---
abstract: 'In this paper we characterize those compact Hausdorff spaces $K$ such that $C(K)\hat{\otimes}C(K)$ (and $C(K)\hat{\otimes}_s C(K)$) have the Dunford-Pettis Property, answering thus in the negative a question posed in [@CaGo] which asked if $\ell_{\infty}\hat{\otimes}\ell_{\infty}$ and $C[0,1]\hat{\otimes}C[0, 1]$ have this property.'
address: |
    Departamento de An lisis Matem tico\
    Facultad de Matem ticas\
    Universidad Complutense de Madrid\
    Madrid 28040 
author:
- Fernando Bombal
- Ignacio Villanueva
title: 'On the Dunford-Pettis property of the tensor product of $C(K) $ spaces '
---

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

In [@CaGo] it is asked if $\ell_{\infty}\hat{\otimes}\ell_{\infty}$ and $C[0,1]\hat{\otimes}C[0,1]$ have the Dunford-Pettis Property (DPP). The purpose of this paper is to show that this is not the case, and to characterize those $C(K) $ spaces such that $\hat{\otimes}_n
C(K)$ or $\hat{\otimes}_{n,s} C(K)$ have the Dunford-Pettis Property, when $n>1$.

Let us recall that a Banach space $E$ has the DPP if, for any Banach space $F$, every weakly compact operator from $E$ into $F$ is completely continuous. In this paper we will need the following facts about the DPP; they are all well known, and can be found, for instance, in [@Di]: (a) $C(K)$ spaces and Schur spaces have the DPP, (b) if a dual Banach space $E^*$ has the DPP, then so does $E$ (but the reciprocal is not true), (c) the DPP is stable under complemented subspaces.

We will now explain our notation. Throughout the paper $K$ will denote a compact Hausdorff space and $C(K)$ the space of continuous scalar functions defined on it. We will write $\mathcal L(E;X)$ to indicate the linear operators from $E$ into $X$, and $\mathcal
L^n(E_1,\ldots, E_n
;X)$ will denote the space of continuous multilinear operators from $E_1 \times \cdots \times E_n$ into $X$. When $E_1= \cdots = E_n
=E$ we will write this as $\mathcal L^{n}(E;X)$ and in this case $\mathcal L^{n}_s (E;X)$ will refer to those of the previous operators which are symmetric. In all the cases, if $X=\Bbb K$, the scalar field, we will not write it. We will write the projective tensor product of $E_1,\ldots, E_n$ as $E_1\hat{\otimes}\cdots
\hat{\otimes} E_n$. We will write $\hat{\otimes}_n E$ to denote the $n$-fold projective tensor product of $E$, and $\hat{\otimes}_{n,s}
E$ will denote the symmetric $n$-fold projective tensor product of $E$. If $n=2$ we will sometimes also write this as $E\hat{\otimes}_s E$.

We consider to be well known that $\mathcal L^n(E_1,\ldots, E_n
;X)$ is naturally isometric to $\mathcal L (E_1\hat{\otimes}\cdots
 \hat{\otimes} E_n;X)$ and that $\mathcal L^{n}_s(E;X)$ is naturally isometric to $\mathcal L (\hat{\otimes}_{n,s} E ;X)$.

We will use the convention $\stackrel{[i]}{\ldots}$ to indicate that the $i^{th}$ coordinate is not involved.

The results
===========

As is well known, every linear operator from any $C(K)$ space into the dual of another $C(K)$ space is weakly compact, and therefore completely continuous. Using that we can prove the following lemma.

\[lema\] Let $K_1$, $K_2 $ be two compact Hausdorff spaces. Let $(f_n)\subset C(K_1)$ be a weakly null sequence and let $(g_n)\subset C(K_2)$ be a bounded sequence. Then the sequence $(f_n \otimes g_n)\subset C(K_1)\hat{\otimes}C(K_2)$ is weakly null.

