# Quotients, $\ell_{\infty}$ and abstract Cesàro spaces

Tomasz Kiwerski  Institute of Mathematics, Poznan University of Technology,
Piotrowo 3A, 60-965 Poznań, Poland<EMAIL_ADDRESS>,  Paweł Kolwicz
Institute of Mathematics, Poznan University of Technology, Piotrowo 3A, 60-965
Poznań, Poland<EMAIL_ADDRESS>and  Jakub Tomaszewski  Institute
of Mathematics, Poznan University of Technology, Piotrowo 3A, 60-965 Poznań,
Poland<EMAIL_ADDRESS>
###### Abstract.

Investigating some re-arrangement properties of the norm in the quotient
spaces $X/X_{a}$ we determine the properties of the spaces $X$ guaranteeing
the existence of a lattice isometric copy of $\ell_{\infty}$ in the abstract
Cesàro spaces $CX$.

Dedicated to the memory of

Professor Henryk Hudzik

(1945–2019)

00footnotetext:  Date: August 27, 2024. 2020 Mathematics Subject
Classification: Primary 46E30; Secondary 46B20, 46B42. Key words and phrases:
Banach lattices; Banach ideal spaces; rearrangement invariant (symmetric)
spaces; Cesàro spaces; Cesàro (Hardy) operator; lattice isometric copies of
$\ell_{\infty}$; order ideals; order continuity; quotients. Research (all
authors) supported by Ministry of Science and Higher Education of Poland,
grant number 0213/SIGR/2154.

##  1\. Introduction and preliminaries

Let ${\mathscr{P}}$ be a property for which it makes sense to aks whether a
Banach space has this property or not. For a Banach ideal space $X$ we can now
pose the following problem: what condition, say $\varepsilon$, should we
impose on the space $X$ so that the implication

($\Delta$) $X\in({\mathscr{P}})+\varepsilon\Longrightarrow
CX\in({\mathscr{P}})$

holds true? Here $CX$ stands for the Cesàro space which can be viewed, for
example, as an optimal domain for the Hardy operator $C\colon f\mapsto
C(f)(x)\coloneqq\frac{1}{x}\int_{0}^{x}f(t)dt$, that is, the biggest in the
sense of inclusion Banach ideal space such that the operator $C$ with fixed
codomain space $X$ is still bounded. We should probably point out that there
is really no reason (maybe, beyond some presonal interests) to restrict this
question to the space $CX$, because the Cesàro spaces themselves constitute
only a special, but important, subclass of a very broad family of spaces
generated via sublinear operators (see [Ast12], [CR06], [CR16], [DS07],
[HW06a], [HW06b] and [Mas91]); this class of spaces includes, for example,
Köthe–Bochner spaces, real interpolation spaces, extrapolation spaces, Besov
spaces, Triebel–Sobolev spaces and some other optimal domains associated with
the kernel operators (like multiplication operator, Volterra operator, Cesàro
operator, Copson operator, Poisson operator or Riemann–Liouville operator),
differential operators, convolutions, the Fourier transform, the (finite)
Hilbert transform and the Sobolev embedding (see [ORS08] and references given
there). Many, more or less classical results, can be seen through the prism
($\Delta$) of the above scheme, where the Cesàro operator $C$ is replaced by
the appropriate sublinear operator, say, $S$; let us give few concrete
examples of this type of results, which, however, remain close enough to the
Cesàro spaces: (a) Astashkin [Ast12] (solving the problem posed in [Mas91])
proved that every non-trivial subspace of a Banach space $D_{p}(S)$ generated
by some positive sublinear operator $S$ and an $L_{p}$-space with $1\leqslant
p<\infty$ contains, for any $\varepsilon>0$, an $(1+\varepsilon)$-copy of
$\ell_{p}$ which is $(1+\varepsilon)$-complemented in $D_{p}(S)$ (note that
this result generalizes the well-known Levy result for the Lions–Petree
interpolation spaces $(X_{0},X_{1})_{\theta,p}$ and, on the other hand, the
earlier Astashkin and Maligranda result [AM09] for the classical Cesàro
function spaces $Ces_{p}$); (b) Hudzik and Wlaźlak in [HW06a] and [HW06b]
studied various convexity and monotonicity properties, say ${\mathscr{G}}$, of
the space $D_{E}(S)$, where $E$ is a Banach ideal space. Essentially, they
were able to show that if $E\in({\mathscr{G}})$ and an injective positive
sublinear operator $S$ has some special geometric properties corresponding to
the property ${\mathscr{G}}$, then also $D_{E}(S)\in({\mathscr{G}})$. This
result implies immediate applications for the Köthe–Bochner spaces and the
Cesàro–Orlicz spaces, cf. [KK18a] for a more direct approach in the case of
different types of the Cesàro function spaces; (c) Mastyło in [Mas91] and
[Mas92] investigated some structural properties of the space $D_{E}(S)$.

In this paper we give a general and rather natural condition to ensure that
the Cesàro space $CX$ contains a lattice isometric copy of $\ell_{\infty}$, as
long as a rearrangement invariant space $X$ contains such a copy as well.

A Banach space $(X,\lVert\cdot\rVert_{X})$ is said to be a Banach lattice if
there is a lattice order on $X$, say $\leqslant$, such that for $x,y\in X$
with $\left\lvert x\right\rvert\leqslant\left\lvert y\right\rvert$ we have
$\lVert x\rVert_{X}\leqslant\lVert y\rVert_{X}$. By $X_{+}$ we denote the
positive cone of a Banach lattice $X$, that is, $X_{+}=\\{x\in X\colon
x\geqslant 0\\}$.

A mapping $T$ between two Banach lattices $X$ and $Y$ is said to be a lattice
(or an order) isomorphism if it is a linear topological isomorphism which
preserves the order $\leqslant$. If, additionally, the operator norm of a
lattice isomorphism $T\colon X\rightarrow Y$ is equal one, then we will
emphasize this fact by calling $T$ a lattice (or an order) isometry.

Let $(\Omega,\Sigma,\mu)$ be a complete $\sigma$-finite measure space. Denote
by $L_{0}(\Omega)=L_{0}(\Omega,\Sigma,\mu)$ the set of all (equivalence
classes of) real-valued $\mu$-measurable functions defined on a measure space
$(\Omega,\Sigma,\mu)$. A Banach ideal space $X=(X,\lVert\cdot\rVert_{X})$ on
$(\Omega,\Sigma,\mu)$ is understood to be a Banach space $X$ such that $X$ is
a linear subspace of $L_{0}(\Omega,\Sigma,\mu)$ satisfying the so-called ideal
property, which means that if $f,g\in L_{0}(\Omega,\Sigma,\mu)$, $\left\lvert
f(t)\right\rvert\leqslant\left\lvert g(t)\right\rvert$ $\mu$-almost everywhere
on $\Omega$ and $g\in X$, then $f\in X$ and $\lVert f\rVert_{X}\leqslant\lVert
g\rVert_{X}$. If it is not stated otherwise we assume that a Banach ideal
space $X$ on $(\Omega,\Sigma,\mu)$ contains a function which is positive
$\mu$-almost everywhere on $\Omega$ (such a function is called the weak unit
in $X$). For a function $f\in L_{0}(\Omega,\Sigma,\mu)$ we define a support of
$f$ as a set $\operatorname{supp}(f)\coloneqq\\{x\in\Omega\colon f(x)\neq
0\\}$. Moreover, by a support $\operatorname{supp}(X)$ of the Banach ideal
space $X$ on $(\Omega,\Sigma,\mu)$ we mean the smallest (in the sense of
inclusion) $\mu$-measurable subset, say $A$, of $\Omega$ such that
$f\chi_{\Omega\setminus A}=0$ for all $f\in X$. The existence of the weak unit
is equivalent to the following condition that $\operatorname{supp}(X)=\Omega$.
We say that a Banach ideal space $X$ is non-trivial if
$X\neq\left\\{0\right\\}$.

For two Banach ideal spaces $X$ and $Y$ on $\Omega$ the symbol
$X\overset{E}{\hookrightarrow}Y$ means that the embedding $X\subset Y$ is
continuous with the norm not bigger than $E>0$, that is to say, $\lVert
f\rVert_{Y}\leqslant E\lVert f\rVert_{X}$ for all $f\in X$. If the embedding
$X\overset{E}{\hookrightarrow}Y$ holds with some (maybe unknown) constant
$E>0$ we simply write $X\hookrightarrow Y$. Moreover, the symbol $X=Y$
($X\equiv Y$) means that the spaces are the same as a sets and the norms are
equivalent (equal, respectively).

We say that an element $x$ in a Banach lattice $X$ is order continuous (or has
order continuous norm) if for any sequence $\\{x_{n}\\}_{n=1}^{\infty}$ in $X$
satisfying $0\leqslant x_{n}\leqslant\left\lvert x\right\rvert$ and
$x_{n}\downarrow 0$ (that is, $x_{n+1}\leqslant x_{n}$ and
$\inf_{n\in\mathbb{N}}\\{x_{n}\\}=0$) we have $\lVert x_{n}\rVert\rightarrow
0$ as $n\rightarrow\infty$. By $X_{a}$ we denote the subspace of all order
continuous elements of $X$. If $X=X_{a}$, that is, every element of $X$ is
order continuous, then the space $X$ is said to be order continuous.

By $X^{*}$ we denote the topological dual space of a Banach lattice $X$. If
$X^{dd}=X$ then we have the Yosida–Hewitt decomposition of the space $X^{*}$,
namely

$X^{*}=X_{a}^{*}\oplus X^{s},$

where $X_{a}^{*}$ is the space of all order continuous functionals, that is to
say, $x^{*}\in X_{a}^{*}$ if and only if $x^{*}(x_{n})\rightarrow 0$ as
$n\rightarrow\infty$ for any sequnce $\\{x_{n}\\}_{n=1}^{\infty}\subset X_{+}$
and $x_{n}\downarrow 0$, and $X^{s}$ is the space of all singular functionals,
that is to say, $x^{*}\in X^{s}$ if and only if $x^{*}(x)=0$ for any $x\in
X_{a}$. For an order continuous Banach lattice $X$ the spaces $X_{a}^{*}$ and
$X^{*}$ coincide. If $X$ is a Banach ideal space on $(\Omega,\Sigma,\mu)$ such
that $\operatorname{supp}(X_{a})=\operatorname{supp}(X)$ then the space
$X_{a}^{*}$ coincide with the Köthe dual space (or associated space)
$X^{\prime}$

$X^{\prime}\coloneqq\left\\{f\in L_{0}(\Omega)\colon\sup\limits_{\lVert
g\rVert_{X}\leqslant 1}\int_{\Omega}\left\lvert f(t)g(t)\right\rvert
d\mu<\infty\right\\}.$

Recall that $X\overset{1}{\hookrightarrow}X^{\prime\prime}$ and we have the
equality $X\equiv X^{\prime\prime}$ if and only if the norm in $X$ has the
Fatou property, that is to say, if the conditions $0\leqslant f_{n}\uparrow
f\in L_{0}$ with $\\{f_{n}\\}_{n=1}^{\infty}\subset X$ and
$\sup_{n\in\mathbb{N}}\|f_{n}\|_{X}<\infty$ imply that $f\in X$ and
$\|f_{n}\|_{X}\uparrow\|f\|_{X}$.

A closed linear subspace $J$ of a Banach lattice $X$ is called an order ideal
(or simply ideal) if it has the so-called ideal property which means that if
$f\in J$, $g\in X$ and $\left\lvert g\right\rvert\leqslant\left\lvert
f\right\rvert$ then also $g\in J$ (subspaces with this property are also
called solid). It is clear that $\\{0\\}$ and the space $X$ itself are order
ideals of $X$. Non-trivial example of an order ideal is the closure (in norm
topology of the Banach ideal space $X$) of the set of simple functions $X_{b}$
(see [BS88, Theorem 3.11, p. 18]) and $X_{a}$, that is, the subspace of all
functions with order continuous norm in $X$ (see [BS88, Theorem 3.8, p. 16]).
An intersection and a sum of any two ideals is again an ideal (cf. [MN91,
Proposition 1.2.2, p. 12]). If $J$ is a closed ideal in a Banach lattice $X$,
then $X/J$ is a Banach lattice with respect to the quotient norm, that is to
say, $\lVert x\rVert_{X/J}=\inf\\{\lVert y\rVert_{X}\colon Q(y)=Q(x)\\}$,
where $Q\colon J\rightarrow X/J$ is the cannonical quotient map (cf. [MN91,
Corollary 1.3.14, p. 32]).

