---
abstract: 'We extend the well-known criterion of Lotz for the dual Radon-Nikodym property (RNP) of Banach lattices to finitely generated Banach $C(K)$-modules and Banach $C(K)$-modules of finite multiplicity.  Namely, we prove that if $X$ is a Banach space from one of these classes then its Banach dual $X^\star$ has the RNP iff $X$ does not contain a closed subspace isomorphic to $\ell^1$.'
address:
- 'Department of Mathematics, Community College of Philadelphia, 1700 Spring Garden St., Philadelphia, PA, USA'
- 'Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA'
author:
- Arkady Kitover
- Mehmet Orhon
title: |
     The dual Radon - Nikodym property\
    for finitely generated Banach $C(K)$-modules
---

Introduction
============

Let us start with reminding the reader about the following equivalences in the class of Banach lattices. 

\[tA\] (Lozanovsky [@Loz] - Lotz [@Lot], see also [@MN Theorem 2.4.15, p.94]) Let $X$ be a Banach lattice.  Then the following conditions are equivalent. 

1.   $X$ is reflexive. 

2.   $X$ does not contain a copy [^1] of either $c_0$ or $\ell^1$.

3.   $X$ does not contain a copy of either $c_0$ or $\ell^1$ as a sublattice. [ ^2]

\[tB\] (Lozanovsky [@Loz1], see also [@MN Theorem 2.5.6, p.104 and Theorem 2.4.12, p. 92]) Let $X$ be a Banach lattice.  Then the following conditions are equivalent. 

1.   $X$ is weakly sequentially complete. 

2.   $X$ does not contain a copy of $c_0$.

3.   $X$ does not contain a copy of $c_0$ as a sublattice. 

Before we remind the reader the next result in this direction let us recall the following definition. 

\[d7\] A Banach space $X$ is said to have the Radon-Nikodym property (RNP) if for every finite measure space $(\Omega, \Sigma, \lambda)$ and for every bounded linear operator $T: L^1(\lambda) \rightarrow X$ there exists a strongly measurable $g \in L^\infty(\lambda, X)$ such that $$Tf = \int \limits_{\Omega} {fg d\lambda}, \; f \in L^1(\lambda),$$ where the integral is the Bochner integral. 

\[tC\] (Lotz, see [@MN Theorem 5.4.14, p.367]) Let $X$ be a Banach lattice.  The following conditions are equivalent. 

1.   The Banach dual $X^\star$ of $X$ has RNP. 

2.   $X$ does not contain a copy of $\ell^1$.

3.   $X^\star$ does not contain a copy of either $c_0$ or $L^1[0,1]$ as a sublattice. 

The statements of Theorems \[tA\] - \[tC\] become false if instead of Banach lattices we consider arbitrary Banach spaces.  For Theorems \[tA\] and \[tB\] a counterexample is provided by the famous James’ space [@Ja] while in the case of Theorem \[tC\] we need to use another example of James in [@Ja1] [^3] where he constructed a separable Banach space $X$ not containing a copy of $\ell^1$ and such that $X^\star$ is not separable.  It follows from a later result of Stegall [@St] that the space $X$ from [@Ja1] does not have the dual RNP. 

But, if instead of the class of all Banach spaces we consider the much smaller classes of finitely generated Banach $C(K)$-modules or Banach $C(K)$-modules of finite multiplicity, which while not contained in the class of all Banach lattices can be considered as its nearest relatives, the analogues of Theorems \[tA\] - \[tC\] remain true. 

We refer the reader for precise definitions of the notions used in the next two statements to the next section of the current paper. 

\[tD\] ([@KO Theorems 1 and 2]) Let $X$ be a finitely generated Banach $C(K)$-module or a Banach $C(K)$-module of finite multiplicity.  Then the following conditions are equivalent. 

1.   $X$ is reflexive. 

2.   $X$ does not contain a copy of either $c_0$ or $\ell^1$.

