Operators preserving $\ell _\infty$

Abstract

Let $Y$ be a Banach space, let the space $\ell _\infty$ be real, let $W$ denote the Banach space $\ell _\infty /c_0,$ and let $Q$ denote the quotient map $\ell _\infty \rightarrow W.$ In 1981, Partington proved there is a topological embedding $J$ of $\ell _\infty$ into $W$ such that the composition $QJ$ is an isometry; in particular, $Q$ preserves $\ell _\infty .$ In this paper we prove that if the kernel of an operator $T:\ell _\infty /c_0 \rightarrow Y$ does not contain an isometric copy of $c_0$ (in particular, if $T$ is injective), then $T$ preserves $\ell _\infty ,$ and hence $T$ is non-weakly compact. This, in turn, allows us to extend Partington’s theorem: we show that natural quotient mappings of some real function spaces preserve $\ell _\infty .$ We also remark that our results apply to some quotients of both Orlicz and Marcinkiewicz spaces.

1 Introduction

In what follows we use notations from the abstract. For notions and notations undefined here we refer the reader to the monographs [2, 3]. All operators are linear and continuous, spaces are of infinite dimension, and $\Gamma$ denotes an infinite set endowed with the discrete topology.

The present paper deals with operators preserving the real Banach space $\ell _\infty$ or, what comes to the same thing, real $C(\beta \mathbf{N}),$ where $\beta \mathbf{N}$ is the Čech–Stone compactification of the discrete space of positive integers $\mathbf{N}.$

Let $X,Y,V$ be Banach spaces, and let $K$ be a compact Hausdorff space. An operator $T:X\rightarrow Y$ preserves [isometrically] $V$ if there is a subspace $U$ of $X,$ isomorphic [isometric, resp.] to $V,$ such that the restricted operator $T_{|U}$ is an isomorphism [isometry, resp.]. The first theorem on such operators is due to Pełczyński [5]. In 1962, he proved that

• (*) Every non-weakly compact operator $T:C(K)\rightarrow Y$ preserves $c_0.$

In 1968 Rosenthal strengthened partially the above-cited Pełczyński’s theorem by showing that

• (**) Every non-weakly compact operator $T:C(\beta \Gamma )=\ell _\infty (\Gamma )\rightarrow Y$ preserves $\ell _\infty$; in particular, the quotient map $Q:\ell _\infty \rightarrow \ell _\infty /c_0$ preserves $\ell _\infty$

(see [7]; cf. [2, Proposition 2.f.4]). In 1970 he published the classical by now result [8, Proposition 1.2 and Remark 1 on p. 30]:

• (R) If an operator $T: \ell _\infty (\Gamma )\rightarrow Y$ is such that $\inf \{\Vert Te_\gamma \Vert :\gamma \in \Gamma \}>0,$ where $\{e_\gamma :\gamma \in \Gamma \}$ is the unit vector basis of $c_0(\Gamma ),$ then there exists $\Gamma ^{\prime }\subset \Gamma$ with $\text{ card}(\Gamma ^{\prime })=\text{ card}(\Gamma )$ such that $T_{|\ell _\infty (\Gamma ^{\prime })}$ is an isomorphism.

Here $\ell _\infty (\Gamma ^{\prime })$ denotes the closed subspace of $\ell _\infty (\Gamma )$ consisting of the elements with support included in $\Gamma ^{\prime }.$

Remark 1

It is easy to see that if, in the hypothesis of $(R),$ the unit vector basis $\{e_\gamma :\gamma \in \Gamma \}$ is replaced by any norm-bounded subset $\{u_\gamma :\gamma \in \Gamma \}$ of $\ell _\infty (\Gamma )$ whose elements are pairwise disjoint, then $T$ is an isomorphism on a subspace $U$ of $\ell _\infty (\Gamma ),$ isomorphic to $\ell _\infty (\Gamma ),$ spanned on the set $\{u_\gamma :\gamma \in \Gamma ^{\prime }\}$ as in $(P)$ below.