In this paper we will deal with Banach lattices but the main focus will be on
rearrangement invariant spaces (or symmetric spaces), that is to say, the
Banach ideal spaces with the additional property that for any two
equimeasurable functions $f,g\in L_{0}$ (which means that they have the same
distribution functions $d_{f}\equiv d_{g}$, where
$d_{f}(\lambda)\coloneqq\mu(\\{t\in\Omega\colon|f(t)|>\lambda\\})$ for
$\lambda\geqslant 0$) and if $f\in X$ then $g\in X$ and $\|f\|_{X}=\|g\|_{X}$.
Due to the Luxemburg representation theorem, it is enough to consider
rearrangement invariant spaces only on three separable measure spaces, namely,
the unit interval $\left(0,1\right)$ or the semi-axis $\left(0,\infty\right)$
with the Lebesgue measure $m$ (in this case we will talk about function
spaces) and the set of positive integers $\mathbb{N}$ with the counting
measure $\\#$ (and then we will refer to sequence spaces). Moreover, by a non-
increasing rearrangement of a function $f\colon\Omega\rightarrow\mathbb{R}$ we
mean the function $f^{*}\colon[0,\infty)\rightarrow[0,\infty]$ defined as

$f^{\ast}(t)=\mathrm{inf}\\{\lambda>0\colon d_{f}(\lambda)\leqslant t\\}\text{
for }t\geqslant 0,$

under the convention $\mathrm{inf}\\{\emptyset\\}=\infty$. In the sequence
case, however, we need a little modification of the above definition, namely
$x_{n}^{*}=\mathrm{inf}\\{\lambda>0\colon d_{x}(\lambda)<n\\}$, where
$x=\\{x_{n}\\}_{n=1}^{\infty}$.

The fundamental function $\varphi_{X}$ of a rearrangement invariant space $X$
on $\Omega$ is defined by the following formula

$\varphi_{X}(t)=\|\chi_{(0,t)}\|_{X}\text{ for }t>0,$

(with a fairly obvious modification when $\Omega=\mathbb{N}$, i.e.,
$\varphi_{X}(n)=\lVert\sum_{k=1}^{n}e_{k}\rVert_{X}$ for $n\in\mathbb{N}$,
where $\\{e_{n}\\}_{n=1}^{\infty}$ is the canonical basic sequence of $X$)
where $\chi_{A}$, throughout, will denote the characteristic function of a set
$A$. It is well-known that fundamental function is quasi-concave on $\Omega$,
that is: $\varphi_{X}(0)=0$; $\varphi_{X}$ is positive and non-decreasing;
$t\mapsto\varphi_{X}(t)/t$ is non-increasing for $t>0$ or, equivalently,
$\varphi_{X}(t)\leqslant\max\\{1,t/s\\}\varphi_{X}(s)$ for all $s,t\in\Omega$.

For some general properties of Banach ideal and rearrangement invariant spaces
we refer, for example, to [BS88], [KPS82], [LT77], [LT79] and [Mal89]. More
information about Banach lattices can be found, for example, in [AB85], [KA82]
and [MN91]

Let us recall some examples of rearrangement invariant spaces. Each increasing
concave function $\varphi$ on $\Omega$ generates the Lorentz function space
$\Lambda_{\varphi}$ on $\Omega$ endowed with the norm

$\lVert
f\rVert_{\Lambda_{\varphi}}=\int_{\Omega}f^{*}(t)d\varphi(t)=\varphi(0^{+})\lVert
f\rVert_{\infty}+\int_{0}^{m(\Omega)}f^{*}(t)\varphi^{\prime}(t)dt<\infty.$

Recall also that for a quasi-concave function $\varphi$ the Marcinkiewicz
function space $M_{\varphi}$ on $\Omega$ is defined in the following way

$M_{\varphi}=\left\\{f\in L_{0}(\Omega)\colon\lVert
f\rVert_{M_{\varphi}}=\sup_{t\in\Omega}\frac{\varphi(t)}{t}\int_{0}^{t}f^{*}(s)ds<\infty\right\\}.$

Lorentz and Marcinkiewicz spaces play quite a special role among rearrangement
invariant spaces, namely they are the smallest and, respectively, the largest
(in a sense of inclusion) rearrangement invariant spaces with a given
fundamental function. More precisely, for a given rearrangement invariant
space $X$ with the fundamental function $\varphi$ (note that every such a
function is equivalent to a concave function) we have the following embeddings

$\Lambda_{\varphi}\overset{2}{\hookrightarrow}X\overset{1}{\hookrightarrow}M_{\varphi}.$

Let $F$ be an non-decreasing convex function on $[0,\infty)$ such that
$F(0)=0$ and let $X$ be a Banach ideal space on $\Omega$. The
Calderón–Lozanovskiĭ space $X_{F}$ on $\Omega$ is defined as the space of all
measurable functions $f\colon\Omega\rightarrow\mathbb{R}$ for which the
followng norm, the so-called Luxemburg–Nakano norm, is finite

$\lVert f\rVert_{X_{F}}=\inf\left\\{\lambda>0\colon\lVert
F(f/\lambda)\rVert_{X}\leqslant 1\right\\}<\infty.$

The spaces $X_{F}$ were introduced by Calderón [Cal64, p. 122] and Lozanovskiĭ
[Loz65]. This is a Banach ideal space. Moreover, if $X$ is a rearrangement
invariant space with the Fatou property, then the space $X_{F}$ is like that
as well. Note also that in the case when $X=L_{1}$ the space $X_{F}$ is just
the Orlicz space $L_{F}$ equipped with the Luxemburg–Nakano norm. On the other
hand, if $X$ is a Lorentz space $\Lambda_{\varphi}$, then $X_{F}$ is the
corresponding Orlicz–Lorentz space $\Lambda_{F,\varphi}$. It is also clear
that in the special situation, when $F(t)=t$, the Orlicz–Lorentz space
$\Lambda_{F,\varphi}$ coincide, up to equality of norms, with the Lorentz
space $\Lambda_{\varphi}$. Finally, if $F(t)=t^{p}$, where $1\leqslant
p<\infty$, then the space $X_{F}$ is the $p$-convexification $X^{(p)}$ of the
space $X$ equipped with the norm $\lVert f\rVert_{X^{(p)}}=\lVert\left\lvert
f\right\rvert^{p}\rVert_{X}^{1/p}$ and if $F(t)=0$ for $0\leqslant t\leqslant
1$ and $F(t)=\infty$ for $t>1$, then $X_{F}\equiv L_{\infty}$.

In the case of sequence spaces we prefer a slight modification of the
introduced notation, that is, if this make sense, we will use small letters to
denote the space, for example, we will just denote the Lorentz space
$\Lambda_{\varphi}$ on $\mathbb{N}$ as $\lambda_{\varphi}$ and the Orlicz
space $L_{F}$ on $\mathbb{N}$ as $\ell_{F}$, etc.

We will use the notation $A\preccurlyeq B$ or $B\succcurlyeq A$ to denote an
estimate of the form $A\leqslant CB$ for some constant $C>0$ depending on the
involved parameters only. We also write $A\asymp B$ for $A\preccurlyeq
B\preccurlyeq A$. Other notations and definitions will be introduced as
needed.

Now we give a brief overview of the paper.

Section 2 is slightly different in nature to the rest of this article. Here we
revisit some Hudzik’s [Hud98] characterizations of Banach lattices containing
a lattice isometric copies of $\ell_{\infty}$ and complete one of them
(Theorem 2.1). This connection between, on the one hand, the existence of a
lattice isometric copy of $\ell_{\infty}$ in a Banach lattice $X$ and, on the
other, with the fact that the unit sphere in the space $X$ must contain an
element, say $x$, with the property that the distance from $x$ to the order
ideal of all order continuous elements in $X$ is equal exactly one, will be
our starting point and the leitmotif to which we will come back in later parts
of our work.

Section 3 is rather technical in character. In short, we show there that the
quotient spaces $X/X_{a}$, where $X_{a}$ is the ideal of all order continuous
elements in a rearrangement invariant space $X$, behave somewhat similar to a
rearrangement invariant space, that is to say, they have the ideal property
(Corollary 3.2) and they are symmetric (Theorem 3.5). We will also explain how
our formulas are related to some of de Jonge’s earlier results from [deJ77].

In Section 4 we will focus on Cesàro spaces. Using the results from the
previous sections we prove some general theorems about a lattice isometric
copies of $\ell_{\infty}$ in these spaces (Theorem 4.1 and Theorem 4.6) and
justify (which is not so easy) that they really generalize the so far known
results obtained in the class of Cesàro–Orlicz spaces in [KamK09] and [KK18b].

Finally, at the end of this paper, we include Appendix, which was created for
the purposes of §4 (and, at the same time, it complements the results obtained
by the first and last author in [KT17]). It contains a fairly satisfactory
description of the ideal of all order continuous functions in the Cesàro
sequence spaces $CX$ expressed in the language of the ideal $X_{a}$.

##  2\. About isometric copies of $\ell_{\infty}$ one more time

Lozanowskiĭ’s well-known result from [Loz69] gives some geometric
characterization of the order continuity property. More precisely, it states
that a Banach lattice $X$ contains a lattice isomorphic copy of
$\ell_{\infty}$ if and only if the space $X$ is not order continuous. Of
course, every isomorphic copy of $\ell_{\infty}$ is automatically complemented
in the ambient space due to the fact that the space $\ell_{\infty}$ is
isometrically injective. Note also that a Banach lattice $X$ contains a
(lattice) almost isometric copy of $\ell_{\infty}$ whenever it contains a
(lattice) isomorphic copy of $\ell_{\infty}$ (see [HM93, Theorem 1] and
[Par81, Theorem 3]). It seems to be worth comparing this statement with the
classical James result [Jam64] (often called simply the James’s distortion
theorem), which describes a similar phenomenon but for spaces $c_{0}$ and
$\ell_{1}$. Moreover, if a Banach space $X$ contains an asymptotically
isometric copy of $\ell_{\infty}$, then it contains even an isometric copy of
$\ell_{\infty}$ (see [Dow00, Theorem 6] and [CHL17, Theorem 2.5] for a
“lattice version”). However, in general, the lack of order continuity does not
guarantee the existence of any isometric copy of $\ell_{\infty}$. Indeed, let
$X$ and $Y$ be two Banach function spaces on $\Omega$ with non-trivial
intersection $X\cap Y$ such that the first one is not order continuous and the
second one is strictly convex (for example, we can take $X=L_{\infty}$ and
$Y=L_{2}$). Then the space

$\mathcal{X}=X\cap Y\text{ with the norm }\lVert f\rVert_{\mathcal{X}}=\lVert
f\rVert_{X}+\lVert f\rVert_{Y},$

is strictly convex and is not order continuous. Consequently, in view of the
Lozanowskiĭ result [Loz69], it contains a lattice isomorphic copy of
$\ell_{\infty}$ but clearly cannot contain an isometric copy of
$\ell_{\infty}$. Note also that the same trick will work if instead of strict
convexity we will require that the second space is strictly monotone. In this
case, however, the space $Z$ will be strictly monotone and not order
continuous, so it will contains a lattice isomorphic but not lattice isometric
copy of $\ell_{\infty}$. Let us now recall the following classical result.

######  Theorem A (Riesz’s lemma).

If $Y$ is a proper closed linear subspace of a normed space
$X=(X,\lVert\cdot\rVert)$, then for any $0<\varepsilon<1$ there exists $x\in
S(X)$ such that $\emph{d}_{X}(x,Y)\geqslant 1-\varepsilon$.