3.   Every cyclic subspace of $X$ does not contain a copy of either $c_0$ or of $\ell^1$.

\[tE\] ([@KO1 Theorems 3.1 and 3.8]) Let $X$ be a finitely generated Banach $C(K)$-module or a Banach $C(K)$-module of finite multiplicity.  Then the following conditions are equivalent. 

1.   $X$ is weakly sequentially complete. 

2.   $X$ does not contain a copy of $c_0$

3.   Every cyclic subspace of $X$ does not contain a copy of $c_0$.

\[r1\] Every cyclic subspace of a Banach $C(K)$-module can be endowed in the unique way with the structure of a Banach lattice compatible with its structure of $C(K)$-submodule (see [@KO; @KO1] for more details).  Therefore we can state that conditions of Theorems \[tD\] and \[tE\] are equivalent to any cyclic subspace not containing either $c_0$ or $\ell^1$ as a sublattice in the case of Theorem \[tD\] and not containing $c_0$ as a sublattice in the case of Theorem \[tE\]. 

The goal of the present paper is to show that similar to Theorems \[tD\] and \[tE\] the result of Theorem \[tC\] can be extended to the classes of finitely generated $C(K)$-modules and $C(K)$-modules of finite multiplicity. 

We will need the following important characterization of the dual RNP property in Banach spaces. 

\[tI\] (Stegall - Uhl).Let $X$ be a Banach space.  The following conditions are equivalent. 

\(1) $X^\star$ has RNP. 

\(2) For every separable subspace $Y$ of $X$ its conjugate $Y^\star$ is also separable. 

\[r5\] The implication (1) $\Rightarrow$ (2) was proved by Stegall (see [@St Theorem2]) and the implication (2) $\Rightarrow$ (1) by Uhl (see [@U Corollary 3])

At the end of the introduction we want to emphasize that one of the reasons to study the dual RNP is the following important result. 

\[tF\] ( [@DU Theorem 1, p.98]) Let $(\Omega, \Sigma, \mu )$ be a finite measure space, $1 \leq p < \infty$, and $X$ be a Banach space.  Then $L^p(\mu,X)^\star = L^q(\mu,X^\star)$, where $1/p+1/q=1$, if and only if $X^\star$ has the Radon-Nikodym property with respect to $\mu$.

\[r2\] (1) We remind the reader that $L^p(\mu, X)$, $1 \leq p < \infty$, is the space of (classes) of all Bochner integrable functions on $\Omega$ endowed with the norm $\|f\| = \big{(} \int \limits_\Omega \|f(\omega)\|_X^p d\mu \big{)}^{1/p}$.

\(2) If $X^\star$ does not have the RNP the description of $L^p(\mu,X)^\star$ becomes rather involved.  For a detailed discussion of it and for the background of Theorem \[tF\] we refer the reader to [@DU]. 

Preliminaries
=============

All the linear spaces will be considered either over the field of real numbers $\mathds{R}$ or the field of complex numbers $\mathds{C}$. If $X$ is a Banach space we will denote its Banach dual by $X^{\star}$.

Let us recall some definitions. 

\[d1\] Let $K$ be a compact Hausdorff space and $X$ be a Banach space.  We say that $X$ is a Banach $C(K)$-module if there is a continuous unital algebra homomorphism $m$ of $C(K)$ into the algebra $L(X)$ of all bounded linear operators on $X$.

\[r3\] Because $\ker{m}$ is a closed ideal in $C(K)$ by considering, if needed, $C(\tilde{K}) = C(K)/\ker{m}$ we can and will assume without loss of generality that $m$ is a contractive homomorphism (see [@KO]) and $\ker{m}
= 0$. Then (see [@HO Lemma 2 (2)]) $m$ is an isometry.  Moreover, when it does not cause any ambiguity we will identify $f    \in C(K)$ and $m(f)    \in L(X)$.

\[d2\] Let $X$ be a Banach $C(K)$-module and $x    \in X$. We introduce the *cyclic* subspace $X(x)$ of $X$ as $X(x) = cl\{fx : f    \in C(K)\}$.

