (1)\) Assume that the space \(C_p(X)\) contains a closed subspace *Z* of infinite codimension such that the quotient space \(E:=C_p(X)/Z\) is Polishable. Denote by \(\tau _p\) the quotient topology of \(C_p(X)/Z\) and by \(\tau _0\supset \tau _p\) a stronger separable Fréchet locally convex topology on *E*. Denote by \(\tau _\infty \) the topology of the quotient Banach space *C*(*X*) / *Z*. Here *C*(*X*) is endowed with the sup-norm \(\Vert f\Vert _\infty :=\sup _{x\in X}|f(x)|\) (which is well-defined as *X* is pseudocompact).

The identity maps between \((E, \tau _0)\) and \((E, \tau _{\infty })\) have closed graphs, since \(\tau _p \subset \tau _0 \cap \tau _{\infty }.\) Using the Closed Graph Theorem we infer that the topologies \(\tau _0\) and \(\tau _{\infty }\) are equal. Let *G* be a countable subset of *C*(*X*) such that the set \(\{g+Z: g\in G\}\) is dense in the Banach space *C*(*X*) / *Z*. Then the set

is dense in *C*(*X*). Let \((g_n)_{n\in \mathbb N}\) be a linearly independent sequence in *G* such that its linear span \(G_0\) has \(G_0\cap Z=\{0\}\) and \(G_0+Z=G+Z\). Let \(f_1=g_1\) and \(\nu _1 \in C_p(X)^{*}\) with \(\nu _1|Z=0\) such that \(\nu _1(f_1)=1.\)

Assume that for some \(n\in \mathbb {N}\) we have chosen

such that

and

Put

Then

and \(\nu _j(f_{n+1})=0\) for \(1\le j \le n.\) Let \(\nu _{n+1}\in C_p(X)^{*}\) with \(\nu _{n+1}|Z=0\) such that \(\nu _{n+1}(f_i)=0\) for \(1\le i \le n\) and \(\nu _{n+1}(f_{n+1})=1.\)

Continuing on this way we can construct inductively a biorthogonal sequence \(((f_n, \nu _n))_{n\in \mathbb N}\) in \(C_p(X)\times C_p(X)^{*}\) such that \(\text{ lin } \{f_n: n\in \mathbb {N}\}= \text{ lin } \{g_n: n\in \mathbb {N}\}\) and \(\nu _n|Z=0\), \(\nu _n(f_m)=\delta _{n,m}\) for all \(n,m \in \mathbb {N}.\) Then \(\text{ lin } \{f_n: n\in \mathbb {N}\} +Z\) is dense in \((C(X), \Vert .\Vert _{\infty })\). Let \(\mu _n=\nu _n/\Vert \nu _n\Vert \) for \( n\in \mathbb {N}.\) Then \(\Vert \mu _n\Vert =1\) and \(\mu _n(f_m)=0\) for all \(n,m \in \mathbb {N}\) with \(n\ne m.\)

We prove that \(\mu _n(f)\rightarrow _n 0\) for every \(f\in C_p(X).\) Given any \(f\in C(X)\) and \(\varepsilon >0\), find \(m\in \mathbb {N}\) and \(g\in \text{ lin }\{f_1, \ldots , f_m\} +Z\) with \(d(f,g)<\varepsilon ;\) clearly \(d(f,g)=\Vert f-g\Vert _{\infty }.\) Then \(\mu _n(g)=0\) for \(n>m,\) so

for \(n>m.\) Thus \(\mu _n(f)\rightarrow _n 0\), which means that the space \(C_p(X)\) has the JNP. \(\square \)

## An example of Plebanek

In this section we describe the following example suggested to the authors by Grzegorz Plebanek [26].

### Example 1

(Plebanek) There exists a compact Hausdorff space *K* such that

- (1)
*K*contain no nontrivial converging sequences but contains a copy of \(\beta \mathbb {N}\); - (2)
the function space \(C_p(K)\) has the JNP.

We need some facts to present the construction of the space *K*. By definition, the *asymptotic density* of a subset \(A\subset \mathbb N\) is the limit

if this limit exists. The family \(\mathcal Z=\{A\subset \mathbb N:d(A)=0\}\) of sets of asymptotic density zero in \(\mathbb N\) is an ideal on \(\mathbb N\). Recall the following standard fact (here \(A\subset ^{*} B\) means that \(A{\setminus } B\) is finite).