It can be demonstrated that the above result is in general not true for
$\varepsilon=0$. Actually, for a Banach space $X$ to have the property that
given a proper closed linear subspace $Y$ of $X$ there exists $x\in S(X)$ with
$\emph{d}_{X}(x,Y)=1$ it is necessary and sufficient that the space $X$ is
reflexive (cf. [Dis84, Notes and Remarks on p. 6]). In fact, James [Jam57]
proved that a Banach space $X$ is reflexive if and only if every continuous
linear functional on $X$ is norm attaining. Therefore, if $X$ is nonreflexive,
then there is a linear functional, say $\varphi$, in $S(X^{*})$ which does not
achieve its norm. Taking $Y=\text{ker}(\varphi)$ we see immediately that $Y$
is a proper closed linear subspace of $X$ and there is no such element $x\in
S(X)$ with $\emph{d}_{X}(x,Y)=1$. On the other hand, if $X$ is reflexive and
$Y$ is a closed linear subspace of $X$, then applying the Hahn–Banach theorem
and bearing in mind the result of James we see that there exists $\psi\in
S(X^{*})$ such that $Y\subset\text{ker}(\psi)$ and $x\in S(X)$ with
$\psi(x)=1$. Consequently, we have the following inequalities:

$\lVert
x-y\rVert\geqslant\left\lvert\psi(x)-\psi(y)\right\rvert=\left\lvert\psi(x)\right\rvert=1\text{
for all }y\in Y,$

that is, $\emph{d}_{X}(x,Y)=1$. On the other hand, the existence of an element
realizing the distance from the ideal $X_{a}$, where now $X$ is a Banach ideal
space, is closely related to the fact that the space $X$ contains a lattice
isometric copy of $\ell_{\infty}$. Anyhow, for $X=\ell_{\infty}$ and
$f=\chi_{\mathbb{N}}$ we have $\lVert
f\rVert_{X}=\emph{d}_{X}(\chi_{\mathbb{N}},X_{a}=c_{0})=1$ and, in a sense,
this is the generic case.

######  Theorem B (H. Hudzik, 1998 [Hud98]).

Let $X$ be a super $\sigma$-Dedekind complete Banach lattice with the semi-
Fatou property such that $(X_{a}^{d})^{d}=X$ ***In less general situation,
e.g. when $X$ is a Banach ideal space, this condition simply means that
$\operatorname{supp}(X_{a})=\operatorname{supp}(X)$ (cf. [Hud98, p. 523]).. If
we can find an element $f\in X$ such that $\lVert
f\rVert_{X}=\emph{d}_{X}(f,X_{a})=1$, then $X$ contains a lattice isometric
copy of $\ell_{\infty}$.

Let’s us consider the space
$\mathfrak{X}=(L_{\infty},\lVert\cdot\rVert_{\mathfrak{X}})$, where
$\lVert\cdot\rVert_{\mathfrak{X}}$ is an equivalent norm on $L_{\infty}$ given
by the formula: $\lVert f\rVert_{\mathfrak{X}}=\lVert
f\rVert_{L_{\infty}}+\lVert f\rVert_{Y}$ for $f\in L_{\infty}$, and $Y$ is
strictly convex space such that $L_{\infty}\hookrightarrow Y$. Then the
subspace $\mathfrak{X}_{a}$ is trivial and

$\emph{d}_{X}(f,\mathfrak{X}_{a})=\emph{d}_{X}(f,\\{0\\})=\lVert
f\rVert_{\mathfrak{X}}=1\text{ for all }f\in S(\mathfrak{X}).$

However, the space $\mathfrak{X}$ cannot contain a lattice isometric copy of
$\ell_{\infty}$ as we have already explained above. Therefore the assumption
$(X_{a}^{d})^{d}=X$ in Theorem B about the support of the ideal $X_{a}$ cannot
be omitted in general.

Needless to say, the question whether the condition from Theorem B is also
sufficient imposes itself. We believe that, this fact belongs to folklore but,
as far as we know, it was never explicitly mentioned (even in [Hud98]). To
fill this gap and avoid the impression (which the authors themselves had) that
this condition may only be necessary, we will give a short proof which
completes Hudzik’s result.

######  Theorem 2.1.

Assume that $X$ is a Banach ideal space over a $\sigma$-finite measure space
with the semi-Fatou property such that
$\operatorname{supp}(X_{a})=\operatorname{supp}(X)$. The space $X$ contains a
lattice isometric copy of $\ell_{\infty}$ if and only if there exists an
element $f\in X$ with $\lVert f\rVert_{X}=\emph{d}_{X}(f,X_{a})=1$.

###### Proof.

Using Theorem 1 from [Hud98] we can find a sequence
$\\{f_{n}\\}_{n=1}^{\infty}\subset S(X)$ with
$\operatorname{supp}(f_{n})\cap\operatorname{supp}(f_{m})=\emptyset$ for
$n\neq m$ such that $\lVert\sum_{n=1}^{\infty}f_{n}\rVert_{X}=1$. Take $g\in
X_{a}$ and observe that

$\displaystyle\lVert\sum_{n=1}^{\infty}f_{n}-g\rVert_{X}$
$\displaystyle\geqslant\sup\limits_{n\in\mathbb{N}}\lVert
f_{n}-g\chi_{\operatorname{supp}(f_{n})}\rVert_{X}$
$\displaystyle\geqslant\left\lvert\lVert f_{n}\rVert_{X}-\lVert
g\chi_{\operatorname{supp}(f_{n})}\rVert_{X}\right\rvert\rightarrow 1,$

because $\lVert g\chi_{\operatorname{supp}(f_{n})}\rVert_{X}\rightarrow 0$ as
$n\rightarrow\infty$. ∎

Reffering to Theorem 2.1, if a Banach ideal space $X$ contains a lattice
isometric copy of $\ell_{\infty}$, then we can find $f\in X$ with $\lVert
f\rVert_{X}=\emph{d}_{X}(f,X_{a})=1$. So, looking from a slightly different
perspective, this means that $\lVert f\rVert_{X/X_{a}}=1$ and there exist a
singular functional $S\in S(X^{*})$ with $S(f)=1$ (note only that
$(X/X_{a})^{*}\approx X_{a}^{\perp}=X^{s}$, where $X^{s}$ is the space of all
singular functionals, that is to say, those $S\in X^{*}$ for which $S(x)=0$
for all $x\in X_{a}$). The converse of this statement is also true but before
we give a short proof let us note the following result.

######  Lemma 2.2.

Let $X$ be a Banach lattice. If $S$ is a singular functional on $X$ then we
have the following formula

$\lVert S\rVert_{X^{*}}=\sup\left\\{\frac{S(f)}{\emph{d}_{X}(f,X_{a})}\colon
f\in X\setminus X_{a}\text{ with }\lVert f\rVert_{X}=1\right\\}.$

###### Proof.

First, observe that $S(f)=0$ for all $f\in X_{a}$, so

$\displaystyle\lVert S\rVert_{X^{*}}$ $\displaystyle=\sup\left\\{S(f)\colon
f\in X\text{ with }\lVert f\rVert_{X}=1\right\\}$
$\displaystyle=\sup\left\\{S(f)\colon f\in X\setminus X_{a}\text{ with }\lVert
f\rVert_{X}=1\right\\}$
$\displaystyle\leqslant\sup\left\\{\frac{S(f)}{\emph{d}_{X}(f,X_{a})}\colon
f\in X\setminus X_{a}\text{ with }\lVert f\rVert_{X}=1\right\\},$

where the last inequality follows from the fact that if $\lVert
f\rVert_{X}=1$, then $\emph{d}_{X}(f,X_{a})\leqslant 1$. Therefore, we only
need to prove the reverse inequality. To do this, take $f\in X\setminus X_{a}$
and let $\\{f_{n}\\}_{n=1}^{\infty}\subset X_{a}$ be a sequence that realizes
the distance of the function $f$ from the ideal $X_{a}$, that is to say,
$\lim_{n\rightarrow\infty}\lVert f-f_{n}\rVert_{X}=\emph{d}_{X}(f,X_{a})$.
Now, since $S(f)=S(f-f_{n})$, so

$S(f)\leqslant\lVert S\rVert_{X^{*}}\lVert f-f_{n}\rVert_{X}\rightarrow\lVert
S\rVert_{X^{*}}\emph{d}_{X}(f,X_{a})\text{ as }n\rightarrow\infty.$

Thus, we obtain that

$\lVert S\rVert_{X^{*}}\geqslant\frac{S(f)}{\emph{d}_{X}(f,X_{a})},$

which ends the proof. ∎

Note that the above lemma for Orlicz–Lorentz spaces was proved in [KLT19]
basing on the modular space structure.

Consequently, if there exists a norm attaining singular functional on $X$, say
$S$, then $S(f)=\lVert S\rVert_{X^{*}}$ for some $f\in B(X)$, i.e.,
$\emph{d}_{X}(f,X_{a})=1$ (cf. Lemma 2.2) and the space $X$ contains a lattice
isometric copy of $\ell_{\infty}$ in view of Theorem 2.1. What we have said so
far may be summarized as follows.

######  Proposition 2.3.

Let $X$ be a super $\sigma$-Dedekind complete Banach lattice with the semi-
Fatou property such that $(X_{a}^{d})^{d}=X$. Then $X$ contains a lattice
isometric copy of $\ell_{\infty}$ if and only if there exists a singular
functional on $X$ which attains its norm.

We will now (seemingly) deviate for a moment from the course choosen so far.
In [GH96] (see also [Wnu84]; cf. [SS83]) Granero and Hudzik extended and
completed the results obtained in [LW83] for the space
$\ell_{\infty}/c_{0}$†††As mentioned in [LW83]: The Banach space
$\ell_{\infty}/c_{0}$ certainly falls into the category of a “classical Banach
space”. Not only has it been around since the time of Banach’s original
monograph (1932), but it is also classical in the sense of Lacey or
Lindenstauss and Tzafriri since it is isomorphic to the space
$C(\beta\mathbb{N}\setminus\mathbb{N})$ (just to be clear $\beta\mathbb{N}$,
as usual, is the Čech–Stone compactification of $\mathbb{N}$ endowed with the
discrete topology, that is, $\beta\mathbb{N}$ is the unique compact Hausdorff
space containing $\mathbb{N}$ as a dense subspace so that every bounded
continuous function on $\mathbb{N}$ extends to a continuous function on
$\beta\mathbb{N}$); we also recommend taking a look at[DR91]. considering a
more general construction $\ell_{F}/(\ell_{F})_{a}$, where $\ell_{F}$ is the
Orlicz sequence space (in fact, they even allowed for $\ell_{F}$ to be a
Fréchet space). Of course, we can go a step futher and consider a neo-
classical Banach space $X/X_{a}$, where $X$ is a Banach lattice. Despite this
somewhat baroque name, it is clear that $X/X_{a}$ is sometimes classical, for
example, if $X$ is a rearrangement invariant Banach sequence space with the
Fatou property such that $\varphi_{X}(\infty)<\infty$, then
$X/X_{a}=\ell_{\infty}/c_{0}\approx C(\beta\mathbb{N}\setminus\mathbb{N})$.

######  Proposition 2.4.

Let $X$ be a Banach sequence space such that $X\hookrightarrow\ell_{\infty}$,
the ideal $X_{a}$ is non-trivial and $X_{a}\neq X$. Then $X/X_{a}$ is not a
dual space.

###### Proof.

It is well-known that the space $\ell_{\infty}/c_{0}$ contains an isomorphic
copy of $c_{0}(\Gamma)$, where $\text{card}(\Gamma)=\mathfrak{c}$ and
$\mathfrak{c}$ denotes the cardinal of the continuum, that is,
$\mathfrak{c}=2^{\aleph_{0}}$. Let us briefly remind that we can proceed as
follows: (1) for each irrational number $r$, take a sequence
$\\{q_{n}^{r}\\}_{n=1}^{\infty}$ of distinct rational numbers converging to
$r$ and put $F_{r}=\\{q_{n}^{r}\colon n\in\mathbb{N}\\}$; (2) the sets
$\\{F_{r}\\}_{r\in\mathbb{R}}$ form an uncountable family of infinite sets
such that $F_{r}\cap F_{r^{\prime}}$ is finite whenever $r\neq r^{\prime}$;
(3) observe that $\ell_{\infty}\equiv\ell_{\infty}(\mathbb{Q})$; (4) the
closed linear span of $\\{\pi(\chi_{F_{r}})\colon r\in\mathbb{R}\\}$, where
$\pi\colon\ell_{\infty}\rightarrow\ell_{\infty}/c_{0}$ is the cannonical
quotient map, is isometric to $c_{0}(\mathbb{R})$ (cf. footnote on page 19 in
[Ros70]). Now, if $X/X_{a}$ were a dual space, then thanks to Rosenthal’s
generalization of the classical Bessaga and Pełczyński result [Ros70,
Corollary 1.5], the space $X/X_{a}$ should have a copy of
$\ell_{\infty}(\Gamma)$. But this is impossible, because

$\text{card}(\ell_{\infty}(\Gamma))=2^{\mathfrak{c}}>\mathfrak{c}=\text{card}(\ell_{\infty})\geqslant\text{card}(X)\geqslant\text{card}(X/X_{a}),$

where the second inequality follows from the fact that
$X\hookrightarrow\ell_{\infty}$. ∎

The above result is a direct generalization of [GH96, Proposition 6.2] which
was proved in the case when $X=\ell_{F}$.