In 1981, Partington added a new information to the Rosenthal result (**): he proved that every automorphism of the quotient space $\ell _\infty /c_0$ preserves $\ell _\infty$ in a particular way [4, Theorem 1]:

• (P) For the real Banach space $\ell _\infty$ and every norm $\Vert | \Vert |$ on $W=\ell _\infty /c_0,$ equivalent to the usual (quotient) norm, there exist pairwise disjoint and norm-bounded elements $\{u_n: n\in \mathbf{N}\}$ in $\ell _\infty$ such that for every $(a_n)\in \ell _\infty$ we have

  $$\begin{aligned} \left\| \left| Q\left( (p) \sum _{n=1}^\infty a_nu_n\right) \right\| \right| = \sup _{n\ge 1}|a_n|, \end{aligned}$$

  (1)

  where $(p) \sum _{n=1}^\infty a_nu_n$ denotes the formal pointwise sum of the $a_nu_n$ in $\ell _\infty .$

The following result of the present author extends partially Partington’s theorem $(P)$ to the Banach lattice-case (see [11, Proof of Theorem 1.1 and Remark on p. 3006]; cf. [12, Proof of Proposition 1]):

• (W) Let $E$ be a Dedekind complete Banach lattice, and let $M$ be its closed ideal that does not contain a copy of $\ell _\infty .$ If $E$ contains a lattice-topological [lattice-isometric, resp.] copy of $\ell _\infty (\Gamma ),$ then the quotient map $E\rightarrow E/M$ preserves [lattice-isometrically, resp.] $\ell _\infty (\Gamma ).$ In particular, the quotient map $Q:\ell _\infty (\Gamma )\rightarrow \ell _\infty (\Gamma )/c_0(\Gamma )$ preserves $\ell _\infty (\Gamma )$ lattice-isometrically.

(The examples of such pairs of $E$ and $M$ can be found both in the class of Orlicz and in the class of Marcinkiewicz spaces [6, Lemma 1 and remarks before Corollary 7]).

For the results on operators on Banach lattices/spaces preserving $c_0,\ell _1,$ or $\ell _\infty$ see [3, pp. 196–199, 324, 341–343]; the case of operators preserving some $C(K)$-spaces is addressed in the survey article [9, pp. 1579–1593].

In the present paper, we extend and supplement the above-cited theorems on operators preserving $\ell _\infty .$ Our main results are included in Theorems 1 and 2, and their applications to concrete cases are given in Example 1, and Corollaries 4 and 5.

2 The results

Our basic theorem is given below. In what follows, for the set $\Gamma$ fixed, the letter $Q$ denotes the natural quotient map from $\ell _\infty (\Gamma )$ onto $\ell _\infty (\Gamma )/c_0(\Gamma ),$ and $\Vert \Vert _W$ denotes the natural (quotient and lattice) norm on $W:=\ell _\infty (\Gamma )/c_0(\Gamma ).$

Theorem 1

Let $W$ denote [an isomorphic copy of] the real space $\ell _\infty (\Gamma )/c_0(\Gamma ).$ If the kernel of an operator $T:W \rightarrow Y$ does not contain an isometric [isomorphic, resp.] copy of $c_0$ then $T$ preserves $\ell _\infty .$ In particular, $T$ is non-weakly compact.

Proof

Since, by Partington’s result $(P),$ the isomorphic case follows from the isometric case, we shall assume without loss of generality that $W$ is endowed with the norm $\Vert \Vert _W.$ Let us fix $\Gamma .$

Let us consider first the case $\Gamma$ countable. Without loss of generality we set $W=\ell _\infty /c_0.$ Our goal is to show that there is a pairwise disjoint and norm-bounded sequence $(u_n)$ in $W$ such that $\inf _{n\ge 1}\Vert T(Qu_n)\Vert _Y>0$; then we shall apply Rosenthal’s theorem $(R).$

We define a new norm $\Vert | \Vert |$ on $W$ by the formula

$$\begin{aligned} \Vert | w\Vert |:= \Vert w\Vert _W + \Vert Tw\Vert _Y, \end{aligned}$$

(2)

where $\Vert \Vert _Y$ is a norm on $Y.$ Now we apply $(P)$ and choose a sequence $(u_n)$ in $\ell _\infty$ as in (1), and we set $x_n:=Qu_n, n=1,2,\ldots .$ Hence

$$\begin{aligned} 1=\Vert | x_n\Vert | = \Vert x_n\Vert _W+\Vert Tx_n\Vert _Y, \quad n=1,2,\ldots . \end{aligned}$$