### Fact 1

For any countable subfamily \(\mathcal C\subset \mathcal {Z}\) there is a set \(B\in \mathcal {Z}\) such that \(C\subset ^{*} B\) for all \(C\in \mathcal C\).

Let \(\mathfrak {A}=\big \{A\subset \mathbb N:d(A)\in \{0,1\}\big \}\) be the algebra of subsets of \(\mathbb {N}\) generated by \(\mathcal {Z}\). We now let *K* be the Stone space of the algebra \(\mathfrak {A}\) so we treat elements of *K* as ultrafilters on \(\mathfrak {A}\). There are three types of such \(x\in K\):

- (1)
\(\{n\} \in x\) for some \(n\in \mathbb {N}\); then \(x=\{A\in \mathfrak {A}: n\in A\}\) is identified with

*n*; - (2)
*x*contains no finite subsets of \(\mathbb {N}\) but \(Z\in x\) for some \(Z\in \mathcal {Z}\); - (3)
\(Z\notin x\) for every \(Z\in \mathcal {Z}\); this defines the unique

$$\begin{aligned} p=\{A\in \mathfrak {A}:d(A)=1\}\in K. \end{aligned}$$

To see that *K* is the required space it is enough to check the following two facts.

### Fact 2

The space *K* contains no nontrivial converging sequence.

### Proof

In fact we check that every infinite \(X\subset K\) contains an infinite set *Y* such that \(\overline{Y}\) is homeomorphic to \(\beta \mathbb {N}\). Note first that for every \(Z\in \mathcal {Z}\), the corresponding clopen set

is homeomorphic to \(\beta \mathbb {N}\) because \(\{A\in \mathfrak {A}: A\subset Z\}= 2^{Z}\).

For an infinite set \(X\subset K\), we have two cases:

*Case 1*, \(X\cap \mathbb {N}\) is infinite. There is an infinite \(Z\subset X\cap \mathbb {N}\) having density zero. Then every subset of *Z* is in \(\mathfrak {A}\), which implies that \(\overline{Z}\cong \beta \mathbb {N}\) .

*Case 2*, \(X\cap ({K{\setminus }\mathbb {N}})\) is infinite. Let us fix a sequence of different \(x_n \in X\cap (K{\setminus }\mathbb {N})\) such that \(x_n\ne p\) for every *n*. Then for every *n* we have \(Z_n\in x_n\) for some \(Z_n\in \mathcal {Z}\). Take \(B\in \mathcal {Z}\) as in Fact 1. Then \(B\in x_n\) because \(x_n\) is a nonprincipial ultrafilter on \(\mathfrak A\) so \(A_n{\setminus }B\notin x_n\). Again, we conclude that \(\overline{\{x_n: n\in \mathbb {N}\}}\) is \(\beta \mathbb {N}\). \(\square \)

### Fact 3

If \(\nu _n=\frac{1}{n}\sum _{k\le n} \delta _k\) and \(\mu _n=\frac{1}{2}(\nu _n - \delta _p)\) for \(n \in \mathbb {N}\), then \(\nu _n(f)\rightarrow _n \delta _p(f)\) and \(\mu _n(f)\rightarrow _n 0\) for every \(f\in C(K)\).

### Proof

Observe \(\nu _n(A)\rightarrow _n d(A)\) for every \(A\in \mathfrak {A}\) since elements of \(\mathfrak {A}\) have asymptotic density either 0 or 1. This means that, when we treat \(\nu _n\) as measures on *K* then \(\nu _n(V)\) converges to \(\delta _p(V)\) for every clopen set \(V\subset K\). This implies the assertion since every continuous function on *K* can be uniformly approximated by simple functions built from clopens. \(\square \)

## Proof of Theorem 2

Let us recall that a topological space *X* is called

*submetrizable*if*X*admits a continuous metric;*hemicompact*if*X*has a countable family \(\mathcal K\) of compact sets such that each compact subset of*X*is contained in some compact set \(K\in \mathcal {K}\);a

*k-space*if a subset \(F\subset X\) is closed if and only if for every compact subset \(K\subset X\) the intersection \(F\cap K\) is closed in*K*.

In order to prove Theorem 2, we should check the equivalence of the following conditions for every Tychonoff space *X*:

- (1)
\(C_k(X)\) is Polishable;

- (2)
\(C_p(X)\) is Polishable;

- (3)
\(C_k(X)\) is Polish;

- (4)
*X*is a submetrizable hemicompact*k*-space.