Let us now recall the well-known Phillips–Sobczyk theorem, which states that
$c_{0}$ is not complemented in $\ell_{\infty}$ (to be precise, what Phillips
proved was that $c$ is not complemented in $\ell_{\infty}$; look at [CSCY00],
cf. [AK06, pp. 44–48]). We will close this paragraph by re-proving a “lattice”
variant of this theorem and exploring the Bourgain’s approach [Bou80] to this
phenomenon.

######  Proposition 2.5 (G. Ja. Lozanovskiĭ, 1973).

Let $X$ be a Banach ideal space on a complete $\sigma$-finite measure space
$(\Omega,\Sigma,\mu)$ such that $X_{a}\neq X$. Then the subspace $X_{a}$ is
not complemented in $X$ except for the case when $X_{a}$ is trivial.

###### Proof.

Suppose that $X$ is as above and consider the following diagram

$c_{0}\hookrightarrow\ell_{\infty}\overset{T}{\longrightarrow}X\overset{Q}{\longrightarrow}X/X_{a},$

where $T$ is a lattice isomorphism and $Q$ is the canonical quotient map. It
follows that $T(c_{0})\subset X_{a}$ (see Proposition 1 and Theorem 1 in
[Wój05]; cf. [PW07]). Thus, if
$q\colon\ell_{\infty}\rightarrow\ell_{\infty}/c_{0}$ is a quotient mapping,
then there exist a unique mapping $U\colon\ell_{\infty}/c_{0}\rightarrow
X/X_{a}$ defined in the following way $U\colon x+c_{0}\mapsto T(x)+X_{a}$. It
is not hard to see that $QT=Uq$. In short, the following diagram commutes

${\ell_{\infty}}$${X}$${\ell_{\infty}/c_{0}}$${X/X_{a}}$$\scriptstyle{T}$$\scriptstyle{q}$Q$\scriptstyle{U}$

and the space $X/X_{a}$ contains a lattice isomorphic copy of
$\ell_{\infty}/c_{0}$. Now, observe that $X$ can be renormed to be a strictly
convex space. In fact, let us consider the multiplication operator $M_{y}$
given by $M_{y}\colon X\ni x\mapsto xy\in L_{1}(\mu)$, where $y\in X^{\prime}$
is chosen so that $\lVert y\rVert_{X^{\prime}}=1$ and $y(t)>0$ for $\mu$-a.e.
$t\in\Omega$. Then we can define the functional $\sharp\cdot\sharp$ on $X$ as
follows

$\sharp x\sharp=\lVert x\rVert_{X}+\lVert M_{y}(x)\rVert_{L_{1}(\mu)},$

Now, since the space $\ell_{\infty}$ is universal for all separable Banach
spaces (see [AK06, Theorem 2.5.7, p. 46]) and admit strictly convex renorming
(see [Dis76, p. 94]), so without losing generality we can assume that
$L_{1}(\mu)$ is already strictly convex; another words, we can always equip
the space $L_{1}(\mu)$ with a new norm, say $\left\lvert\cdot\right\rvert$,
such that the space $(L_{1}(\mu),\left\lvert\cdot\right\rvert)$ is strictly
convex. But this gives that $\sharp\cdot\sharp$ is an equivalent norm on $X$
and that the space $(X,\sharp\cdot\sharp)$ is strictly convex as well (this
follows from the well-known fact that if $T\colon X\rightarrow Y$ is an
injective operator and $Y$ is strictly convex Banach space then $X$ can be
renormed to be also strictly convex; see [Dis76, p. 94] for more details). On
the other hand, Bourgain [Bou80] proved that the space $\ell_{\infty}/c_{0}$
does not admit an equivalent strictly convex norm while, as we showed above,
$X$ does. Consequently, the space $X/X_{a}$, which contains an isomorphic copy
of $\ell_{\infty}/c_{0}$, is not isomorphic to a subspace of $X$ and this
implies that $X_{a}$ is not complemented in $X$. ∎

Another proof of this fact can be found, for example, in [Wnu99, Theorem 1.21,
pp. 62–63].

Note also that Partington [Par81, Theorem 1] proved that the space $X/X_{a}$
not only cannot be renormed to be strictly convex, but even contains a lattice
isometric copy of $\ell_{\infty}$ (cf. [GH96, Proposition 5.1] and [Wój05,
Theorem 2]).

##  3\. Some properties of the norm in the quotient space

It is clear that the quotient space $X/M$, where $X$ is a normed space and $M$
is a closed subspace of $X$, is also a normed space with respect to the norm
$\lVert\cdot\rVert_{X/M}$ defined as

$\lVert x+M\rVert_{X/M}\coloneqq\inf\\{\lVert y\rVert_{X}\colon y\in x+M\\}.$

In simple words, this norm measures the distance from a coset to the origin of
the space $X/M$, that is

$\lVert
x+M\rVert_{X/M}=\emph{d}_{X}(x+M,0+M)=\emph{d}_{X}(x,M)\coloneqq\inf\\{\lVert
x-m\rVert_{X}\colon m\in M\\}.$

Therefore, the study of the function $x\mapsto\emph{d}_{X}(x,M)$ is parallel
to the study of certain properties of the quotient space $X/M$ and vice versa.
For example, de Jonge [deJ77] proved that a Banach lattice $X$ is a semi-$M$
space (without going into detail, this class includes Orlicz spaces $L_{F}$
with both the Orlicz and the Luxemburg–Nakano norm and Lorentz spaces
$\Lambda_{\varphi}$; see [deJ77, Examples 2.4 and 9.6]) if and only if
$X/X_{a}$ is an $AM$-space (or, which is one thing, $X^{s}$ is an $AL$-space),
that is to say,

$\emph{d}_{X}(\sup\\{x,y\\},X_{a})=\max\\{\emph{d}_{X}(x,X_{a}),\emph{d}_{X}(y,X_{a})\\}\text{
for }x,y\in X_{+}.$

From this perspective, we will show that if $X$ is a rearrangement invariant
Banach space, then the quotient space $X/X_{a}$ is, in a sense, also
“rearrangement invariant” (Theorem 3.5). This result will turn out to be
crucial in the next section, where we will deal with abstract Cesàro spaces
looking for isometric copies of $\ell_{\infty}$. But let’s start with some
auxiliary lemmas.

######  Lemma 3.1.

Let $X$ be a Banach ideal space on $(\Omega,\Sigma,\mu)$ and let $J$ be an
order ideal of $X$. If $f\in X$ then we have the following formula

$\emph{d}_{X}(f,J)=\inf\\{\lVert f-g\rVert_{X}\colon g\in J\text{ {\it with}
}\left\lvert g\right\rvert\leqslant\left\lvert f\right\rvert\\}.$

###### Proof.

Without loss of generality we can assume that $f>0$. Of course, to show the
above equality we need only to prove that

$\inf\\{\lVert f-g\rVert_{X}\colon g\in J\text{ such that }0\leqslant
g\leqslant f\\}\leqslant\emph{d}_{X}(f,J).$

Let $\\{f_{n}\\}_{n=1}^{\infty}\subset J^{+}$ be a sequence realizing the
distance, that is, $\lim_{n\rightarrow\infty}\lVert
f-f_{n}\rVert_{X}=\emph{d}_{X}(f,J)$. Set

$\Omega_{n}=\\{x\in\Omega\colon f_{n}(x)\leqslant f(x)\\}\text{ for
}n\in\mathbb{N},$

and put

$\widetilde{f}_{n}=f_{n}\chi_{\Omega_{n}}+f\chi_{\Omega\setminus\Omega_{n}}\text{
for }n\in\mathbb{N}.$

Observe that $\widetilde{f}_{n}\leqslant f$ for every $n\in\mathbb{N}$.
Consequently,

$\lVert
f-\widetilde{f}_{n}\rVert_{X}=\lVert(f-f_{n})\chi_{\Omega_{n}}\rVert_{X}\leqslant\lVert\left\lvert
f-f_{n}\right\rvert\rVert_{X}=\lVert f-f_{n}\rVert_{X}.$

Taking limits on both sides of the above inequality, we see immediately that

$\lim\limits_{n\rightarrow\infty}\lVert
f-\widetilde{f}_{n}\rVert_{X}\leqslant\lim\limits_{n\rightarrow\infty}\lVert
f-f_{n}\rVert_{X}=\emph{d}_{X}(f,J).$

Moreover, since $J$ is an order ideal of $X$, it follows that
$\\{\widetilde{f}_{n}\\}_{n=1}^{\infty}\subset J$ and the proof is finished. ∎

The immediate conclusion from the above lemma is the following

######  Corollary 3.2.

Let $X$ be a Banach ideal space on $(\Omega,\Sigma,\mu)$ and let $J$ be an
order ideal of $X$. If $0\leqslant g\leqslant f\in X$ then

$\emph{d}_{X}(g,J)\leqslant\emph{d}_{X}(f,J).$

###### Proof.

Take $0<g\leqslant f\in X$. Let $\\{f_{n}\\}_{n=1}^{\infty}\subset J$ be a
sequence that realizes the distance, that is to say,
$\lim_{n\rightarrow\infty}\lVert f-f_{n}\rVert=\emph{d}_{X}(f,J)$. It follows
from Lemma 3.1 that we can additionally assume that $f_{n}\leqslant f$ for
every $n\in\mathbb{N}$. Put

$g_{n}=\min\\{g,f_{n}\\}\text{ for }n\in\mathbb{N}.$

Since $J$ is an order ideal of $X$, it is clear that
$\\{g_{n}\\}_{n=1}^{\infty}\subset J$. Observe also that $0<g-g_{n}\leqslant
f-f_{n}$ for all $n\in\mathbb{N}$, so due to the ideal property of $X$ we get
that

$\lVert g-g_{n}\rVert_{X}\leqslant\lVert f-f_{n}\rVert_{X}.$

But this simply means that $\emph{d}_{X}(g,J)\leqslant\emph{d}_{X}(f,J)$. ∎

######  Lemma 3.3.

Let $X$ be a Banach ideal space on $(\Omega,\Sigma,\mu)$ and let $J$ be an
order ideal of $X$. If $f\in X$ then

$\emph{d}_{X}(f,J)=\emph{d}_{X}(\left\lvert f\right\rvert,J).$

###### Proof.

To show one of the inequalities let’s take $f\in X$ and notice that

$\displaystyle\emph{d}_{X}(f,J)\geqslant\inf\\{\lVert\left\lvert
f\right\rvert-\left\lvert g\right\rvert\rVert_{X}\colon g\in
J\\}\geqslant\inf\\{\lVert\left\lvert f\right\rvert-h\rVert_{X}\colon h\in
J\\}=\emph{d}_{X}(\left\lvert f\right\rvert,J).$

On the other hand, for each $g\in J_{+}$ there is $h\in J$ such that $\lVert
f-h\rVert_{X}\leqslant\lVert\left\lvert f\right\rvert-g\rVert_{X}$. Indeed,
for $g\in J_{+}$ set

$h\left(x\right)=\left\\{\begin{array}[]{ccc}g\left(x\right)&\text{if}&f\left(x\right)\geqslant
0,\\\ -g\left(x\right)&\text{if}&f\left(x\right)<0.\end{array}\right.$

Then $h\in J$ and $\lVert f-h\rVert_{X}=\lVert\left\lvert
f\right\rvert-g\rVert_{X}$, whence $\emph{d}_{X}(\left\lvert
f\right\rvert,J_{+})\geqslant\emph{d}_{X}(f,J)$. Since
$\emph{d}_{X}(\left\lvert f\right\rvert,J)=\emph{d}_{X}(\left\lvert
f\right\rvert,J_{+})$ we conclude that
$\emph{d}_{X}(f,J)\leqslant\emph{d}_{X}(\left\lvert f\right\rvert,J)$ and this
finishes the proof. ∎

It will turn out right away that the formula expressing the distance of the
non-increasing rearrangement of $f$ from the ideal $X_{a}$ can be given in a
fairly computable form.

