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Josefson–Nissenzweig property for \(C_{p}\)-spaces

  • T. Banakh
  • J. Ka̧kol
  • W. ŚliwaEmail author
Open Access
Original Paper
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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.

Keywords

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

Mathematics Subject Classification

46E10 54C35 

1 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. (1)

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

     
  2. (2)

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

     
  3. (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 \(\mathbb {R}^{\mathbb {N}}\).

2 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. (1)

    If a compact spaceKcontains 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. (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. (3)

    There exists a compact spaceKcontaining 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. (1)

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

     
  2. (2)

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

     
  3. (3)

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

     
  4. (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. (5)

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

     
  6. (6)

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

     
  7. (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. (1)

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

     
  2. (2)

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

     
  3. (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. (1)

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

     
  2. (2)

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

     
  3. (3)

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

     
  4. (4)

    has no Polishable infinite-dimensional quotients;

     
  5. (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. (1)

    \(C_p(X)\) is Polishable;

     
  2. (2)

    \(C_k(X)\) is Polishable;

     
  3. (3)

    \(C_k(X)\) is Polish;

     
  4. (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].

3 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. (1)

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

     
  2. (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
  1. (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 \(C_p(X)\);

     
  2. (b)

    the quotient space L / Z is isomorphic to the subspace \(c_0\) of \(\mathbb {R}^\mathbb {N}\);

     
  3. (c)

    L contains a complemented subspace isomorphic to \(c_0\);

     
  4. (d)

    the quotient space \(C_p(X)/Z\) is infinite-dimensional and metrizable (and so, separable).

     

Proof

  1. (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 \(C(K)^{*}\).

     
  2. (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
  1. (A1)

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

     
  2. (A2)

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

     
  3. (A3)

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

     
  1. (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 \(n>m.\)
     
  2. (IV)
    Let \(i_k= \min N_k, \lambda _k=\nu _{i_k}\) and \(V_k=W_{i_k}\) for \(k\in \mathbb {N}.\) Then
    1. (B1)

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

       
    2. (B2)

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

       
    3. (B3)

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

       
     