(3)

By $(P),$ the sequence $(u_n)$ consists of pairwise disjoint elements and “spans” a copy $U$ of $\ell _\infty$ in $\ell _\infty$ in the same way as in $(P)$; we thus have

$$\begin{aligned} \left\| \left| Q\left( (p) \sum _{n=1}^\infty a_nu_n\right) \right\| \right| = \sup _{n\ge 1}|a_n|, \end{aligned}$$

(4)

for every $(a_n)\in \ell _\infty .$ So that $Q(U)$ is an isometric copy of $\ell _\infty$ in $W.$ Now we claim that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\Vert Tx_n\Vert _Y= \limsup _{n\rightarrow \infty }\Vert TQ(u_n)\Vert _Y >0. \end{aligned}$$

(5)

Let us suppose the contrary, i.e.,

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert TQ(u_n)\Vert _Y=0. \end{aligned}$$

(6)

Let $(N_k)$ be a sequence of infinite and pairwise disjoint subsets of $\mathbf{N}$ such that $\mathbf{N}=\bigcup _{k=1}^\infty N_k,$ and let us set

$$\begin{aligned} y_k:=Q\left( (p)\sum _{n\in N_k} u_n\right) , \quad k=1,2,\ldots . \end{aligned}$$

(7)

By (4), the elements $y_k$ are well defined (in $W$) and $\Vert |y_k |\Vert =1$ for all $k.$ Since $Q$ is a lattice homomorphism and the elements $(u_n)$ are pairwise disjoint, we have

$$\begin{aligned} |y_k|&= \left| Q\left( (p) \sum _{n\in N_k} u_n\right) \right| = Q\left( \left| (p) \sum _{n\in N_k} u_n \right| \right) \\&= Q\left( (p) \sum _{n\in N_k} |u_n| \right) \ge Q(|u_n|)= |Qu_n|=|x_n|, \end{aligned}$$

for all $n\in N_k$ (the latter inequality $\ge$ follows from the inequality $|a+b|\ge |a|$ for the elements $a$ and $b$ disjoint).

Hence, by (2) (and since $\Vert \Vert _W$ is a lattice norm), we obtain

$$\begin{aligned} 1=\Vert | y_k|\Vert \ge \Vert y_k\Vert _W \ge \Vert x_n\Vert _W, \quad {\text{ for} \text{ all} }\ n\in N_k. \end{aligned}$$

(8)

By (2), (3), (6) and (8), we obtain that $1\ge \Vert y_k\Vert _W\ge \lim _{n\rightarrow \infty }\Vert x_n\Vert _W=1,$ i.e.,

$$\begin{aligned} \Vert y_k\Vert _W=1, \quad {\text{ for} \text{ all} }\ k=1,2,\ldots , \end{aligned}$$

(9)

whence, by (2) again, $Ty_k=0$ for all $k.$ But, by (7) and (9), the elements $(y_k)$ are pairwise disjoint in $W=C(\beta \mathbf{N}{\setminus }\mathbf{N})$ and are of norm one, thus they span an isometric copy of $c_0.$ Hence the kernel of $T$ contains an isometric copy of $c_0,$ but this contradicts the hypothesis of our theorem. Thus our claim (6) is false, so (5) must be true.

Now we apply the general case of Rosenthal’s result $(R)$ (see Remark 1) to the space $U_1$ isomorphic to $\ell _\infty$ and spanned—as in $(P)$—by an infinite subsequence $(u_{n_j})$ of $(u_n)$ such that $\inf _{j\rightarrow \infty }\Vert TQ(u_{n_j})\Vert _Y>0$; and without loss of generality we may assume that

$$\begin{aligned} {\text{ the} \text{ operator} }\ TQ { \text{ acts} \text{ on} }\ U_1 { \text{ as} \text{ an} \text{ isomorphism}}. \end{aligned}$$

Set $Q_1:=Q_{|U_1},W_1:=Q_1(U_1),T_1:=T_{|W_1},$ and let $S$ denote the composition $T_1\circ Q_1.$ We thus have that $U_1$ and $W_1$ are isomorphic copies of $\ell _\infty ,$ and that $S$ is an isomorphism from $U_1$ onto $Y_1:=T_1(W_1)$. Moreover, by (4), $Q_1$ is an isomorphism from $U_1$ onto $W_1$.