######  Theorem 3.4.

Let $X$ be a rearrangement invariant space on $\Omega$, where $\Omega=(0,1)$,
$\Omega=(0,\infty)$ or $\Omega=\mathbb{N}$, such that the ideal $X_{a}$ is
non-trivial. Then for $f\in X$ we have the following equality

$\emph{d}_{X}(f^{*},X_{a})=\lim\limits_{n\rightarrow\infty}\lVert
f^{*}\chi_{\Omega_{n}}\rVert_{X},$

where $\Omega_{n}=\Omega\cap\left((0,\frac{1}{n})\cup(n,\infty)\right)$.

###### Proof.

We will only give a proof when $\Omega=(0,\infty)$, because the other cases
are completely analogous.

To begin with, due to the fact that the sequence $\\{\lVert
f^{*}\chi_{(0,1/n)\cup(n,\infty)}\rVert_{X}\\}_{n=1}^{\infty}$ is non-
increasing and bounded from below by 0, the limit of the above sequence must
exists. Moreover, it is clear that we can assume that $X\neq X_{a}$, because
otherwise there is nothing to prove.

Denote

$L=\lim\limits_{n\rightarrow\infty}\lVert
f^{*}\chi_{(0,1/n)\cup(n,\infty)}\rVert_{X}.$

First of all, it is not hard to see that

$\emph{d}_{X}(f^{*},X_{a})\geqslant L,$

or, equivalently, $\lVert f^{*}-g\rVert_{X}\geqslant L$ for each $g\in X_{a}$.
Indeed,

$\displaystyle\lVert f^{*}-g\rVert_{X}$
$\displaystyle\geqslant\lim\limits_{n\rightarrow\infty}\lVert(f^{*}-g)\chi_{(0,1/n)\cup(n,\infty)}\rVert_{X}$
$\displaystyle\geqslant\lim\limits_{n\rightarrow\infty}\left\lvert\lVert
f^{*}\chi_{(0,1/n)\cup(n,\infty)}\rVert_{X}-\lVert
g\chi_{(0,1/n)\cup(n,\infty)}\rVert_{X}\right\rvert=L,$

where the equality follows from the fact that $g\in X_{a}$.

On the other hand, our assumption that $X_{a}\neq\\{0\\}$ implies that
$\\{f^{*}\chi_{(1/n,n)}\\}_{n=1}^{\infty}\subset X_{a}$ (cf. [deJ77, Lemma 7.3
(ii)] or [KT17, Theorem B]), so

$\emph{d}_{X}(f^{*},X_{a})=\inf\\{\lVert f^{*}-g\rVert\colon g\in
X_{a}\\}\leqslant\lim\limits_{n\rightarrow\infty}\lVert
f^{*}-f^{*}\chi_{(1/n,n)}\rVert_{X}=L,$

and the proof is done. ∎

Now we prove the main result of this section.

######  Theorem 3.5.

Let $X$ be a rearrangement invariant space on $\Omega$, where $\Omega=(0,1)$
or $\Omega=(0,\infty)$, with the Fatou property. Then for $f\in X$ we have the
following equality

$\emph{d}_{X}(f^{*},X_{a})=\emph{d}_{X}(f,X_{a}).$

###### Proof.

We can assume that $X_{a}\neq\left\\{0\right\\}$, $X_{a}\neq X$ and $f\notin
X_{a}$, because otherwise there is nothing to prove. Moreover, in view of
Lemma 3.3, we can also assume that $f>0$.

Let’s start by noticing that the inequality

($\heartsuit$) $\emph{d}_{X}(f,X_{a})\geqslant\emph{d}_{X}(f^{*},X_{a})$

is true even without additional assumptions on the function $f\in X$. In fact,
we have

$\displaystyle\emph{d}_{X}(f,X_{a})$ $\displaystyle=\inf\\{\lVert
f-g\rVert_{X}\colon g\in X_{a}\\}\geqslant\inf\\{\lVert
f^{*}-g^{*}\rVert_{X}\colon g\in X_{a}\\}$ $\displaystyle=\inf\\{\lVert
f^{*}-g\rVert_{X}\colon g\in X_{a}\text{ and
}g=g^{*}\\}\geqslant\emph{d}_{X}(f^{*},X_{a}),$

where the first inequality follows essentially from the Calderón–Ryff theorem
(see [BS88, Theorem 2.10, p. 114]; cf. also Lemma 4.6 in [KPS82, p. 95]) and
the second equality follows from Lemma 2.6 in [CKP14]. Consequently, it
remains to prove the reverse inequality. To do this, we divide the proof into
several parts.

$1^{\circ}$. Suppose that $f^{\ast}\left(\infty\right)=0$. Then, as a
consequence of Ryff’s theorem (see [BS88, Corollary 7.6, p. 83]), there is a
measure preserving transformation $\sigma$ from the support of $f$ onto the
support of $f^{*}$ such that $f^{\ast}\circ\sigma=f$ $m$-almost everywhere on
the support of $f$. If we were able to show that for each $g\in X_{a}$ there
exists $h_{g}\in X_{a}$ such that $\lVert f-h_{g}\rVert_{X}\leqslant\lVert
f^{*}-g\rVert_{X}$, the proof of this part would be complete. For this, let’s
take $g\in X_{a}$ and put $h_{g}=g\circ\sigma$. Due to [CKP14, Lemma 2.6] and
the fact that $h_{g}^{*}=(g\circ\sigma)^{*}$, we conclude that $h_{g}\in
X_{a}$. Now, since we have the equalities

$\lVert f^{*}-g\rVert_{X}=\lVert(f^{*}-g)\circ\sigma\rVert_{X}=\lVert
f-h_{g}\rVert_{X},$

it follows that $\emph{d}_{X}(f^{*},X_{a})\geqslant\emph{d}_{X}(f,X_{a})$.
Combining this inequality with the inequality ($\heartsuit$), we see that we
have just finished the proof in the first case.

$2^{\circ}$. Moving on to the second case, let’s assume that
$f^{*}(\infty)>0$. Denote

$\Omega_{\Box}=\\{x\in\Omega\colon f(x)\Box f^{*}(\infty)\\},$

where $\Box$ is one of the following relations: $\geqslant$, $>$, $=$, $<$ or
$\leqslant$ (for example, $\Omega_{>}=\\{x\in\Omega\colon
f(x)>f^{*}(\infty)\\}$). Let us consider three subcases.

Case 1: Assume that $m(\Omega_{>})=\infty$. Then there exists a measure
preserving transformation $\omega\colon\Omega_{>}\rightarrow[0,\infty)$ such
that $f^{\ast}\circ\omega=f$ $m$-almost everywhere on $\Omega_{>}$. Take $g\in
X_{a}$ and put

$h_{g}=(g\circ\omega)\chi_{\Omega_{>}}.$

We claim that $d_{f\chi_{\Omega_{>}}-h_{g}}=d_{f-h_{g}}$. Indeed, for all
$0\leqslant x<f^{*}(\infty)$ we have

$d_{f\chi_{\Omega_{>}}-h_{g}}(x)=\infty=d_{f-h_{g}}(x),$

because $h_{g}\in X_{a}$, whence $h^{*}(\infty)=0$. Furthermore, for
$x\in\Omega_{\leqslant}$

$h_{g}(x)=0\text{ and }f(x)\leqslant f^{*}(\infty),$

so for each $x\geqslant f^{*}(\infty)$ we obtain that

$\displaystyle d_{f-h_{g}}(x)$ $\displaystyle=m\\{t\in\Omega\colon\left\lvert
f(t)-h_{g}(t)\right\rvert>x\\}=m\\{t\in\Omega_{>}\colon\left\lvert
f(t)-h_{g}(t)\right\rvert>x\\}$
$\displaystyle=m\\{t\in\Omega_{>}\colon\left\lvert
f(t)\chi_{\Omega_{>}}-h_{g}(t)\right\rvert>x\\}=d_{f\chi_{\Omega_{>}}-h_{g}},$

and the claim follows. In consequence, we have

$\displaystyle\lVert f^{*}-g\rVert_{X}$
$\displaystyle=\lVert(f^{*}-g)\circ\omega\rVert_{X}=\lVert
f\chi_{\Omega_{>}}-h_{g}\rVert_{X}$
$\displaystyle=\lVert(f\chi_{\Omega_{>}}-h_{g})^{*}\rVert_{X}=\lVert(f-h_{g})^{*}\rVert_{X}=\lVert
f-h_{g}\rVert_{X}.$

Case 2: Suppose now that $m(\Omega_{>})<\infty$ and $f\chi_{\Omega_{>}}\in
X_{a}$. Put

$\Gamma_{f}=\\{g\in X_{a}\colon g(x)=f(x)\text{ for }x\in\Omega_{>}\text{ and
}0\leqslant g(x)\leqslant f^{*}(\infty)\text{ otherwise }\\}.$

It follows directly from the above definition that
$f\chi_{\Omega_{>}}\in\Gamma_{f}$ and $\Gamma_{f}\subset X_{a}$. We claim that

($\spadesuit$) $\emph{d}_{X}(f,X_{a})=\emph{d}_{X}(f,\Gamma_{f}).$

To see this, for each $g\in X_{a}$ let us define a function $g_{f}$ in the
following way

$g_{f}(x)=\left\\{\begin{array}[]{ccc}f\left(x\right)&\text{if}&x\in\Omega_{>}\\\
f\left(x\right)&\text{if}&x\in\Omega\setminus\Omega_{>}\text{ and
}g(x)>f^{*}(\infty)\\\ g\left(x\right)&&\text{otherwise}\end{array}\right..$

Now, it is clear that $\lVert f-g_{f}\rVert_{X}\leqslant\lVert f-g\rVert_{X}$
and, due to the fact that $m\\{x\in\Omega\setminus\Omega_{>}\colon
g(x)>f^{*}(\infty)\\}<\infty$, also $g_{f}\in\Gamma_{f}$. But this means that
$\emph{d}_{X}(f,X_{a})\geqslant\emph{d}_{X}(f,\Gamma_{f})$. Since
$\Gamma_{f}\subset X_{a}$, the opposite inequality is evident and the equality
$(\spadesuit)$ is proved. Consequently, since $g\in X_{a}$, so

$\emph{d}_{X}(f,X_{a})=\inf\\{\lVert f-g\rVert\colon g\in\Gamma_{f}\\}=\lVert
f^{*}\chi_{(m(\Omega_{>}),\infty)}\rVert_{X}=f^{*}(\infty)\lVert\chi_{(0,\infty)}\rVert_{X}.$

On the other hand, since $m(\Omega_{>})<\infty$ and $f\chi_{\Omega_{>}}\in
X_{a}$, applying Theorem 3.4, we conclude that

$\emph{d}_{X}(f^{*},X_{a})=\lim\limits_{n\rightarrow\infty}\lVert
f^{*}\chi_{(0,1/n)\cup(n,\infty)}\rVert=f^{*}(\infty)\lVert\chi_{(0,\infty)}\rVert_{X},$

that is to say, $\emph{d}_{X}(f,X_{a})=\emph{d}_{X}(f^{*},X_{a})$.

