1 Preliminaries

Throughout this paper \(X\) will stand for a Banach space over the real or complex field \(\mathbb {K}\), \(X^{*}\) for its dual space and \(\left( \Omega ,\Sigma ,\mu \right) \) for a finite measure space. As usual \(ca\left( \Sigma ,X\right) \) will denote the Banach space over \(\mathbb {K}\) of all \(X\)-valued countably additive measures \(F\) on \(\Sigma \) equipped with the semivariation norm \(\left\| F\right\| \) whereas \(bvca\left( \Sigma ,X\right) \) will stand for the Banach space of all \(X\)-valued countably additive measures \(F\) of bounded variation on \(\Sigma \) equipped with the variation norm \(\left| F\right| \).

Let us recall that a weakly \(\mu \)-measurable function \(f\): \(\Omega \rightarrow X\) is said to be Dunford integrable if \(x^{*}f\in \mathcal {L}_1\left( \mu \right) \) for every \(x^{*}\in X^{*}\). If \(f\) is Dunford integrable and \(E\in \Sigma \), the map \(x^{*}\mapsto \int _Ex^{*}f\,d\mu \), usually denoted by \( \left( D\right) \int _Ef\,d\mu \), is a continuous linear form on \(X^{*}\). If \( \left( D\right) \int _Ef\,d\mu \in X\) for each \(E\in \Sigma \) then \(f\) is said to be Pettis integrable and we write \(\left( P\right) \int _Ef\,d\mu \) instead of \(\left( D\right) \int _Ef\,d\mu \). The Pettis space of all weakly measurable (classes of scalarly equivalent) Pettis integrable functions \( f\): \(\Omega \rightarrow X\) is denoted by \(\mathcal {P}_1\left( \mu ,X\right) \) whereas the subspace of all those strongly measurable (classes of) functions is represented by \(P_1\left( \mu ,X\right) \). We will consider these two spaces \(\mathcal {P}_1\left( \mu ,X\right) \) and \(P_1\left( \mu ,X\right) \) endowed with the semivariation norm

$$\begin{aligned} \left\| f\right\| _{\mathcal {P}_1(\mu ,X)}=\sup \left\{ \int _\Omega \left| x^{*}f\left( \omega \right) \right| d\mu \left( \omega \right) : x^{*}\in X^{*},\,\left\| x^{*}\right\| \le 1\right\} . \end{aligned}$$

By a result of Pettis, if \(f\): \(\Omega \rightarrow X\) is weakly measurable and Pettis integrable, the map \(F\): \(\Sigma \rightarrow X\) defined by \(F\left( E\right) =\left( P\right) \int _Ef\,d\mu \) is a \(\mu \)-continuous countably additive \(X\)-valued measure and the linear operator \(S\): \(\mathcal {P}_1(\mu ,X)\rightarrow ca\left( \Sigma ,X\right) \) defined by \(Sf=F\) is a linear isometry from \(\mathcal {P}_1(\mu ,X)\) into \(ca\left( \Sigma ,X\right) \), that is, it holds that \(\left\| Sf\right\| =\left\| f\right\| _{\mathcal {P} _1(\mu ,X)}\). If in addition \(f\) is strongly measurable, then \( Sf\left( \Sigma \right) \) is a relatively compact subset of \(X\) [5, Chapter VIII].

We also recall that a subspace \(M\) of \(X^{*}\) is said to be norming over a subspace \(Y\) of \(X\) if for some \(c>0\) it holds that for all \(y\in Y\), \( \sup _{f\in S\left( M\right) }\left| f\left( y\right) \right| \ge c\left\| y\right\| \) where \(S\left( M\right) \) denotes the unit sphere of \(M.\) For short \(M\) is said to be norming when \(M\) happens to be norming on \(X\).