From all this follows that $T_1=T_{|W_1}=S\circ Q_1^{-1}$ is an isomorphism from $W_1$ (=an isomorphic copy of $\ell _\infty$) into $Y.$ In other words, $T$ preserves $\ell _\infty .$

If the set $\Gamma$ is uncountable, then the space $\ell _\infty (\Gamma )/c_0(\Gamma )$ contains an isometric copy of $\ell _\infty /c_0$ (see [13, Corollary 2]). Now we apply the just proved result for $\Gamma$ countable.

Now let us consider the case when the operator $T: \ell _\infty (\Gamma )/c_0(\Gamma )\rightarrow Y$ is injective. If $\Gamma$ is countable, Theorem 1 implies that $T$ preserves $\ell _\infty .$ Moreover, if $\Gamma$ is uncountable, from the above-cited results $(W)$ and $(R)$ we obtain immediately that $T$ preserves $\ell _\infty (\Gamma ).$ More exactly: from the proof of Theorem 1 it follows that, in each of the either cases, there is a set $\{u_\gamma : \gamma \in \Gamma \}$ of pairwise disjoint and norm bounded elements of $\ell _\infty (\Gamma )$ such that $\inf _{\gamma \in \Gamma }\Vert TQ(u_\gamma )\Vert _Y>0.$

Hence, by Theorem 1, we have the following result.

Corollary 1

Every injective operator $T: \ell _\infty (\Gamma )/c_0(\Gamma )\rightarrow Y$ preserves $\ell _\infty (\Gamma )$: we have

$$\begin{aligned} \inf _{\gamma \in \Gamma }\Vert TQ(u_\gamma )\Vert _Y>0, \end{aligned}$$

where $(u_\gamma )_{\gamma \in \Gamma }$ are pairwise disjoint and norm bounded elements of $\ell _\infty (\Gamma )$; hence Rosenthal’s result $(R)$ applies to the operator $T$ and the family $(Qu_\gamma )_{\gamma \in \Gamma }.$

Moreover, if $Y$ is a WCG-space, then the kernel of every operator $T:\ell _\infty /c_0\rightarrow Y$ contains an isometric copy of $c_0$; in particular, $T$ is not injective.

(The second part of Corollary 1 follows from the well known fact that a weakly compactly generated (WCG) Banach space cannot contain an isomorphic copy of $\ell _\infty$).

In the next corollary we supplement Rosenthal’s result $(R)$ in a particular case of the condition $\inf _{\gamma \in \Gamma }\Vert Te_\gamma \Vert = 0.$

Corollary 2

Let $T$ be an operator from $\ell _\infty (\Gamma )\rightarrow Y$ such that $\mathrm{ker}T=c_0(\Gamma ).$ Then $T$ preserves $\ell _\infty (\Gamma ).$

Proof

The operator $\overline{T}:\ell _\infty (\Gamma )/c_0(\Gamma ) \rightarrow Y$ of the form $\overline{T}(Qx):=Tx,$ is injective. By Corollary 1, from the form of $\overline{T}$ we obtain $\inf _{\gamma \in \Gamma }\Vert Tu_\gamma \Vert _Y>0,$ where $\{u_\gamma : \gamma \in \Gamma \}$ is a norm-bounded subset of $\ell _\infty (\Gamma )$ whose elements are pairwise disjoint. By Remark 1, the operator $T$ preserves $\ell _\infty (\Gamma ).$

The following example illustrates Corollary 2 (for another application of this corollary see the proof of Theorem 2).