The following proposition was proved in [@Ve] (see also [@Ra]) in the case when the compact space $K$ is extremally disconnected and announced for an arbitrary compact Hausdorff space $K$ in [@Ka].  It follows as a special case from [@HO Lemma 2 (2)]. 

\[p1\] Let $X$ be a Banach $C(K)$-module, $x    \in X$, and $X(x)$ be the corresponding cyclic subspace.  Then, endowed with the cone $X(x)_{+} = cl \{fx
: f    \in C(K), f \geq0\}$ and the norm inherited from $X$, $X(x)$ is a Banach lattice with positive quasi-interior point $x$.

Our next proposition follows from Theorem 1 (3) in [@Or]

\[p2\] The center $Z(X(x))$ of the Banach lattice $X(x)$ is isometrically isomorphic to the weak operator closure of $m(C(K))$ in $L(X(x))$.

Now we can introduce one of the two main objects of interest in the current paper. 

\[d3\] Let $X$ be a Banach $C(K)$-module.  We say that $X$ is finitely generated if there are an $n    \in \mathds{N}$ and $x_{1}, \ldots, x_{n}    \in X$ such that the set $\sum\limits_{i=1}^{n} X(x_{i})$ is dense in $X$.

\[d4\] Let $X$ be a Banach space and $\mathcal{B}$ be a Boolean algebra of projections on $X$. The algebra $\mathcal{B}$ is called Bade complete (see [@DS XVII.3.4, p.2197] and [@Sch V.3, p.315]) if

\(1) $\mathcal{B}$ is a complete Boolean algebra. 

\(2) Let $\{\chi_{\gamma }\}_{\gamma   \in \Gamma }$ be an increasing net in $\mathcal{B}$, $\chi$ be the supremum of this net, and $x   \in X$. Then the net $\{\chi_{\gamma }x\}$ converges to $\chi x$ in norm in $X$.

\[d5\] Let $\mathcal{B}$ be a Bade complete Boolean algebra of projections on $X$. $\mathcal{B}$ is said to be of *uniform multiplicity* $n,$ if there exist a set of nonzero pairwise disjoint idempotents $\{e_{\alpha }\}$ in $\mathcal{B}$ with $\sup e_{\alpha }=1$ such that for any $e_{\alpha }$ and for any $e   \in \mathcal{B}$, $e\leq e_{\alpha }$ the $C(K)$-module $eX$ has exactly $n$ generators. 

We will need the following result of Rall [@Ra1] (for a proof, see Lemma 2 in [@Or1]). 

\[l3\] Let $\mathcal{B}$ be of uniform multiplicity one on $X$. Then $X$ may be represented as a Banach lattice with order continuous norm such that $\mathcal{B}$ is the Boolean algebra of band projections on $X$.

\[d6\] A Bade complete Boolean algebra of projections $\mathcal{B}$ on $X$ is said to be of finite multiplicity on $X$ if there exists a collection of disjoint idempotents $\{e_{\alpha }\}$ in $\mathcal{B}$ such that, for each $\alpha,$ $e_{\alpha }X$ is $n_{\alpha }$-generated and $\sup e_{\alpha }=1.$

\[r4\] The collection $\{n_{\alpha }\}$ of positive integers in Definition \[d6\] need not be bounded. 

\[tJ\] (Bade [@DS XVIII.3.8, p. 2267]).  Let $X$ be a Banach $C(K)$-module of finite multiplicity.  Then there exists a sequence of disjoint idempotents $\{e_{n}\}$ in $\mathcal{B}$ such that, for each $n,$ $\mathcal{B}$ is of uniform multiplicity $n$ on $e_{n}X$ and $\sup e_{n}=1.$ Also the norm closure of the sum of the sequence of the spaces $\{e_{n}X\}$ is equal to $X.$

The main Results
================

\[t1\] Let $X$ be a finitely generated Banach $C(K)$-module.  Then the following are equivalent:(1) $X^\star$ has the Radon-Nikodym property.(2) $X$ does not contain any copy of $\ell^1$.

The implication $(1)   \Rightarrow   (2)$ follows from Theorem \[tI\].  It can also be proved in the following way. 