Case 3: Let $m(\Omega_{>})<\infty$ and $f\chi_{\Omega_{>}}\in X\setminus
X_{a}$. Observe that $f^{*}(0^{+})=\infty$, because otherwise, keeping in mind
that $m(\Omega_{>})<\infty$, we would get $f\chi_{\Omega_{>}}\in X_{a}$ which
is undoubtedly a contradiction. We will show that

($\diamondsuit$)
$\emph{d}_{X}(f,X_{a})=L\coloneqq\lim\limits_{n\rightarrow\infty}\lVert
f^{*}\chi_{(0,\frac{1}{n})}+f^{*}(\infty)\chi_{(\frac{1}{n},\infty)}\rVert_{X}.$

Take $n_{0}\in\mathbb{N}$ satisfying $f^{*}(1/n_{0})>f^{*}(\infty)$ and note
that this limit exists because the sequence $\left\\{\lVert
f^{*}\chi_{(0,\frac{1}{n})}+f^{*}(\infty)\chi_{(\frac{1}{n},\infty)}\rVert_{X}\right\\}_{n=1}^{\infty}$
is non-increasing for $n\geqslant n_{0}$ and bounded from below. Now, there is
a measure preserving transformation
$\tau\colon\Omega_{>}\rightarrow(0,m(\Omega_{>}))$ such that
$f^{*}\circ\tau=f$. Put $T_{n}=\tau^{-1}((0,1/n))$ for $n=n_{0},n_{0}+1,...$
and denote

$f_{n}=f^{*}\left(\frac{1}{n_{0}}\right)\chi_{T_{n}}$

It is clear that $f_{n}\in X_{a}$ for $n\geqslant n_{0}$. Moreover, since
$f^{*}(0^{+})=\infty$, so

$\left(f^{*}-f^{*}\left(\frac{1}{n_{0}}\right)\right)\chi_{(0,\frac{1}{n})}\geqslant
f^{*}(\infty)\chi_{(0,\frac{1}{n})}\text{ for }n\in\mathbb{N}\text{ big
enough},$

and, consequently, we have

$\emph{d}_{X}(f,X_{a})\leqslant\lim\limits_{n\rightarrow\infty}\lVert
f-f_{n}\rVert_{X}=\lim\limits_{n\rightarrow\infty}\lVert\left(f^{*}-f^{*}\left(\frac{1}{n_{0}}\right)\right)\chi_{(0,\frac{1}{n})}+f^{*}(\infty)\chi_{(\frac{1}{n},\infty)}\rVert_{X}\leqslant
L.$

On the other hand, we get

$\displaystyle L\geqslant\emph{d}_{X}(f,X_{a})$ $\displaystyle=\inf\\{\lVert
f-g\rVert_{X}\colon g\in X_{a}\text{ and }\left\lvert
g\right\rvert\leqslant\left\lvert f\right\rvert\\}$
$\displaystyle\geqslant\inf\\{\lVert f^{*}-g^{*}\rVert_{X}\colon g\in
X_{a}\text{ and }\left\lvert g\right\rvert\leqslant f\\}$
$\displaystyle\geqslant\inf\\{\lVert f^{*}-g^{*}\rVert_{X}\colon g\in
X_{a}\text{ and }0\leqslant g^{*}\leqslant f^{*}\\},$

where the first equality follows from Lemma 3.1 and second inequality follows
from the Calderón–Ryff theorem. Thus, to finish the proof of $(\diamondsuit)$,
it is enough to show that

($\clubsuit$) $\text{for every }\varepsilon>0\text{ and }h\in X_{a}\text{ with
}0\leqslant h^{*}\leqslant f^{*}\text{ we have }\lVert
f^{*}-h^{*}\rVert_{X}\geqslant L-\varepsilon.$

To see this, take $\varepsilon>0$ and $h\in X_{a}$ as above. We can find a
number, say $m\in\mathbb{N}$, such that

$\lVert h^{*}\chi_{(0,\frac{1}{m})\cup(m,\infty)}\rVert<\epsilon,$

because $h^{*}\in X_{a}$ (see [CKP14, Lemma 2.6]). Hence,

$\displaystyle\lVert f^{*}-h^{*}\rVert_{X}$ $\displaystyle=\lVert
f^{*}-h^{*}\chi_{(\frac{1}{m},m)}-h^{*}\chi_{(0,\frac{1}{m})\cup(m,\infty)}\rVert_{X}$
$\displaystyle\geqslant\lVert
f^{*}-h^{*}\chi_{(\frac{1}{m},m)}\rVert_{X}-\lVert
h^{*}\chi_{(0,\frac{1}{m})\cup(m,\infty)}\rVert_{X}$ $\displaystyle>\lVert
f^{*}-h^{*}\chi_{(\frac{1}{m},m)}\rVert_{X}-\varepsilon$
$\displaystyle\geqslant\lVert
f^{*}\chi_{(0,\frac{1}{m})\cup(m,\infty)}+(f^{*}-h^{*})\chi_{(\frac{1}{m},m)}\rVert_{X}-\varepsilon\geqslant\lVert
f^{*}\chi_{(0,\frac{1}{m})\cup(m,\infty)}\rVert_{X}-\varepsilon,$

and $(\clubsuit)$ follows. But this means that the proof of $(\diamondsuit)$
is finished as well. However, using Theorem 3.4 again, we see that

$\emph{d}_{X}(f^{*},X_{a})=\lim\limits_{n\rightarrow\infty}\lVert
f^{*}\chi_{(0,1/n)\cup(n,\infty)}\rVert_{X},$

that is, $\emph{d}_{X}(f,X_{a})=\emph{d}_{X}(f^{*},X_{a})$. But this, finally,
completes the proof of the last case and, at the same time, also of the whole
theorem. ∎

######  Theorem 3.6.

Let $X$ be a rearrangement invariant space on $\mathbb{N}$ with the Fatou
property. Then for $f\in X$ we have the following equality

$\emph{d}_{X}(f^{*},X_{a})=\emph{d}_{X}(f,X_{a}).$

###### Proof.

The argument in the sequence case (which is probably not a suprise) is
analogous to the proof of Theorem 3.5 and, in fact, it is even easier. But we
want to give a few clues that (we hope) will allow to easily modify the
mentioned proof.

$1^{\circ}$. For a rearrangement invariant sequence space $X$ the ideal
$X_{a}$ is always non-trivial.

$2^{\circ}$. Since $X\hookrightarrow\ell_{\infty}$ (cf. [BS88, Corollary 6.8,
p. 78]), so $X_{a}\hookrightarrow c_{0}$ and this means that
$f^{*}(\infty)=0$, whenever $f\in X_{a}$.

$3^{\circ}$. Every function supported on the set of finite measure belongs to
$X_{a}$.

$4^{\circ}$. The inequality (coming from the Calderón–Ryff theorem):

$\lVert f^{*}-g^{*}\rVert_{X}\leqslant\lVert f-g\rVert_{X},$

works also for every rearrangement invariant sequence space $X$ (cf. [BS88,
Corollary 7.6, p. 83]). ∎

######  Corollary 3.7.

Let $X$ be a rearrangement invariant space on $\Omega$, where $\Omega=(0,1)$,
$\Omega=(0,\infty)$ or $\Omega=\mathbb{N}$, such that the ideal $X_{a}$ is
non-trivial. Then for $f\in X$ we have the following equality

$\emph{d}_{X}(f,X_{a})=\lim\limits_{n\rightarrow\infty}\lVert
f^{*}\chi_{\Omega_{n}}\rVert_{X},$

where $\Omega_{n}=\Omega\cap\left((0,\frac{1}{n})\cup(n,\infty)\right)$.

The above corollary can be interpreted in the following way: the function
$f\in X$ can be approximated by a bounded function supported in set of finite
measure to within $\emph{d}_{X}(f,X_{a})+\varepsilon$, where $\varepsilon>0$
can be arbitrarily small, and this function can be selected in an explicit
way. A similar result, but for Orlicz spaces equipped with an Orlicz norm, can
be found in [KR61, Lemma 10.1, p. 84].

######  Corollary 3.8 (E. de Jonge, 1977 [deJ77]).

Let $X$ be a rearrangement invariant space on $(0,\infty)$ with the Fatou
property such that $L_{\infty}\hookrightarrow X$. Suppose that

(${\mathscr{D}}_{\infty}$) $\forall f\in X\text{ }\forall
A\subset(0,\infty)\textit{ with }m(A)<\infty\colon f\chi_{A}\in X_{a}.$

Then for all $f\in X$ we have the following formula

$\emph{d}_{X}(f,X_{a})=f^{*}(\infty)\varphi_{X}(\infty).$

Before we provide the proof, let us note that the above condition
(${\mathscr{D}}_{\infty}$) itself plays in rearrangement invariant spaces an
analogous role to the $\Delta_{2}(\infty)$-condition in the case of Orlicz
spaces. In fact, if $X$ is an Orlicz space $L_{F}$ on $(0,\infty)$ generated
by the Orlicz function $F$ which vanishes outside zero and
$F\in\Delta_{2}(\infty)$, then $L_{\infty}\hookrightarrow X$ and
$X\in({\mathscr{D}}_{\infty})$. However, the assumptions of the above
corollary are true also in the case of Lorentz spaces $\Lambda_{\varphi}$
provided $\varphi(0^{+})=0$ and $\varphi(\infty)<\infty$.

###### Proof.

Suppose that $X$ is as above and observe that the sequence
$\left\\{(f^{*}\chi_{(n,\infty)})^{*}\right\\}_{n=1}^{\infty}$ is bounded and
converges uniformly to a bounded function $f^{*}(\infty)\chi_{(0,\infty)}$, so
in view of the embedding $L_{\infty}\hookrightarrow X$ we get

$\emph{d}_{X}(f,X_{a})=\lim\limits_{n\rightarrow\infty}\lVert
f^{*}\chi_{(n,\infty)}\rVert_{X}=\lim\limits_{n\rightarrow\infty}\lVert(f^{*}\chi_{(n,\infty)})^{*}\rVert_{X}=\lVert
f^{*}(\infty)\chi_{(0,\infty)}\rVert_{X}=f^{*}(\infty)\varphi_{X}(\infty),$

where the first equality follows from Corollary 3.7. ∎

Although we believe that the below results (Corollaries 3.9 and 3.11) are
expected, we will present proofs of these facts as illustrations of the
techniques we developed in this paragraph.

######  Corollary 3.9.

Let $\varphi$ be a quasi-concave function such that $\lim_{t\rightarrow
0^{+}}t/\varphi(t)=0$. Then the Marcinkiewicz space $M_{\varphi}$ contains a
lattice isometric copy of $\ell_{\infty}$.

###### Proof.

Let us consider the function $\psi$ defined in the following way

$\psi(0)=0\text{ and }\psi(t)=\frac{t}{\varphi(t)}\text{ for }t>0.$

It follows from [BS88, Theorem 5.2, p. 66] that $\psi$ is also a quasi-concave
function as a fundamental function of the space $X^{\prime}$. In addition, due
to the condition $\lim_{t\rightarrow 0^{+}}t/\varphi(t)=0$, the function
$\psi$ is continuous (cf. [BS88, Corollary 5.3, p. 67]). Therefore, we can
find $0<a<1$ such that the function $\psi$ is strictly increasing on $(0,a)$.
Consequently, the function $\psi^{\prime}$ is non-increasing and positive
$m$-almost everywhere on $(0,a)$. Put $g=\psi^{\prime}\chi_{(0,a)}$. Then

$\frac{\varphi(t)}{t}\int_{0}^{t}g^{*}(s)ds=\frac{\varphi(t)}{t}\int_{0}^{t}g(s)ds=\frac{\varphi(t)}{t}\psi(t)=1\text{
for }0<t<a,$

and, because the function $t\mapsto\varphi(t)/t$ is non-increasing, we also
have that

$\frac{\varphi(t)}{t}\int_{0}^{t}g^{*}(s)ds=\frac{\varphi(t)}{t}\frac{a}{\varphi(a)}\leqslant
1\text{ for }t\geqslant a.$

This means that $\lVert g\rVert_{M_{\varphi}}=1$. Moreover, we can repeat the
above argument with the same result for the function
$\psi_{\varepsilon}=\psi\chi_{(0,\varepsilon)}$, where $0<\varepsilon<a$, in
place of the function $\psi$. Therefore, using Corollary 3.7, we get

$d_{M_{\varphi}}(g,(M_{\varphi})_{a})\geqslant\lim_{n\rightarrow\infty}\lVert
g^{*}\chi_{(0,\frac{1}{n})}\rVert_{M_{\varphi}}=\lim_{n\rightarrow\infty}\lVert
g\chi_{(0,\frac{1}{n})}\rVert_{M_{\varphi}}\geqslant\lim_{n\rightarrow\infty}\lVert\psi_{\varepsilon_{n}}\rVert_{M_{\varphi}}=1,$

for a properly choosen subsequence $\\{\varepsilon_{n}\\}_{n=1}^{\infty}$. ∎

######  Lemma 3.10.

Let $X$ be a rearrangement invariant space on $(0,\infty)$. If
$L_{\infty}\hookrightarrow X$, then $X$ contains a lattice isometric copy of
$\ell_{\infty}$.