Let $\phi \in (C(K_1)\hat{\otimes}C(K_2))^* $, and let us consider the linear operator $S\in \mathcal L(C(K_1);C(K_2)^*) $ associated to it defined by $$S(f)(g)=\phi(f\otimes g) \quad.$$ Let us suppose without loss of generality that $\sup_n \|g_n\| \leq
1$. Since $S$ is completely continuous, we get that $$\lim_{n\rightarrow\infty} |\phi(f_n\otimes g_n)| \leq
    \lim_{n\rightarrow\infty} \|S(f_n)\| =0 \quad.$$

In [@BoVi] it can be seen that, if $K$ is scattered, then $(\hat{\otimes}_n C(K))^*$ and $(\hat{\otimes}_{n,s} C(K))^*$ are Schur spaces for every $n\in \Bbb N$. With analogous proof it can be proved that, in case $K_1,\ldots K_n$ are scattered, $(C(K_1)\hat{\otimes}\cdots\hat{\otimes}C(K_n))^*$ is a Schur space.

We state now our main result.

\[teoprincipal\] Let $K_1 , K_2$ be two infinite compact Hausdorff spaces. Then $C(K_1)\hat{\otimes} C(K_2)$ has the DPP if and only if both $K_1$ and $K_2$ are scattered.

If both $K_1$ and $K_2$ are scattered, then $(C(K_1)
\hat{\otimes}C(K_2))^*$ is a Schur space and therefore $C(K_1)
\hat{\otimes}C(K_2)$ has the DPP. Now, let us suppose that one of them, say $K_2$, is not scattered. Since $K_1$ is infinite, $C(K_1)$ is not Schur, and therefore there exist two sequences $(f_n)\subset B_{C(K_1)}$ and $(\xi_n)\subset B_{(C(K_1))^*}$ such that $(f_n)$ is weakly null and $\xi_n(f_n)=1$ for every $n\in \Bbb
N$. Also, since $K_2$ is not scattered, $C(K_2)$ contains an isomorphic copy of $\ell_1$, and therefore there exists a continuous surjective operator $q: C(K_2)\mapsto \ell_2$ ([@DiJT Corollary 4.16]). Then let us consider the trilinear form $$T: C(K_1) \times C(K_2) \times C(K_2) \mapsto \Bbb K$$ defined by $$T(f,g,h)=\sum_{n=1}^{\infty} \xi_n(f) q(g)_n q(h)_n$$ and let us consider the linear operator $$\hat{T}^1:C(K_1) \hat{\otimes} C(K_2) \mapsto
(C(K_2))^*$$ canonically associated to it given by $$\hat{T}^1(f\otimes g)(h)=T(f,g,h) \quad.$$

It is clear that $\hat{T}^1 = q^* \circ \psi \circ \phi $ where $$\phi:C(K_1) \hat{\otimes} C(K_2)\mapsto \ell_2$$ is given by $$\phi(f\otimes g) = (\xi_n(f) q(g)_n)_n\quad.$$ and $\psi\in\mathcal L(\ell_2;\ell_2^*)  $ is the canonical linear isometry identifying both spaces. Since $\psi$, $\phi$ and $q^*$ are all of them weakly compact, so is $\hat{T}^1$.

So we now just have to see that $\hat{T}^1$ is not completely continuous. Let us consider a sequence of bounded functions $(g_n)_n\subset C(K_2)$ such that $q(g_n)=e_n$, where $(e_n)$ is the canonical basis of $\ell_2$. Then, according to Lemma \[lema\], the sequence $(f_n \otimes g_n)_n\subset
C(K_1)\hat{\otimes}C(K_2)$ weakly converges to zero, but, for each $n\in \Bbb N$, $$\| \hat{T}^1(f_n \otimes g_n)\| \sup_n \|g_n\| \geq |\hat{T}^1
    (f_n \otimes g_n)(g_n)| = |T(f_n, g_n, g_n)| = 1$$ a contradiction.

In [@ArGa], the following definition is stated: a multilinear form $T\in \mathcal L^n(E_1, \ldots, E_n)$ is said to be

regular

if every one of the associated linear operators $$T_1^i: E_i \mapsto \mathcal L^{n-1}(E_1,
    \stackrel{[i]}{\ldots}, E_n)$$ is weakly compact, and this is shown to be equivalent to every one of the associated $(n-1)$-linear operators $$T_{n-1}^i: E_1 \times \stackrel{[i]}{\cdots} \times E_n
     \mapsto \mathcal L(E_i)$$ being weakly compact. From the proof it is clear that, given an $i\in (1,\ldots, k)$, $T_1^i$ is weakly compact if and only if so is $T_{n-1}^i$, but there is not reason to believe that, for not symmetric multilinear forms, the fact that $T_1^i $ is weakly compact should imply that $T_1^j$ is weakly compact, too. The main idea behind our proof is to find a trilinear form such that (using the notation of [@ArGa]) $T_1^1$ is weakly compact but $T_1^2$ is not. In [@GoGu], a trilinear form on $\ell_{\infty}$ is used, which is a slight modification of a trilinear form defined in [@ArChLl]. This form does exactly what we want it to do.