Suppose (1) holds.  Then it is well known that $X^\star$ does not contain any copy of $L^{1}(0,1)$ ( [@MN 5.4.4(ii), p.363 and 5.4.8, p.365] ).  Then by a result of Hagler [@Ha], we have that $X$ does not contain a copy of $\ell^1$. Hence (1) implies (2) for all Banach spaces $X$.

Conversely, suppose that (2) holds.  We will complete the proof by induction on the number of generators of $X$. Let $X$ have one generator.  That is $X$ is cyclic.  Then it follows from Proposition \[p1\] and Theorem \[tC\] that $X^\star$ has the RNP.  Now suppose that whenever $X$ has $r$ generators for some fixed $r\geq1$ and satisfies (2), then $X^\star$ has the RNP.  Let $X$ have $r+1$ generators and satisfy (2).  Suppose $\{x_{0},x_{1},\ldots,x_{r}\}$ is a set of generators of $X$. Let $Y=X(x_{1},x_{2},\ldots,x_{r})$. Then $Y$ is a Banach $C(K)$-module with $r$ generators and as a subspace of $X$, it satisfies (2).  Hence, by the induction hypothesis, $Y^\star$ has the RNP.  Now consider the cyclic space  $X/Y=X/Y([x_{0}])$ where $[x_{0}]=x_{0}+Y$. We have that $\left(  X/Y\right)  ^\star = Y^{o}   \subset X^\star$ where $Y^{o}$ is the polar (annihilator) of $Y$ in $X^\star$. Since $X$ satisfies (2), by Hagler’s Theorem [@Ha], $X^\star$ does not contain a copy of $L^{1}(0,1)$ and the same is true for $(X/Y)^\star$ as a subspace of $X^\star$. Again by Hagler’s Theorem, this means that $X/Y$ satisfies (2).  Since $X/Y$ is a cyclic Banach space, it is representable as a Banach lattice.  Therefore, by Lotz’s Theorem \[tC\], $Y_0 = (X/Y)^\star$ has the RNP.  Recall also that $ X^\star/Y^0 = Y^\star$. Since the RNP is a three space property [@CG 6.5.b, p. 202], we have that $X^\star$ has the Radon-Nikodym property. 

It is possible to sharpen the result of Theorem \[t1\] if we assume that the cyclic subspaces of $X$ have order continuous norm when represented as Banach lattices.  Initially, we need a result which guarantees that a Banach lattice $X$ which does not contain a copy of $\ell^1$ as a sublattice does not contain a copy of $\ell^1$ as a subspace.  This is not true in general.  For example, $\ell^1$ is not isomorphic to any sublattice of $\ell^{\infty}$, however any separable Banach space (hence in particular $\ell^1$) may be embedded isometrically into $\ell^{\infty}$.

To proceed we need to recall the following important result by Lotz and Rosenthal. 

\[tH\] (Lotz and Rosenthal [@LR], see also [@MN Theorem 5.2.15, p. 345])

Let $X$ be a Banach lattice.  The following conditions are equivalent. 

\(1) For each $x \in X_+$ the order interval $[0,x]$ is weakly sequentially precompact. 

\(2) $X^\star$ does not contain any copy of $L^1[0,1]$ as a sublattice. 

The Lotz - Rosenthal theorem leads to the following lemma. 

\[l1\] Let $X$ be a Banach lattice with order continuous norm then $X$ contains a copy of $\ell^1$ if and only if $X$ contains a copy of $\ell^1$ as a sublattice. 

Suppose $X$ does not contain a copy of $\ell^1$ as a sublattice.  Then $X^\star$ does not contain any copy of $c_{0}$ as a sublattice [@MN Proposition 2.3.12, p. 83].  Also since $X$ has order continuous norm any order interval in $X$ is weakly compact [@MN Theorem 2.4.2, p. 86].  Hence by the Eberlein - Šmulian theorem any order interval in $X$ is weakly sequentially compact.  Then, by Theorem \[tH\] of Lotz and Rosenthal $X^\star$ does not contain any copy of $L^{1}(0,1)$ as a sublattice.  Finally, by the equivalence (2) $\Leftrightarrow$ (3) in Theorem \[tC\] by Lotz we conclude that $X$ does not contain any copy of $\ell^1$.

\[t2\] Let $K$ be a hyperstonian compact space and let $X$ be a finitely generated Banach $C(K)$-module such that the algebra $\mathcal{B}$ of the idempotents in $C(K)$, is a Bade complete Boolean algebra of projections on $X$. Then the following conditions are equivalent

1.   $X^\star$ has the Radon-Nikodym property. 