The implication \((1)\Rightarrow (2)\) follows from the continuity of the identity map \(C_k(X)\rightarrow C_p(X)\).

\((2)\Rightarrow (3)\) Assume that the space \(C_p(X)\) is Polishable and fix a stronger Polish locally convex topology \(\tau \) on \(C_p(X)\). Let \(C_\tau (X)\) denote the separable Fréchet space \((C_p(X),\tau )\). By \(\tau _{k}\) denote the compact open topology of \(C_{k}(X)\). Taking into account that the space \(C_p(X)\) is a continuous image of the Polish space \(C_\tau (X)\), we conclude that \(C_p(X)\) has countable network and by [2, I.1.3], the space *X* has countable network and hence is Lindelöf. By the normality (and the Lindelöf property) of *X*, each closed bounded set in *X* is countably compact (and hence compact). So *X* is a \(\mu \)-space. By Theorem 10.1.20 in [25, Theorem 10.1.20] the function space \(C_{k}(X)\) is barrelled. The continuity of the identity maps \(C_k(X)\rightarrow C_p(X)\) and \(C_\tau (X)\rightarrow C_p(X)\) implies that the identity map \(C_k(X)\rightarrow C_\tau (X)\) has closed graph. Since \(C_k(X)\) is barelled and \(C_\tau (X)\) is Fréchet, we can apply the Closed Graph Theorem 4.1.10 in [25] and conclude that the identity map \(C_k(X)\rightarrow C_\tau (X)\) is continuous.

Next, we show that the identity map \(C_\tau (X)\rightarrow C_k(X)\) is continuous. Given any compact set \(K\subset X\) and any \(\varepsilon >0\) we have to find a neighborhood \(U\subset C_\tau (X)\) of zero such that

The continuity of the restriction operator \(R:C_p(X)\rightarrow C_p(K)\), \(R:f\mapsto f{\upharpoonright }K\), and the continuity of the idenity map \(C_\tau (X)\rightarrow C_p(X)\) imply that the restriction operator \(R:C_\tau (X)\rightarrow C_p(K)\) is continuous and hence has closed graph. The continuity of the identity map \(C_k(K)\rightarrow C_p(K)\) implies that *R* seen as an operator \(R:C_\tau (X)\rightarrow C_k(K)\) still has closed graph. Since the spaces \(C_\tau (X)\) and \(C_k(K)\) are Fréchet, the Closed Graph Theorem 1.2.19 in [25] implies that the restriction operator \(R:C_\tau (X)\rightarrow C_k(K)\) is continuous. So, there exists a neighborhood \(U\subset C_\tau (X)\) of zero such that

Then \(U\subset \{f\in C(X):f(K)\subset (-\varepsilon ,\varepsilon )\}\) and we are done. Hence \(\tau =\tau _{k}\) is a Polish locally convex topology as claimed.

The implication \((3)\Rightarrow (1)\) is trivial.

\((3)\Rightarrow (4)\) If the function space \(C_k(X)\) is Polish, then by Theorem 4.2 in [24], *X* is a hemicompact *k*-space. Taking into account that the space \(C_p(X)\) is a continuous image of the space \(C_k(X)\), we conclude that \(C_p(X)\) has countable network and by [2, I.1.3], the space *X* has countable network. By [17, 2.9], the space *X* is submetrizable.

\((4)\Rightarrow (3)\) If *X* is a submetrizable hemicompact *k*-space, then \(X=\bigcup _{n\in \omega }X_n\) for some increasing sequence \((X_n)_{n\in \omega }\) of compact metrizable spaces such that each compact subset of *X* is contained in some compact set \(X_n\). Then the function space \(C_k(X)\) is Polish, being topologically isomorphic to the closed subspace

of the countable product \(\prod _{n=1}^\infty C_k(X_n)\) of separable Banach spaces. \(\square \)

The authors thank to the referee for his/her valuable comments and remarks.

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Banakh, T., Ka̧kol, J. & Śliwa, W. Josefson–Nissenzweig property for \(C_{p}\)-spaces.
*RACSAM* **113, **3015–3030 (2019). https://doi.org/10.1007/s13398-019-00667-8

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### Keywords

- The separable quotient problem
- Spaces of continuous functions
- Quotient spaces
- The Josefson–Nissenzweig theorem
- Efimov space

### Mathematics Subject Classification

- 46E10
- 54C35