Although $\hat{T}^1$ is not completely continuous when considered as a linear operator, its bilinear counterpart $$T^1: C(K_1) \times
C(K_2) \mapsto (C(K_2))^*$$ given by $$T^1(f,g)(h)=T(f,g,h)$$

is

completely continuous, according to the usual definition of completely continuous

bilinear

operator (i.e., if $(f_1^n)\subset C(K_1)$ and $(f_2^n)\subset C(K_2)$ are weakly Cauchy sequences, then $T^1(f_1^n, f_2^n)$ is a norm Cauchy sequence), as follows from [@Pe2] (indeed, it follows from [@Vi] that

every

bilinear continuous operator from $C(K_1) \times C(K_2)$ into $(C(K_2))^*$ is completely continuous). This proves a conjecture of [@Vi] that states that the fact that a multilinear operator from the product of $C(K)$ spaces is completely continuous (considered as a multilinear mapping) does not imply that the same operator, when considered as a linear mapping from the projective tensor product of the spaces, has to be completely continuous.

It is well known that, for every Banach spaces $E_1, \ldots, E_n$, $(n>1)$, $E_1 \hat{\otimes} \cdots \hat{\otimes} E_{n-1}$ is complemented in $E_1 \hat{\otimes} \cdots \hat{\otimes} E_{n}$. Using this, the next corollary follows easily.

Let $K_1,\ldots K_n$ be infinite compact Hausdorff spaces. Then $C(K_1)\hat{\otimes}\cdots\hat{\otimes}C(K_n) $ has the DPP if and only if $K_1$,…,$K_n$ are all scattered.

The theorem and corollary above remain true for the

symmetric

projective tensor product.

Let $K$ be a compact Hausdorff space. Then $C(K)\hat{\otimes}_s
C(K)$ has the DPP if and only if $K$ is scattered.

If $K$ is scattered, then $(\hat{\otimes}_{n,s} C(K))^*$ is a Schur space for every $n\in \Bbb N$, and therefore $C(K)\hat{\otimes}_s
C(K)$ has the DPP. Now, if $K$ is not scattered, we can consider the trilinear form $$T: C(K) \times C(K) \times C(K) \mapsto \Bbb
K$$ defined by $$T(f,g,h)=\sum_{n=1}^{\infty} \frac{1}{2}\left( \xi_n(f) q(g)_n +
\xi_n(g) q(f)_n \right) q(h)_n \quad,$$ that is, the symmetrized respect to the two first variables of the trilinear form used in Theorem \[teoprincipal\]. Now we can apply analogous reasonings as before to conclude that the linear operator $$\hat{T}^1:C(K) \hat{\otimes} C(K) \mapsto
(C(K))^*$$ defined by $$\hat{T}^1(f,g)(h)=T(f,g,h)$$ is weakly compact and not completely continuous.

Again, it is well known (see [@Bl]) that, for every Banach space $E$, $\hat{\otimes}_{n-1,s} E$ is complemented in $\hat{\otimes}_{n,s} E$. So, the following corollary follows.

Let $K$ be a compact Hausdorff space. Then, for every $n>1$, $\hat{\otimes}_{n,s}C(K) $ has the DPP if and only if $K$ is scattered.

As suggested by J. Guti‚rrez, the next theorem can be proved with the same proof as above.

Let $E $ be a Banach space such that

i\) Every linear operator from $E$ into $E^*$ is completely continuous,

ii\) $E$ is not Schur, and

iii\) $E$ has an isomorphic copy of $\ell_1$,

then $E\hat{\otimes}E$ and $E\hat{\otimes}_s E$ do not have the DPP.

There are several spaces which are not $C(K)$ spaces and verify the conditions of the proposition above, among then, the space $H^\infty$ [@Bo1; @Bo2], the disc algebra [@De; @Sa], the space of analytic uniformly convergent Fourier series on the unit circle [@Sac].

The authors want to thank Ver¢nica Dimant and Joaqu¡n Guti‚rrez for some clarifying conversations.

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