2.   $X$ does not contain any copy of $\ell^1$.

3.   No cyclic subspace of $X$ contains a copy of $\ell^1$.

4.   No cyclic subspace of $X$, when represented as a Banach lattice, contains a copy of $\ell^1$ as a sublattice. 

It is clear that (2)$   \Rightarrow   $(3).  Let us show that (3)$   \Rightarrow   $(2).  We will use induction on the number of generators of $X$. When $n=1$, $X$ is cyclic, so (3)$   \Rightarrow   $(2) trivially.  Suppose for some $r\geq1,$ for all $X$ with $r$ generators we have (3)$   \Rightarrow   $(2).  Suppose $X$ has $r+1$ generators $\{x_{0},x_{1},\ldots,x_{r}\}$ and satisfies (3).  Let $Y=X(x_{1},\ldots,x_{r})$. Then $Y$ satisfies (2) by the induction hypothesis.  On the other hand, $X/Y$ $=X/Y([x_{0}])$ where $[x_{0}]=x_{0}+Y.$ Since $\mathcal{B}$ is Bade complete on $X,$ by [@KO Lemma 1], $\mathcal{B}$ is Bade complete on the cyclic space $X/Y$. By Lemma \[l3\], $X/Y$ can be represented as a Banach lattice with order continuous norm and quasi-interior point $[x_{0}]$ such that the algebra $\mathcal{B}$ is isometrically isomorphic to the algebra of band projections on the Banach lattice $X/Y$. Now (3) implies that $\ell^1$ is not contained in $X/Y$ as a sublattice (see the proof of Theorem 1 in [@KO p. 480]).  Then by Lemma \[l1\], we have that $X/Y$ does not contain a copy of $l^{1}.$ Since not containing $\ell^1$ is a three space property [@CG Theorem 3.2.d, p.96], we have that $X$ does not contain any copy of $\ell^1$.

The implication (3) $\Rightarrow$ (4) is trivial. 

Assume (4).  Every cyclic subspace of $X$ when represented as a Banach lattice has order continuous norm.  Thus the implication (4) $\Rightarrow$ (3) follows from Lemma \[l1\].  It completes the proof. 

The purpose of the next two examples is to show that condition $(3)$ in Theorem \[t2\] cannot be weakened as follows.  Let $x_1, \ldots , x_n$ be some system of generators of $X$. Assume that none of the cyclic subspaces $X(x_i), i= 1, \ldots , n$ contains a copy of $\ell^1$. Then $X^\star$ has RNP. ( Compare Examples 1 and 2 on page 482 in [@KO])

\[e1\] Let $X = L^1(0,1) \oplus L^2(0,1)$ considered as a $L^\infty(0,1)$ Banach module.  Let $x_1=(\mathbf{0}, \mathbf{1})$ and $x_2= (-\mathbf{1}, \mathbf{1})$. Then $\{x_1, x_2\}$ is a system of generators of $X$. Clearly $X(x_1) = L^2$ and $X(x_2)$ is isomorphic to $L^2$ while $X^\star$ does not have RNP. 