Let us note in addition that the converse is not true, for example, the
Zygmund space $e^{L}$ (recall that $e^{L}=L_{F}$, where $F(t)=e^{t}-1$)
contains a lattice isometric copy of $\ell_{\infty}$, but it is clear that
$L_{\infty}\not\hookrightarrow e^{L}$.

###### Proof.

Take $f=\chi_{(0,\infty)}/\lVert\chi_{(0,\infty)}\rVert_{X}$ and let
$\\{N_{i}\\}_{i=1}^{\infty}$ be a pairwise disjoint family of an infinite
subsets of $\mathbb{N}$ such that $\bigcup_{i=1}^{\infty}N_{i}=\mathbb{N}$.
Moreover, let $\\{\omega_{j}\\}_{j=1}^{\infty}$ be a sequence of real numbers
such that $0=\omega_{1}<\omega_{2}<...$ and
$\lim_{j\rightarrow\infty}\omega_{j}=\infty$. Put
$\Omega_{j}^{i}=(\omega_{j},\omega_{j+1})$ for $i\in\mathbb{N}$ and $j\in
N_{i}$. Consider the following sequence $\left\\{f\chi_{\bigcup_{j\in
N_{i}}\Omega_{j}^{i}}\right\\}_{i=1}^{\infty}$. It is clear that this sequence
is pairwise disjoint, $\lVert\sum_{i=1}^{\infty}f\chi_{\bigcup_{j\in
N_{i}}\Omega_{j}^{i}}\rVert_{X}=1$ and $\lVert
f\chi_{\bigcup_{j=1}^{\infty}\Omega_{j}^{i}}\rVert_{X}=1$, because
$\left(f\chi_{\bigcup_{j\in N_{i}}\Omega_{j}^{i}}\right)^{*}=f$ for every
$i\in\mathbb{N}$ and $\lVert f\rVert_{X}=1$. Consequently, in order to
complete the proof we can refer to [Hud98, Theorem 1]. ∎

######  Corollary 3.11.

Let $\varphi$ be a concave function on $(0,\infty)$ such that
$\varphi(0^{+})=0$. Then the Lorentz space $\Lambda_{\varphi}$ contains a
lattice isometric copy of $\ell_{\infty}$ if and only if
$\varphi(\infty)<\infty$.

###### Proof.

It follows from [KPS82, Lemma 5.1] that the space $\Lambda_{\varphi}$ is order
continuous if $\varphi(\infty)=\infty$. Therefore, due to the well-known
Lozanovskiĭ result, it does not contain even an order isomorphic copy of
$\ell_{\infty}$ and proves the necessity.

Now, suppose that $\varphi(\infty)<\infty$. But this means that
$L_{\infty}\hookrightarrow X$ and it is enough to use Lemma 3.10. ∎

Let us add that it may happen that the Lorentz space $\Lambda_{\varphi}$
contains an order isomorphic copy of $\ell_{\infty}$ but not an order
isometric copy of $\ell_{\infty}$. Indeed, if $\varphi(0^{+})>0$ and
$\varphi(\infty)=\infty$, then the space $\Lambda_{\varphi}$ is not order
continuous but is strictly monotone.

##  4\. On isometric copies of $\ell_{\infty}$ in abstract Cesàro spaces

For a Banach function space on $(0,1)$ or $(0,\infty)$ we define the abstract
Cesàro function space $CX$ as a set

$CX=\\{f\in L_{0}\colon C(\left\lvert f\right\rvert)\in X\\}\text{ with the
natural norm }\lVert f\rVert_{CX}=\lVert C(\left\lvert
f\right\rvert)\rVert_{X},$

where $C$ denotes the continuous Cesàro operator defined as

$C\colon f\mapsto C(f)(x)=\frac{1}{x}\int_{0}^{x}\left\lvert f(t)\right\rvert
dt\text{ for }x>0.$

However, in the sequence case, by the discrete Cesàro operator we will
understand the operator $C_{d}$ defined as

$C_{d}\colon\\{x_{n}\\}_{n=1}^{\infty}\mapsto\left\\{\frac{1}{n}\sum_{k=1}^{n}x_{k}\right\\}_{n=1}^{\infty},$

and then the corresponding abstract Cesàro sequence space $CX$ is defined in
an analogous way as above, that is to say, as a set of all sequences
$x=\\{x_{n}\\}_{n=1}^{\infty}$ such that $C_{d}(x)\in X$ for which the
following norm

$\lVert x\rVert_{CX}=\lVert C_{d}(\left\lvert x\right\rvert)\rVert_{X}$

is finite

Because it always should be clear from the context which Cesàro construction
we use, we will abuse the notation a little by writing simply $C$ for the
continuous as well as the discrete Cesáro operator. Moreover, to avoid some
trivialities and unnecessary complications, let’s assume that we are only
interested in non-trivial Cesàro spaces. Incidentally, if the Cesàro operator
$C$ is bounded on $X$, then $X\hookrightarrow CX$ and, consequently, the space
$CX$ is non-trivial.

The classical Cesàro spaces $ces_{p}$ and $Ces_{p}$ are only a special case of
this general construction $CX$ when $X=\ell_{p}$ or $X=L_{p}$, respectively.
We will not say more about the history and the current development of the
Cesaro space theory, referring to [AM14] and the references given there (see
also [Ben96], [AM09], [ALM17], [CR16], [DS07], [Jag74], [KamK09], [KT17],
[KKM21], [LM15] and [NP10]; cf. [KMP07]).

Let us now formulate and prove the main result of this section.

######  Theorem 4.1.

Let $X$ be a rearrangement invariant Banach space on $(0,1)$, $(0,\infty)$ or
$\mathbb{N}$ with the Fatou property such that the Cesàro operator $C$ is
bounded on $X$ and the ideal $X_{a}$ is non-trivial. Suppose there exists an
element $f\in X$ and $0\leqslant a<b\leqslant\infty$ such that $\lVert
f\rVert_{X}=\emph{d}_{X}(f,X_{a})=1$ and $\lVert
f^{*}\chi_{(0,a)\cup(b,\infty)}\rVert_{CX}=1$. Then $CX$ contains a lattice
isometric copy of $\ell_{\infty}$.

The proof of the above theorem makes strong use of Theorem 3.5 from §3 and
Theorem 16 from [KT17]. Note that Theorem 16 from [KT17] has been proved for
rearrangement invariant function, but not sequence, spaces. For the missing
proof, we refer to Appendix.

###### Proof.

Let $f\in X$ be as above. Put $A=(0,a)\cup(b,\infty)$ and $A^{c}=(a,b)$. In
view of Theorem 3.5, we have the equality

$\emph{d}_{X}(f^{*},X_{a})=\emph{d}_{X}(f,X_{a})=1.$

Note also that $f^{*}\chi_{A^{c}}\in(CX)_{a}$ (see [KT17, Lemma 8] in the case
of function spaces; however, simple modification of this argument works in the
sequence case as well). Consequently, taking $h=f^{*}\chi_{A}$, we have

$\displaystyle\emph{d}_{CX}(h,(CX)_{a})$ $\displaystyle=\inf\\{\lVert
h-g\rVert_{CX}\colon g\in(CX)_{a}\text{ and }0<g\leqslant h\\}$
$\displaystyle=\inf\\{\lVert Ch-Cg\rVert_{X}\colon g\in C(X_{a})\text{ and
}0<g\leqslant h\\}$ $\displaystyle\quad\quad\quad(\text{because }h-g\geqslant
0\text{ and }CX_{a}=C(X_{a})\text{, see \cite[cite]{[\@@bibref{}{KT17}{}{},
Theorem 16]}})$ $\displaystyle=\inf\\{\lVert
C(h+f^{*}\chi_{A^{c}})-C(g+f^{*}\chi_{A^{c}})\rVert_{X}\colon C(g)\in
X_{a}\text{ and }0<g\leqslant h\\}$ $\displaystyle\geqslant\inf\\{\lVert
C(f^{*})-C(g^{\prime})\rVert_{X}\colon g^{\prime}\in CX\text{ and
}C(g^{\prime})\in X_{a}\\}$ $\displaystyle\quad\quad\quad(\text{since
}g+f^{*}\chi_{A^{c}}\in C(X_{a}))$ $\displaystyle\geqslant\inf\\{\lVert
C(f^{*})-{g^{\prime\prime}}\rVert_{X}\colon{g^{\prime\prime}}\in X_{a}\\}$
$\displaystyle\quad\quad\quad(\text{we take the infimum on a larger set})$
$\displaystyle=\emph{d}_{X}(C(f^{*}),X_{a})\geqslant\emph{d}_{X}(f^{*},X_{a})=1.$

where the last inequality follows from Corollary 3.2 and the fact that
$f^{*}\leqslant C(f^{*})$. Now, since
$\operatorname{supp}((CX)_{a})=\operatorname{supp}(X)$ (see, again, [KT17,
Lemma 8]) we can apply Theorem B and finish the proof. ∎

It should be probably mentioned here that Theorem 4.1 covers all previous
results concerning the problem of existence of a lattice isometric copy of
$\ell_{\infty}$ in Cesàro spaces $CX$. To be more precise, Kamińska and Kubiak
[KamK09] considered this problem for $X=\ell_{F}$ and Kiwerski and Kolwicz
[KK18b] for $X=L_{F}$. A common feature of these results is a certain
assumption about the Orlicz class, namely, about the closedness of the Orlicz
class under the action of the Cesàro operator, that is to say,

$\left\\{\\{x_{n}\\}_{n=1}^{\infty}\in\ell_{F}\colon\sum_{n=1}^{\infty}F(x_{n})<\infty\right\\}\subset\left\\{\\{x_{n}\\}_{n=1}^{\infty}\in\ell_{F}\colon\sum_{n=1}^{\infty}F\left(\frac{1}{n}\sum_{k=1}^{n}x_{k}\right)<\infty\right\\},$

in the sequence case and the modular-type inequality of the form

$\int_{0}^{\infty}F\left(\frac{1}{x}\int_{0}^{x}\left\lvert f(t)\right\rvert
dt\right)dx\leqslant M\int_{0}^{\infty}F(\left\lvert f(x)\right\rvert)dx,$

in the function case (which looks a little like the boundedness‡‡‡Orlicz class
is not even a linear space, therefore is makes little sense to speak about
such an object as bounded operator on it. of the Cesàro operator on the Orlicz
class or like the Hardy inequality with the Orlicz function $F$ instead of the
power function; cf. [KMP07, §3]). Note that both assumptions imply the
boundedness of the Cesàro operator on the space $\ell_{F}$ (see [KamK09,
Proposition 2]) and, respectively, $L_{F}$ (see [KK18b]). Our Theorem 4.1, at
least formally, improves both of these results. However, it is not so easy to
see that Theorem 4.1 does in fact imply the earlier results from [KamK09] and
[KK18b]. To show this, we will take an even more general point of view,
because we will consider the Calderón–Lozanovskiĭ spaces $X_{F}$ (recall, that
the spaces $(L_{1})_{F}$ and $(\ell_{1})_{F}$ coincide, up to the equality of
norms, with the Orlicz spaces $L_{F}$ and $\ell_{F}$, respectively).

Let us also remind that by the Calderón–Lozanovskiĭ class we understand the
set $\\{f\in X_{F}\colon\lVert F(\left\lvert
f\right\rvert)\rVert_{X}<\infty\\}$ (cf. [Mal89, §3] for some comments and key
properties in the particular case when $X=L_{1}$).

######  Corollary 4.2.

Let $X$ be an order continuous rearrangement invariant function space with the
Fatou property and let $F$ be an Orlicz function. Assume that there exists
$M>0$ such that for all functions, say $f$, from the Calderón–Lozanovskiĭ
class we have the following inequality:

(${\mathscr{M}}$) $\lVert F(C(\left\lvert f\right\rvert))\rVert_{X}\leqslant
M\lVert F(\left\lvert f\right\rvert)\rVert_{X}.$

If $F$ does not satisfy the $\Delta_{2}^{X}$-condition§§§Let’s just mention
that the $\Delta_{2}^{X}$-condition is an appropriate modification of the
well-known condition $\Delta_{2}$ from the Orlicz space theory. Roughly
speaking, it is defined in such a way as to correspond to the order continuity
of the Calderón–Lozanovskiĭ space $X_{F}$. For a more precise definitions we
refer, for example, to [HKM96, p. 642]., then the space $C(X_{F})$ contains a
lattice isometric copy of $\ell_{\infty}$.