Clearly, the set
$$\begin{aligned} \Omega =\{n\in \mathbb {N}:\mu _n=\lambda _k\; \text{ for } \text{ some }\; k\in \mathbb {N}\} \end{aligned}$$
is infinite. Put
$$\begin{aligned} Z=\bigcap _{n\in \mathbb {N}} \{f\in C_p(X): \lambda _n(f)=0\} \end{aligned}$$
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
$$\begin{aligned} S: L \rightarrow c_0,\; S:f\mapsto (\lambda _n(f))_{n}, \end{aligned}$$
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
$$\begin{aligned} \varphi _k(s)=\lambda _k(V_k)/|\lambda _k(V_k)| \end{aligned}$$
for \(s\in V_k \cap \mathrm {supp}(\lambda _k)\) and \(\varphi _k(s)=0\) for \(s\in (K{\setminus } V_k).\) Then
$$\begin{aligned} \lambda _k(\varphi _k)=|\lambda _k(V_k)|>\epsilon , \end{aligned}$$
\(\lambda _n(\varphi _k)=0\) for all \(n,k \in \mathbb {N}\) with \(n<k\) and
$$\begin{aligned} |\lambda _n(\varphi _k)| \le |\lambda _n|(V_k)\le \epsilon / 3^k \end{aligned}$$
for all \(n,k \in \mathbb {N}\) with \(n>k.\)
  1. (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
$$\begin{aligned} M_n=\max \left\{ 2, \max _{1\le k<n} |x'_k|\right\} \end{aligned}$$
for \(n\ge 2.\) For every \(n>m\) we get
$$\begin{aligned} |x'_n|=|x_n-\sum _{1\le k<n}x'_k\lambda _n(\varphi _k)|/\lambda _n(\varphi _n)\le \left[ \epsilon +M_n\sum _{1\le k<n} \epsilon /3^k\right] /\epsilon \le 1+M_n/2\le M_n. \end{aligned}$$
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 \(n\ge m\)
$$\begin{aligned} |x_n|<\delta \epsilon \ \text{ and } \ d\sum _{1\le k\le v} |\lambda _n(\varphi _k)|<\delta \epsilon . \end{aligned}$$
Then for \(n\ge m\) we obtain
$$\begin{aligned} |x'_n|\le & {} \bigg [|x_n|+ \sum _{1\le k\le v} |x'_k|{\cdot }|\lambda _n(\varphi _k)| + \sum _{v< k<n} |x'_k|{\cdot }|\lambda _n(\varphi _k)|\bigg ]/\lambda _n(\varphi _n)\\\le & {} \bigg [\delta \epsilon +\sum _{1\le k\le v} d{\cdot }|\lambda _n(\varphi _k)| + \sum _{v< k<n} d{\cdot }|\lambda _n(\varphi _k)|\bigg ]/\epsilon<\delta + \delta \\&+\sum _{v< k<n} d/3^k< 2\delta +d/3^v<3\delta . \end{aligned}$$
Thus \((x'_n)\in c_0.\)
Clearly, the operator
$$\begin{aligned} \Theta : c_0 \rightarrow c_0,\;\;\Theta :(x_n) \mapsto (x'_n), \end{aligned}$$
is linear and continuous. We prove that \(\Theta \) is surjective. Let \((y_n)\in c_0\). Set \(t=\sup _n |y_n|.\) Let
$$\begin{aligned} x_n=\sum _{k=1}^n \lambda _n (\varphi _k) y_k\ \text{ for }\ n\in \mathbb {N}. \end{aligned}$$
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
$$\begin{aligned} |x_n|\le & {} \sum _{k=1}^n |\lambda _n(\varphi _k)|{\cdot }|y_k| \le \sum _{k=1}^v t{\cdot }|\lambda _n(\varphi _k)| + \sum _{v<k<n} t{\cdot }|\lambda _n(\varphi _k)| + |\lambda _n(\varphi _n)|{\cdot }|y_n|\\< & {} \delta + \sum _{v<k<n} t\epsilon /3^k + ||\lambda _n||{\cdot }|y_n|< \delta +t\epsilon /3^v +\delta <3\delta . \end{aligned}$$
Thus \((x_n)\in c_0.\) Clearly, \(\Theta ((x_n)_{n\in \mathbb N})=(y_n)_{n\in \mathbb N};\) so \(\Theta \) is surjective.
  1. (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
$$\begin{aligned} \Phi =T\circ \Theta : c_0 \rightarrow C_p(X) \end{aligned}$$
is continuous.
Let \(x=(x_k) \in c_0\) and \(x'=(x'_k)=\Theta (x).\) Then
$$\begin{aligned} \Phi (x)=T(x')=\sum _{k=1}^{\infty } x'_k \varphi _k|X. \end{aligned}$$
Using (B3) and the definition of \(\Theta \), we get for every \(n\in \mathbb {N}\)
$$\begin{aligned} \lambda _n(\Phi (x))= & {} \sum _{k=1}^{\infty } x'_k\lambda _n(\varphi _k)=x'_n\lambda _n(\varphi _n)+ \sum _{1\le k<n} x'_k \lambda _n(\varphi _k)\\= & {} \left( x_n-\sum _{1\le k<n} x'_k \lambda _n(\varphi _k)\right) +\sum _{1\le k<n} x'_k \lambda _n(\varphi _k)=x_n; \end{aligned}$$
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
$$\begin{aligned} V=S\circ \Phi (V) \subset S(U). \end{aligned}$$
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 \(C_p(X).\)
  1. (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
$$\begin{aligned} \max _{1\le k \le n}|y_k| \le 1/n \end{aligned}$$
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
$$\begin{aligned} F\cap \bigcup _{k=1}^{\infty }\mathrm {supp}(\lambda _k) \subset \bigcup _{k=1}^n \mathrm {supp}(\lambda _k). \end{aligned}$$
Let \(f\in U_n.\) Choose a function \(h\in C_p(K)\) such that \(h(x)=f(x)\) for every
$$\begin{aligned} x\in F{\setminus }\bigcup _{k=1}^{\infty }\mathrm {supp}(\lambda _k) \end{aligned}$$
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
$$\begin{aligned} x\in \bigcup _{k=1}^n \mathrm {supp}\lambda _k, \end{aligned}$$
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
$$\begin{aligned} S(\varsigma )=S(g)-S\circ \Phi \circ S (g)= S(g)-S(g)=0, \end{aligned}$$
so \(\varsigma \in Z.\) Moreover \(f-\varsigma \in U.\) Indeed, we have
$$\begin{aligned} |f(x)-\varsigma (x)|=|f(x)-g(x)+\xi (x)|=|\xi (x)|<\delta \end{aligned}$$
for \(x\in F{\setminus }\bigcup _{k=1}^{\infty }\mathrm {supp}(\lambda _k)\) and
$$\begin{aligned} |f(x)-\varsigma (x)|=|f(x)-g(x)+\xi (x)|\le |f(x)|+|g(x)|+|\xi (x)|<\tfrac{1}{n}+0+\tfrac{1}{2}\delta <\delta \end{aligned}$$
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
$$\begin{aligned} f:X\rightarrow \mathbb {R}^\mathbb {N},\;\;f:x\mapsto (f_n(x))_{n\in \mathbb {N}}. \end{aligned}$$
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
$$\begin{aligned} C_pf:C_p(M)\rightarrow C_p(X),\;\;C_pf:\varphi \mapsto \varphi \circ f. \end{aligned}$$
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
$$\begin{aligned} h:X\rightarrow M\times \mathbb {R},\,\,\, h:x\mapsto (f(x),\varphi (x)). \end{aligned}$$
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
$$\begin{aligned} \mathrm {pr}_\mathbb {R}\circ h=\varphi =\psi \circ f=\psi \circ \mathrm {pr}_M\circ h, \end{aligned}$$
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
$$\begin{aligned} O_\varphi :=\{\phi \in C_p(X):\phi (x)\ne \phi (y)\} \end{aligned}$$
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. (1)

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

     
  2. (2)