In Example \[e1\], $X$ is a non-atomic Banach lattice.  It is easy to provide a similar example when $X$ is a discrete BL. 

\[e2\] Let $w_n$ be an increasing sequence of positive real numbers such that $\sum \limits_{n=1}^\infty \frac{1}{w_n}<\infty$. We consider the Hilbert space $$\ell^2(w_n) = \{\{a_n\}: \sum \limits_{n=1}^\infty |a_n|^2w_n^2 < \infty \}.$$ with the norm $\|\{a_n\}\| = \sqrt{\sum \limits_{n=1}^\infty |a_n|^2w_n^2}$. Then $\ell^2(w_n) \subset \ell^1$. Let $X= \ell^1 \oplus \ell^2(w_n)$ considered as a $\ell^\infty$-module.  Let $x_1 = (\mathbf{0}, \{\frac{1}{n w_n}\})$ and $x_2 = (\{-\frac{1}{n w_n}\}, \{\frac{1}{n w_n}\})$. Then, as in Example \[e1\] we see that the system $\{x_1, x_2\}$ generates $X$ and that the cyclic subspaces $X(x_1)$ and $X(x_2)$ are reflexive.  However $X^\star$ does not have RNP. 

\[r6\] Already mentioned example of James in [@Ja1] shows that, in general, the condition that $X$ is a finitely generated $C(K)$-module cannot be dropped.  Indeed, every Banach space can be considered as a Banach module over $\mathds{C}$. Still there is a large class of Banach $C(K)$-modules which are in general not finitely generated but for which the conclusion of Theorem \[t2\] remains valid. 

We proceed now to extend the result of Theorem \[t2\] to Banach $C(K)$-modules of finite multiplicity. 

\[l4\] Let $X$ be a Banach $C(K)$-module of uniform multiplicity $n$ and let $x \in X$. Let $\{e_\alpha\}$ be the system of pairwise disjoint idempotents from Definition \[d5\].  Then the set $\{\alpha : e_\alpha x \neq 0\}$ is at most countable. 

Consider the cyclic subspace $Y$ of $X$ generated by $x$. By Lemma \[l3\] $Y$ can be represented as a Banach lattice with order continuous norm.  Assume, contrary to our claim, that the set $\{\alpha : e_\alpha x \neq 0\}$ is uncountable.  Then for some $k \in \mathds{N}$ there are $e_1, e_2, \ldots \in \{e_\alpha\}$ such that $\|e_n x\| \geq 1/k, n \in \mathds{N}$. Notice that the elements $e_n x$ are pairwise disjoint in the Banach lattice $Y$. Then, by Proposition 0.5.5 in [@Wn page 36] $Y$ contains $\ell^\infty$ as a sublattice.  But then by the well known result of Lozanovsky [@Loz2] (see also [@KA Theorem 4, page 295]) $Y$ cannot have order continuous norm, a contradiction. 

\[c1\] Let $X$ be a Banach $C(K)$-module of uniform multiplicity $n$ and let $Y$ be a separable closed subspace of $X$. Then the set $\{\alpha : e_\alpha Y \neq 0\}$ is at most countable. 

The proof of the following lemma repeats verbatim the proof of statement $(\star)$ on page 485 in [@KO]). 

\[l5\] Let $X$ be a Banach $C(K)$-module of finite multiplicity such that any cyclic subspace of $X$ does not contain a copy of $\ell^1$. Suppose that $\{e_n\}$ is a system of pairwise idempotents in $C(K)$ such that $\sup \limits_n e_n = 1$. Let $\chi_n = e_1 + \ldots + e_n$. Then for any $f \in X^\star$ we have $\|f - \chi_n^\star f\| \rightarrow 0$.

\[l2\] Let $X$ be a Banach $C(K)$-module of uniform multiplicity $n$. Assume that no cyclic subspace of $X$, represented as a Banach lattice, contains $\ell^1$ as a sublattice.  Then $X^\star$ has RNP. 

By Theorem \[tI\] it is enough to prove that for any closed separable subspace $Y$ of $X$ the conjugate $Y^\star$ is also separable.  Thus, let $Y$ be a closed separable subspace of $X$. By Corollary \[c1\] there is an at most countable subset $\{e_i, i \in \mathds{N}\}$ of $\{e_\alpha\}$ such that $e_i Y \neq 0, i \in \mathds{N}$ and $e_\alpha Y  =0$ for any $\alpha$ not in this subset.  Then $Y \subseteq Z \subseteq X$ where $Z$ is the closure in $X$ of the direct sum $\sum \limits_{i} e_i Y$ (we remind the reader that the idempotents $e_i$ are pairwise disjoint in $C(K)$).  For every $i$, $e_iX$ is a finitely generated Banach $e_iC(K)$-module (with $n$ generators) and by Theorem \[t2\] its conjugate $(e_i X)^\star$ has RNP.  Therefore by Theorem \[tI\] the conjugate $(e_iY)^\star$ is separable. 