Example 1

Let $c$ denote the Banach space of all real convergent sequences, let $y^*$ be the “lim” functional on $c,$ and let $x^*$ be any fixed continuous extension of $y^*$ to $\ell _\infty .$ Further, let $\mathcal F$ denote the set (with $\text{ card}(\mathcal F )= 2^{\aleph _0}$) of all strictly increasing functions $f:\mathbf{N}\rightarrow \mathbf{N},$ and let $\xi$ be any function $\mathcal F \rightarrow (0,1].$ By $x_f^*$ we denote the element of $\ell _\infty ^*$ defined by the formula $x_f^*(x):=x^*(x\circ f)$; here $x\in \ell _\infty ,$ and $(x\circ f)(n)=x(f(n)), n\ge 1.$ The operator $T_\xi :\ell _\infty \rightarrow \ell _\infty (\mathcal F )$ defined by the formula

$$\begin{aligned} T_\xi (x)=(\xi (f)\cdot x_f^*(x))_{f\in \mathcal F } \end{aligned}$$

is obviously continuous, and it is easy to check that $\text{ ker}T_\xi = c_0.$ From Corollary 2 we obtain that $T_\xi$ preserves $\ell _\infty .$

The next two corollaries also follow from Corollary 1.

Let $Y$ be a closed subspace of a Banach space $X,$ and set $W:=\ell _\infty (\Gamma )/c_0(\Gamma ).$ Let $S:\ell _\infty (\Gamma )$ be an operator such that $S(c_0(\Gamma ))=Y\cap \text{ Im}S,$ and let $q$ and $Q$ denote the natural quotient maps $\ell _\infty (\Gamma )\rightarrow W$ and $X\rightarrow X/Y,$ respectively. By [6, Theorem 2], the induced operator $R : W\rightarrow X/Y$ defined by the rule $R \circ q = Q \circ S$ is injective. From Corollary 1 we thus obtain:

Corollary 3

With the notations as above, and with the hypothesis

$$\begin{aligned} S(c_0(\Gamma ))=Y\cap \text{ Im}S, \end{aligned}$$

the quotient space $X/Y$ contains an isomorphic copy of $\ell _\infty .$

Now let us consider the quotient space $X^{**}/\iota (X),$ where $\iota$ denotes the canonical embedding of $X$ into $X^{**}.$ Assume there is an isomorphic embedding $S_0$ of $c_0$ into $X.$ Then $S:=S_0^{**}$ embeds $\ell _\infty$ into $X^{**}$ with

$$\begin{aligned} \iota (X)\cap \text{ Im}S = \iota (S_0(c_0))=S(c_0). \end{aligned}$$

By [6, Corollary 2], for every separable subspace $V$ of $\ell _\infty$ containing a copy of $c_0,$ the quotient space $X=\ell _\infty /V$ contains a copy of $c_0(\Gamma ),$ where $\text{ card}(\Gamma )= 2^{\aleph _0}.$ Hence, by Corollary 3, we obtain:

Corollary 4

If $X$ contains a copy of $c_0,$ then the quotient space $X^{**}/\iota (X)$ contains an isomorphic copy of $\ell _\infty .$

In particular, for the Banach space $X=\ell _\infty /C[0,1],$ the quotient space $X^{**}/X$ contains an isomorphic copy of $\ell _\infty .$

The next theorem has an application to some quotient mappings.

Theorem 2

Let $U$ be a closed subspace of a Banach space $X.$ If $U$ is isomorphic to $\ell _\infty$ and $T:X\rightarrow Y$ is an operator such that the subspace $U_0:= U\cap \mathrm{ker}T$ is isomorphic to $c_0,$ then $T$ preserves $\ell _\infty .$

Proof

Let $R$ and $S$ be two isomorphisms from $\ell _\infty$ onto $U$ and from $c_0$ onto $U_0,$ respectively. Then the subspace $X_0:=R^{-1}(U_0)$ of $\ell _\infty$ is isomorphic to $c_0,$ and the operator $\tau _0:=R^{-1}S$ maps $c_0$ onto $X_0.$ By [2, Theorem 2.f.12(i)], there is an automorphism $\tau$ of $\ell _\infty$ extending $\tau _0.$ It follows that

$$\begin{aligned} {\text{ the} \text{ operator} }\ \widetilde{S}:=R \tau { \text{ is} \text{ an} \text{ extension} \text{ of} }\ S, { \text{ and} }\ \widetilde{S} \ {\text{ maps} }\ \ell _\infty \ {\text{ onto} }\ U. \end{aligned}$$