The existence of copies of \(c_0\) in the Pettis space \(P_1\left( \mu ,X\right) \) has been studied by a number of authors, e.g. Freniche [10, 11] showed that \(P_1\left( \mu ,X\right) \) contains a copy of \(c_0\) if and only if \(X\) does (see also [7]). In [9, Theorem 1.2] we find that if \(X^{*}\) contains a norming sequence and \(X\) has the weak Radon-Nikodým property (WRNP) with respect to each positive and finite measure defined on \(\Sigma \), then the space \(bvca\left( \Sigma ,X\right) \) contains a copy of \(c_0\) if and only if \(X\) does. Recently, Ferrando [8, Theorem 2.2] has shown that the Musiał space \(\mathcal {M}\left( \mu ,X\right) \) formed by all those functions \(f\): \(\Omega \rightarrow X\) of \( \mathcal {P}_1\left( \mu ,X\right) \) whose associated measure \(Sf\) has bounded variation, endowed with the variation norm of the integral, enjoys the property that if each \(f\in \mathcal {M}(\mu ,X)\) has a Pettis integral with separable range, then \(\mathcal {M}(\mu ,X)\) contains a copy of \(c_0\) if and only if \(X\) does. This happens in a number of situations, for instance if: (A) \(\left( \Omega ,\Sigma ,\mu \right) \) is a perfect space; (B) \(X\) has the weak\(^{**}\) Radon-Nikodým property (W\(^{**}\)RNP); or (C) \(X\) is weakly compactly generated (WCG).

In this note, by making a timing use of [13, Proposition 3.2], under which the existence of a norming sequence in \(X^{*}\) provides that each \( f\in \mathcal {M}\left( \mu ,X\right) \) satisfies that \(\left\| f\left( \cdot \right) \right\| \in L_1(\mu )\) although it may happen that \(f\notin L_1(\mu )\) as \(f\) is not strongly measurable in general, we provide a proof of the fact that in the particular case that \(X^{*}\) contains a norming sequence, then the space \(\mathcal {M}\left( \mu ,X\right) \) contains a copy of \(c_0\) if and only if \(X\) does. This result is not included in those covered by [8, Theorem 2.2] as we will provide examples of some Banach spaces with a norming sequence that fail to have the W\(^{**}\)RNP and are not WCG.

2 Copies of \(c_{0}\) in \(\mathcal {M}\left( \mu ,X\right) \)

Our main theorem reads as follows,

Theorem 2.1

Assuming that \(X^{*}\) has a norming sequence, then the normed space \(\mathcal {M}\left( \mu ,X\right) \) contains a copy of \(c_0\) if and only if \(X\) does.

Proof

Let \(S\) be the isometric embedding map of \(\mathcal {M}\left( \mu ,X\right) \) in \(bvca\left( \Sigma ,X\right) \) defined by

$$\begin{aligned} Sf\left( E\right) =\left( P\right) \int _Ef\left( \omega \right) \,d\mu \left( \omega \right) ,\;E\in \Sigma . \end{aligned}$$

Since \(Sf\in bvca\left( \Sigma ,X\right) \) for each \(f\in \mathcal {M}\left( \mu ,X\right) \), by [13, Proposition 3.2] the fact that \(X^{*}\) has a norming sequence implies that the function \(\omega \mapsto \left\| f\left( \omega \right) \right\| \) belongs to \(L_1\left( \mu \right) \) and

$$\begin{aligned} \left| Sf\right| =\int _\Omega \left\| f\left( \omega \right) \right\| \,d\mu \left( \omega \right) . \end{aligned}$$
(2.1)

Let \(J:c_0\rightarrow \mathcal {M}\left( \mu ,X\right) \) be an isomorphism from \(c_0\) into \(\mathcal {M}\left( \mu ,X\right) \). Set \(f_n:=Je_n\) and

$$\begin{aligned} F_n\left( E\right) :=\left( P\right) \int _EJe_n\,d\mu \end{aligned}$$

for each \(n\in \mathbb {N}\), so that \(F_n=\left( S\circ J\right) e_n\) for each \(n\in \mathbb {N}\).