Before we go to next step in the proof let us notice that for any $i \in \mathds{N}$ we have $(e_i Y)^\star = e_i^\star Z^\star$. Indeed, $(e_i Y)^\star = Z^\star /(e_i Y)^0$ and $e_i$ is a projection on $Z$ with $e_i Z = cl(e_i Y)$. Consequently, $e_i^\star$ is a projection on $Z^\star$ and $(1-e_i^\star)Z^\star = (e_iY)^0$. Therefore $(e_i Y)^\star = e_i^\star Z^\star$. Because no cyclic subspace of $X$ contains a copy of $\ell^1$, Lemma \[l5\] guarantees that $Z^\star = cl \sum \limits_{i}  (e_i Y)^\star$. Indeed, let $f \in Z^\star$, and $m \in \mathds{N}$. Then $\chi_m f \in \sum \limits_{i=1}^m (e_iY)^\star$ and $\chi_m f$ converges to $f$ by norm.  Thus $Z^\star$ is separable.  Then $Y^\star$, being a factor of $Z^\star$ by $Y^0$, the annihilator of $Y$ in $Z^\star$, is separable as well. 

\[t3\] Let $X$ be a Banach $C(K)$-module of finite multiplicity.  Then the following conditions are equivalent. 

\(1) $X^\star$ has RNP. 

\(2) $X$ does not contain a copy of $\ell^1$.

\(3) Any cyclic subspace of $X$ does not contain a copy of $\ell^1$.

\(4) Any cyclic subspace of $X$ represented as a Banach lattice does not contain $\ell^1$ as a sublattice. 

The implication $(1)   \Rightarrow   (2)$ has been already established in the proof of Theorem \[t1\]. 

The implications $(2)   \Rightarrow   (3)   \Rightarrow   (4)$ are trivial. 

The implication $(4) \Rightarrow (3)$ follows from Lemma \[l1\]. 

It remains to prove that $(3)   \Rightarrow   (1)$. Assume (3).  Let $Y$ be a closed separable subspace of $X$. Let $e_n$ be the idempotents from Theorem \[tJ\] and let $Z= cl \sum \limits_{n}  e_nY$. Then $Z$ is a separable closed subspace of $X$ and, applying Theorem \[tJ\] we see that $Y   \subseteq Z$. Because any cyclic subspace of $X$ does not contain a copy of $\ell^1$ by Lemma \[l5\], as explained in the proof of Lemma \[l2\], $Z^\star = cl \sum \limits_{n}  (e_n Y)^\star$.

Next notice that by Lemma \[l2\] the space $(e_nX)^\star$ has RNP and by Theorem \[tI\] the space $(e_nY)^\star$ is separable.  Therefore $Z^\star$ is separable, $Y^\star$ is separable as a factor of $Z^\star$, and $X^\star$ has RNP by Theorem \[tI\]. 

\[r3\] A slight modification of Example 4.2(3) on page 752 in [@KO1] ( where one replaces $\ell^p$ by $c_0$) provides an example of a Banach $C(K)$-module $X$ with the following properties. 

1.   $X$ is of uniform multiplicity $n$, $n > 1$.

2.   $X$ is not separable. 

3.   Every cyclic subspace of $X$ is separable and has separable dual.  In particular, $X$ cannot be finitely (or even countably) generated. 

4.   There are cyclic subspaces of $X$ that are not weakly sequentially complete. 

Thus, while Theorem \[t2\] cannot be applied, by Theorem \[t3\] $X$ has dual RNP. 