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

     
  3. (3)

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

     
  4. (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
$$\begin{aligned} \{e^{*}_n\}_{n\in \mathbb {N}}\subset c_0^{*} \end{aligned}$$
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
$$\begin{aligned} \lambda _n\in C_p(X)^{*},\;\;\lambda _n(f)= e^{*}_n(Tf), \end{aligned}$$
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
$$\begin{aligned} \max \varphi (S_{n_k}) > 3+ \max \varphi (S_{n_i}) \end{aligned}$$
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
$$\begin{aligned} \varphi (x)-\varphi (y)\ge \varphi (x_k)-1-(\max \varphi (S_{n_i})+1)>3-2=1, \end{aligned}$$
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
$$\begin{aligned} \lambda _{n_k}(\psi _k)>1+\sum _{i<k}|\lambda _{n_k}(\psi _i)| \end{aligned}$$
for all \(k\in \mathbb {N}\).
Since the family \((U_k)_{k\in \mathbb {N}}\) is discrete, the function
$$\begin{aligned} \psi :X\rightarrow \mathbb {R}, \,\,\,\psi :x\mapsto \sum _{k=1}^\infty \psi _k(x) \end{aligned}$$
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
$$\begin{aligned} |\lambda _{n_k}(\psi )|=\big |\sum _{i\le k}\lambda _{n_k}(\psi _i)\big |\ge |\lambda _{n_k}(\psi _k)|-\sum _{i<k}|\lambda _{n_k}(\psi _i)|>1. \end{aligned}$$
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
$$\begin{aligned} \mu _n=\frac{\lambda _n}{\Vert \lambda _n\Vert }\in C_p(X)^{*} \end{aligned}$$
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
  1. (5)

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

     
  2. (6)

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

     
  3. (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
$$\begin{aligned} G+Z=\{g+z:g\in G,\; z\in Z\} \end{aligned}$$
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
$$\begin{aligned} (f_1, \nu _1), \ldots , (f_n, \nu _n) \in C_p(X)\times C_p(X)^{*} \end{aligned}$$
such that
$$\begin{aligned} \text{ lin }\{f_1, \ldots , f_n\}=\text{ lin }\{g_1, \ldots , g_n\} \end{aligned}$$
and
$$\begin{aligned} \nu _j|Z=0,\;\; \nu _j(f_i)=\delta _{j,i}\ \quad \mathrm{for~all} \ \, i,j\in \{1, \ldots , n\}. \end{aligned}$$
Put
$$\begin{aligned} f_{n+1}=g_{n+1}-\sum _{i=1}^n \nu _i(g_{n+1})f_i. \end{aligned}$$
Then
$$\begin{aligned} \text{ lin }\{f_1, \ldots , f_{n+1}\}=\text{ lin }\{g_1, \ldots , g_{n+1}\} \end{aligned}$$
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
$$\begin{aligned} |\mu _n(f)|=|\mu _n(f-g)| \le \Vert \mu _n\Vert {\cdot }\Vert f-g\Vert _{\infty }<\varepsilon \end{aligned}$$
for \(n>m.\) Thus \(\mu _n(f)\rightarrow _n 0\), which means that the space \(C_p(X)\) has the JNP. \(\square \)

4 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. (1)

    K contain no nontrivial converging sequences but contains a copy of \(\beta \mathbb {N}\);

     
  2. (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
$$\begin{aligned} d(A):=\lim _{n\rightarrow \infty }\frac{|A\cap [1,n]|}{n} \end{aligned}$$
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. (1)

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

     
  2. (2)

    x contains no finite subsets of \(\mathbb {N}\) but \(Z\in x\) for some \(Z\in \mathcal {Z}\);

     
  3. (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
$$\begin{aligned} \widehat{Z}=\{x\in K: Z\in x\}, \end{aligned}$$
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. \(\square \)

5 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. (1)

    \(C_k(X)\) is Polishable;

     
  2. (2)

    \(C_p(X)\) is Polishable;

     
  3. (3)

    \(C_k(X)\) is Polish;

     
  4. (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
$$\begin{aligned} U\subset \{f\in C(X):f(K)\subset (-\varepsilon ,\varepsilon )\}. \end{aligned}$$
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
$$\begin{aligned} R(U)\subset \{f\in C_k(K):f(K)\subset (-\varepsilon ,\varepsilon )\}. \end{aligned}$$
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
$$\begin{aligned} \left\{ (f_n)_{n\in \omega }\in \prod _{n\in \omega }C_k(X_n):\forall n\in \omega \;\;f_{n+1}{\upharpoonright }X_n=f_n\right\} \end{aligned}$$
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.

Notes

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Authors and Affiliations

  1. 1.Ivan Franko National University of LvivLvivUkraine
  2. 2.Jan Kochanowski University in KielceKielcePoland
  3. 3.Faculty of Mathematics and InformaticsA. Mickiewicz UniversityPoznanPoland
  4. 4.Institute of Mathematics Czech Academy of SciencesPragueCzech Republic
  5. 5.Faculty of Mathematics and Natural SciencesUniversity of RzeszówRzeszowPoland

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