Since the series \(\sum _{n=1}^\infty F_n\) in \(bvca\left( \Sigma ,X\right) \) is weakly unconditionally Cauchy (wuC), there is \(C>0\) such that \(\left| \sum _{i=1}^n\varepsilon _iF_i\right| <C\) for all finite set of signs \( \varepsilon _i\). Using the fact that \(S\) is a linear map onto its range, then

$$\begin{aligned} \sum _{i=1}^n\varepsilon _iF_i=\sum _{i=1}^n\varepsilon _iSf_i=S\left( \sum _{i=1}^n\varepsilon _if_i\right) \end{aligned}$$
(2.2)

for each \(n\in \mathbb {N}\). Then Eqs. (2.1) and (2.2) provide

$$\begin{aligned} \int _\Omega \left\| \sum _{i=1}^n\varepsilon _if_i\left( \omega \right) \right\| d\mu \left( \omega \right) =\left| \sum _{i=1}^n\varepsilon _iF_i\right| <C \end{aligned}$$
(2.3)

for each \(\varepsilon _i\in \left\{ -1,1\right\} \), \(1\le i\le n\) and \( n\in \mathbb {N}\).

The rest of the proof follows a similar argument to that of [8, Theorem 2.2], but working with norms instead of seminorms by properly using Rosenthal’s disjointification lemma [5] and Bourgain averaging theorem [1] in order to provide a basic sequence in \(X\) equivalent to the unit vector basis of \(c_0\). \(\square \)

With the aim of providing some example concerning the Theorem 2.1 hypothesis, let us recall that for a dual Banach space \(X^{*}\), the complementarity of \(X^{*}\) in \(X^{***}\) guarantees that \(X^{*}\) has the W\( ^{**}\)RNP if and only if \(X^{*}\) has the WRNP [14].

Example 2.1

A Banach space with a norming sequence that fails to have the W \(^{**}\) RNP and is not WCG. The dual unit vector sequence \(\left( e_n^{*}\right) _n\) of \(\ell _\infty ^{*}\) is norming for \(\ell _\infty ,\) the latter failing to have the WRNP, see [16, Example 5.13]. The space \(\ell _\infty \) is not WCG either as every weakly compact set in \(\ell _\infty \) is norm separable.

This given example contains a copy of \(c_0\) and so does \(\mathcal {M}\left( \mu ,\ell _\infty \right) \) under this note’s setting. Next example does not contain a copy of \(c_0\) and keeps all the other characteristics of the aforementioned example. It is worth while recalling that according to [2, Corollary 6.8] a normed space \(X\) has got a norming sequence if and only if \(X\) is isometric to a subspace of \(\ell _\infty ,\) and, according to [16], every dual Banach space \(X^{*}\) failing the WRNP, consequently the W\(^{**}\)RNP, must enjoy \(\ell _\infty \) as a quotient or, equivalently, \(X\) must contain a copy of \(\ell _1.\)

Example 2.2

A Banach space, not containing a copy of \(c_0,\) with a norming sequence that fails to have the W \(^{**}\) RNP and is not WCG. Let \(X\) be the Banach space \(\mathcal {C}[0,1]\) of real-valued continuous functions defined on the closed unit interval \([0,1]\) of \(\mathbb {R}\) equipped with the supremum norm. Since \(\mathcal {C}[0,1]\) is separable, \(X^{*}\) is topologically isomorphic to a (weakly closed) linear subspace of \(\ell _\infty \) , see [12, 22.4.(4)]. Hence, the isomorphic copy \(E\) of \(X^{*}\) in \(\ell _\infty \) has a norming sequence. On the other hand, since the compact set \([0,1]\) is not scattered, a classic result of Pełczyńsky and Semadeni (see [3, Theorem 3.1.1]) guarantees that \(X\) has a copy of \( \ell _1.\) According to the previous considerations, this fact ensures that the dual Banach space \(E\) does not have the W \(^{**}\) RNP. Furthermore, a result of Saab and Saab [15], (see [3, Theorem 3.1.4]) prevents any copy of \(\ell _1\) in \(X\) to be complemented, which implies that\(X^{*}\), hence \(E\), does not contain a copy of \(c_0\), see [4, Chapter V, Theorem 10]. Finally, since \(E\) has a quotient isomorphic to the non WCG Banach space \(\ell _\infty \) and the class of WCG Banach spaces is stable under quotients, it follows that \(E \) is not WCG. Consequently the space \(E\) fulfills the stated properties. As a consequence of Theorem 2.1, \(\mathcal {M}\left( \mu ,E\right) \) does not contain a copy of \(c_0\).