[99]{} Castillo, J.M.F., Gonzalez, M.: Three-space problems in Banach space theory.  Lecture Notes in Mathematics, v. **1667**, Springer (1997) Diestel, J., Uhl, J.J. : Vector Measures,AMS, Providence (1977) Dunford, N., Schwartz, J.T.: Linear Operators, Part III: Spectral Operators.  Wiley, New York (1971) Hagler, J. : Some more Banach spaces which contain $L^1$. Studia Math. ** 46**, 35 - 42 (1973) Hadwin, D., Orhon, M.: A noncommutative theory of Bade functionals.  Glasgow Math.  J., **33**, 73-81 (1991) James, R.C.: Bases and reflexivity of Banach spaces.  Ann.  Math. ** 52**(3), 518 - 527 (1950) James, R. C.: A separable somewhat reflexive Banach space with nonseparable dual.  Bull.  Amer.  Math.  Soc. ** 80** , 738 - 743 (1974) Kaijser, S.: Some representation theorems for Banach lattices.  Ark. Mat. ** 16**(2), 179 - 193 (1978) Kantorovich, L.V., Akilov, G. P., Functional Analysis.  Pergamon Press (1982) Kitover, A., Orhon, M.: Reflexivity of Banach C(K)-modules via the reflexivity of Banach lattices.  Positivity. ** 18**(3),475-488 (2014) Kitover, A., Orhon, M.: Weak sequential completeness in Banach C(K)-modules of finite multiplicity.  Positivity. ** 21** (2), 739 - 753 (2017) Lotz, H.P.: Minimal and reflexive Banach lattices.  Math.  Ann. ** 209**, 117 - 126 (1973) Lotz, H.P., Rosenthal H.P. : Embeddings of $C(\Delta )$ and $L^1(0,1)$ in Banach lattices.  Israel J. Math. ** 31**.  169 - 179 (1978) Lozanovsky, G.Ya.:  Some topological conditions on Banach lattices and reflexivity conditions on them.  Soviet Math.  Dokl. ** 9**, 1415 - 1418 (1968) Lozanovsky, G.Ya.:  Banach structures and bases.  Funct.  Anal.  Appl.  294(1) (1967) Losanovsky G. Ya. :  On isomorphic Banach lattices.  Sibirsk.  Mat.  Zh., ** 10**(1), 93- 98 (1969) Meyer-Nieberg, P.: Banach Lattices.  Universitext.  Springer, Berlin (1991) Orhon, M.: The ideal center of the dual of a Banach lattice.  Positivity, **14**(4), 841 - 847 (2010) Orhon, M.: Algebras of operators containing a Boolean algebra of projections of finite multiplicity.  In: Operators in Indefinite Metric Spaces, Scattering Theory And Other Topics (Bucharest, 1985).  Oper.  Theory: Adv.  Appl.,  vol. ** 24**, 265-281.  Birkhäuser, Basel (1987) Rall, C.: Über Boolesche Algebren von Projektionen.  Math Z., **153**, 199 - 217 (1977) Rall, C.: Boolesche Algebren von Projectionen auf Banachräumen.  PhD Thesis, Universität Tübingen (1977) Schaefer, H.H.: Banach lattices and positive operators.  Springer, Berlin (1974) Stegall, C. : The Radon-Nikodym property in conjugate Banach spaces.  Trans.  Amer.  Math.  Soc. ** 206**, 213 - 223 (1975) Uhl, J.J., A note on the Radon-Nikodym property for Banach spaces.  Rev. Roumaine Math.  Pures Appl. ** 17**, 113-115 (1972) Veksler, A.I.: Cyclic Banach spaces and Banach lattices.  Soviet Math.  Dokl. ** 14**, 1773-1779 (1973) Wnuk, W.: Banach lattices with order continuous norms.  PWN, Warszawa (1999)

[^1]: If $X$ and $Y$ are Banach spaces we say that $X$ contains a copy of $Y$ if there is a closed subspace of $X$ linearly isomorphic to $Y$.

[^2]: If $X$ and $Y$ are Banach lattices we say that $X$ contains a copy of $Y$ as a sublattice if there is a closed sublattice of $X$ lattice isomorphic to $Y$.

[^3]: The James’ space from [@Ja], being quasi-reflexive, has the dual RNP.