(10)

Now let us consider the restricted operator $T_{|U}.$ By (10), the composition $T_{|U}\widetilde{S}$ maps $\ell _\infty$ into $Y,$ and $\text{ ker}(T_{|U}\widetilde{S})= c_0$ because $\text{ ker}(T_{|U})=U_0.$ By Corollary 2, the space $\ell _\infty$ contains an isomorphic copy $V$ of $\ell _\infty$ such that $T_{|U}\widetilde{S}$ acts on $V$ as an isomorphism. But since $\widetilde{S}$ is an isomorphism and the subspace $U_1:=\widetilde{S}(V)$ of $U$ is also a copy of $\ell _\infty ,$ the operator $T$ acts on $U_1$ as an isomorphism; that is, $T$ preserves $\ell _\infty ,$ as claimed.

In the corollary below, we show how Theorem 2 works in concrete cases.

Let $X$ denote one of the real Banach spaces, endowed with the “sup”-norm and built on the interval $[0,1]$: $X=\mathcal L _\infty ^b[0,1]$—of all bounded and Lebesgue-measurable functions, or $X=\mathcal B _1^b[0,1]$—of all bounded functions of Baire class one. In [6, Corollary 5] it has been proved that the quotient space $X/C[0,1]$ contains a complemented copy of $\ell _\infty /c_0.$ From Theorem 2 we obtain additional information about the quotient map $q: X\rightarrow X/C[0,1].$

Corollary 5

With the notations as above, $q$ preserves $\ell _\infty .$

Proof

Let $(x_n)$ be a sequence of positive, pairwise disjoint and norm-one elements of $C[0,1].$ Then the closed linear span $U_0\!:=\![x_n]$ of $(x_n)$ is an isometric copy of $c_0,$ and its pointwise “span” $U,$ defined in $X$ as in $(P),$ is isometric to $\ell _\infty .$ Moreover, by Dini’s theorem, the pointwise sum $(p)\sum _{n=1}^\infty t_nx_n,$ where $t_n\ge 0, n\!=\!1,2,\ldots ,$ lies in $C[0,1]$ if and only if $t_n\!\rightarrow \! 0$ as $n\!\rightarrow \! \infty .$ It follows that $U\cap \text{ ker}q \!=\! U_0,$ and hence, by Theorem 2, $q$ preserves $\ell _\infty .$

3 Further applications of Theorem 1

Let $E$ denote a real Banach lattice and let $E_a$ be the order continuous part of $E$ (for unexplained in this section notions and fundamental facts concerning the theory of Banach lattices and Orlicz and Marcinkiewicz spaces we refer the reader to [1–3, 10]; cf. [6, Section 4]). In [13, Theorem 1 and Corollary 3], it has been proved that if $E$ is Dedekind complete with $E\not = E_a$ then the quotient Banach lattice $E/E_a$ contains an isomorphic (or isometric) copy of $W=\ell _\infty /c_0$ (cf. [6, Corollary 6]).

For example, if $\ell _\phi$ denotes a sequence Orlicz space such that the Orlicz function $\phi$ does not fulfil the $\Delta _2$-condition at $0,$ then the quotient space $\ell _\phi /h_\phi$ contains an isometric copy of $W$ (see [6, Corollary 7]).

Similarly, if $\Psi$ denotes a Marcinkiewicz function, then the quotient Marcinkiewicz space $M(\Psi )/M_0(\Psi )$ contains a copy of $W$ whenever $M(\Psi )\not = M_0(\Psi )$ (see [6, Corollary 9]).

Hence, by Theorem 1 and Corollary 1, we obtain:

Corollary 6

Let $E$ be a Dedekind complete Banach lattice such that $E\not = E_a$ (i.e., the norm on $E$ is not order continuous). If the kernel of an operator $T:E/E_a\rightarrow Y$ does not contain an isomorphic copy of $c_0$ (e.g., if $T$ is injective), then $T$ preserves $\ell _\infty .$

In particular, every injective operator $T:E/E_a\rightarrow Y$ is non-weakly compact.