###### Proof.

Since $F\notin\Delta_{2}^{X}$, it follows from Theorem 1 and Theorem 2 in
[HKM96] that the Calderón–Lozanovskiĭ space $X_{F}$ contains a lattice
isometric copy of $\ell_{\infty}$. Therefore, in view of Theorem 2.1 (let us
note the fact that
$\operatorname{supp}((X_{F})_{a})=\operatorname{supp}(X_{F})$) and Theorem
3.5, there exists a function $f\in X_{F}$ such that

$\lVert
f\rVert_{X_{F}}=d_{X_{F}}(f,(X_{F})_{a})=d_{X_{F}}(f^{*},(X_{F})_{a})=1.$

Because the space $X_{F}$ is rearrangement invariant, so $\lVert
f^{*}\rVert_{X_{F}}=1$ and, consequently, $\lVert F(f^{*})\rVert_{X}\leqslant
1$. Now, we divide the proof into two parts¶¶¶It follows from Theorem 2.1 that
a rearrangement invariant space $X$ on $(0,\infty)$ with
$\operatorname{supp}(X_{a})=(0,\infty)$ contains a lattice isometric copy of
$\ell_{\infty}$ if and only if there is a function, say $f$, such that $\lVert
f\rVert=\emph{d}_{X}(f,X_{a})=1$. In many situations (for example, if $X$ is a
Calderón–Lozanovskiĭ space) the function $f$ is given in a simple form, i.e.,
$f=\sum_{n=1}^{\infty}f_{n}$, where $f_{n}=a_{n}\chi_{\Omega_{n}}$,
$a_{n}\in\mathbb{R}_{+}$, $\Omega_{n}$ are a pairwise disjoint measurable
subsets of $(0,\infty)$ and $f_{n}\in S(X)$ for $n\in\mathbb{N}$. In this
case, however, we have essentially two possibilities, namely either $f$ is
bounded (and has a support of infinite measure) or $f$ is unbounded and has a
support of a finite measure. Indeed, it is easy to see that
$1\leqslant\emph{d}_{X}(\sum_{k=1}^{\infty}f_{n_{k}},X_{a})\leqslant\emph{d}_{X}(\sum_{n=1}^{\infty}f_{n},X_{a})=1,$
that is to say, the subsequence $\\{f_{n_{k}}\\}_{k=1}^{\infty}$ still builds
a lattice isometric copy of $\ell_{\infty}$. Therefore, if we can find a
subsequence, say $\\{f_{n_{k}}\\}_{k=1}^{\infty}$, which is uniformly bounded,
then it is enough to put $f=\sum_{k=1}^{\infty}f_{n_{k}}$. Clearly, $f\in
L_{\infty}$. On the other hand, if some subsequence
$\\{f_{n_{k}}\\}_{k=1}^{\infty}$ is unbounded, then since $X\hookrightarrow
L_{1}+L_{\infty}$, so (taking further subsequence, if needed) we see that
$\sum_{k=1}^{\infty}m(\Omega_{n_{k}})<\infty$. .

$1^{\circ}$. Suppose that $f^{*}\chi_{(c,\infty)}\in(X_{F})_{a}$ for some
(equivalently, for all) $c>0$. Consequently, we have that

$1=d_{X_{F}}(f^{*},(X_{F})_{a})=\lim\limits_{n\rightarrow\infty}\lVert
f^{*}\chi_{(0,\frac{1}{n})\cup(n,\infty)}\rVert_{X_{F}}=\lim\limits_{n\rightarrow\infty}\lVert
f^{*}\chi_{(0,\frac{1}{n})}\rVert_{X_{F}},$

where the second equality follows from Theorem 3.4. Since $X\in(OC)$, so we
can find $b>0$ with

$\lVert(F(f^{*}))\chi_{(0,b)}\rVert_{X}\leqslant\frac{1}{M},$

where the constant $M>0$ is from the condition (${\mathscr{M}}$). Therefore,
we have the following inequalities

$\lVert F(C(f^{*}\chi_{(0,b)}))\rVert_{X}\leqslant M\lVert
F(f^{*}\chi_{(0,b)})\rVert_{X}=M\lVert(F(f^{*}))\chi_{(0,b)}\rVert_{X}\leqslant
1.$

But this means that $\lVert f^{*}\chi_{(0,b)}\rVert_{C(X_{F})}\leqslant 1$.
What’s more, thanks to our assumptions,
$f^{*}\chi_{(b,\infty)}\in(X_{F})_{a}$, whence

$1=d_{X_{F}}(f^{*},(X_{F})_{a})\leqslant\lVert
f^{*}-f^{*}\chi_{(b,\infty)}\rVert_{X_{F}}=\lVert
f^{*}\chi_{(0,b)}\rVert_{X_{F}}\leqslant\lVert
C(f^{*}\chi_{(0,b)})\rVert_{X_{F}}\leqslant 1,$

that is, $\lVert C(f^{*}\chi_{(0,b)})\rVert_{X_{F}}=1$. Now we can apply
Theorem 4.1 and finish the proof of this part.

$2^{\circ}$. Assume that $f^{*}\chi_{(c,\infty)}\in X_{F}\setminus(X_{F})_{a}$
and $f^{*}\chi_{(0,c)}\in(X_{F})_{a}$ for some (equivalently, for all) $c>0$.
Referring once again to Theorem 3.4, we have the following equalities

$1=d_{X_{F}}(f^{*},(X_{F})_{a})=\lim\limits_{n\rightarrow\infty}\lVert
f^{*}\chi_{(0,\frac{1}{n})\cup(n,\infty)}\rVert_{X_{F}}=\lim\limits_{n\rightarrow\infty}\lVert
f^{*}\chi_{(n,\infty)}\rVert_{X_{F}}.$

Just as before, since $X\in(OC)$, so

$\lVert(F(f^{*}))\chi_{(b,\infty)}\rVert_{X}\leqslant\frac{1}{M}\text{ for
some }b>0\text{ big enough},$

where $M>0$ is from the condition (${\mathscr{M}}$). Moreover, using the
condition (${\mathscr{M}}$), we get

$\lVert F(C(f^{*}\chi_{(b,\infty)}))\rVert_{X}\leqslant M\lVert
F(f^{*}\chi_{(b,\infty)})\rVert_{X}=M\lVert(F(f^{*}))\chi_{(b,\infty)}\rVert_{X}\leqslant
1.$

that is, $\lVert f^{*}\chi_{(b,\infty)}\rVert_{C(X_{F})}\leqslant 1$. Now, we
claim that

($=$) $\lVert f^{*}\chi_{(b,\infty)}\rVert_{C(X_{F})}=1.$

To prove ($=$ ‣ 4) it is enough to show that

$\lVert F(C((1+\varepsilon)f^{*}\chi_{(b,\infty)}))\rVert_{X}>1\text{ for all
}\varepsilon>0.$

Take $\varepsilon>0$. Note that

$C\left((1+\varepsilon)f^{*}\chi_{(b,\infty)}\right)(x)=(1+\varepsilon)\frac{1}{x}\int_{b}^{x}f^{*}(t)dt\geqslant(1+\varepsilon)\left(1-\frac{b}{x}\right)f^{*}(x),$

for $x\geqslant b$. Moreover, there exists $B>b$ such that

$(1+\varepsilon)\left(1-\frac{b}{x}\right)f^{*}(x)\geqslant\left(1+\frac{\varepsilon}{2}\right)f^{*}(x)\text{
for }x\geqslant B.$

In consequence, we have

$\lVert F(C((1+\varepsilon)f^{*}\chi_{(b,\infty)}))\rVert_{X}\geqslant\lVert
F(C((1+\varepsilon)f^{*}\chi_{(B,\infty)}))\rVert_{X}\geqslant\lVert
F\left(\left(1+\frac{\varepsilon}{2}\right)f^{*}\chi_{(B,\infty)}\right)\rVert_{X}>1,$

where the last inequality follows from the fact that $\lVert
f^{*}\chi_{(B,\infty)}\rVert_{X_{F}}=1$ (to see this just note that due to our
assumptions $f^{*}\chi_{(0,B)}\in(X_{F})_{a}$ and then
$1=d_{X_{F}}(f^{*},(X_{F})_{a})\leqslant\lVert
f^{*}-f^{*}\chi_{(0,B)}\rVert_{X_{F}}=\lVert
f^{*}\chi_{(B,\infty)}\rVert_{X_{F}}\leqslant 1$) and the definition of the
Luxemburg–Nakano norm. But this proves the claim ($=$ ‣ 4). Again, we are left
to use Theorem 4.1. We have therefore completed the proof. ∎

######  Corollary 4.3.

Let $X$ be an order continuous rearrangement invariant sequence space with the
Fatou property and let $F$ be an Orlicz function. Assume that the
Calderón–Lozanovskiĭ class is closed under the Cesàro operator $C$. If $F$
vanishes only at zero and does not satisfy the $\Delta_{2}(0)$-condition, then
the space $C(X_{F})$ contains a lattice isometric copy of $\ell_{\infty}$.

###### Proof.

To begin with, since $X=X_{a}\hookrightarrow
c_{0}\hookrightarrow\ell_{\infty}$ and the Orlicz function $F$ vanishes only
at 0, it follows that the condition $\Delta_{2}(0)$ is equivalent to the
condition $\delta_{2}^{X}$ (see [HN05, Lemma 4])∥∥∥We leave the definition of
the condition $\delta_{2}^{X}$ due to the fact that it concerns the
Musielak–Orlicz functions, which we will not consider here. At the same time,
we refer for details to [FH99, p. 525].. Due to the lack of the
$\delta_{2}^{X}$-condition and Lemma 2.4 from [FH99], the Calderón–Lozanovskiĭ
space $X_{F}$ contains a lattice isometric copy of $\ell_{\infty}$ (note only
that the results from [FH99] have been proven for a wider than $X_{F}$ class
of spaces, the so-called generalized Calderón–Lozanovskiĭ spaces $X_{M}$, in
which the Orlicz function $F$ is replaced by the much more general
Musielak–Orlicz function $M$). Exactly as in the function’ case (cf. Corollary
4.2), citing Theorem B along with Theorem 3.5, we can find a sequence $f\in
X_{F}$ with

$\lVert
f\rVert_{X_{F}}=d_{X_{F}}(f,(X_{F})_{a})=d_{X_{F}}(f^{*},(X_{F})_{a})=1.$

What’s more $\lVert f^{*}\rVert_{X_{F}}=1$ and, consequently, $\lVert
F(f^{*})\rVert_{X}\leqslant 1$. Now, because the Calderón–Lozanovskiĭ class is
closed under the Cesàro operator $C$ we conclude that $\lVert
F(C(f^{*}))\rVert_{X_{F}}<\infty$. Since $X\in(OC)$, there exists
$n\in\mathbb{N}$ such that

$\lVert(F(C(f^{*})))\chi_{\\{n,n+1,...\\}}\rVert_{X}\leqslant 1.$

Moreover, it is not hard to see that

$C(f^{*}\chi_{\\{n,n+1,...\\}})\leqslant(C(f^{*}))\chi_{\\{n,n+1,...\\}},$

whence also

$\lVert F\left(C(f^{*}\chi_{\\{n,n+1,...\\}})\right)\rVert_{X}\leqslant\lVert
F\left((C(f^{*}))\chi_{\\{n,n+1,...\\}}\right)\rVert_{X}\leqslant 1.$

But this means that $\lVert
f^{*}\chi_{\\{n,n+1,...\\}}\rVert_{C(X_{F})}\leqslant 1$. Note also that
$f^{*}\chi_{\\{1,2,...,n-1\\}}\in(X_{F})_{a}$, so

$1=d_{X_{F}}(f^{*},(X_{F})_{a})\leqslant\lVert
f-f^{*}\chi_{\\{1,2,...,n-1\\}}\rVert_{X_{F}}=\lVert
f^{*}\chi_{\\{n,n+1,...\\}}\rVert_{X_{F}}\leqslant 1.$

Now, after some obvious modifications (such as converting integrals with sums
or using the discrete Cesàro operator in place of his continuous counterpart)
in the proof of equality ($=$ ‣ 4) from Corollary 4.2, we can show that

$\lVert f^{*}\chi_{\\{n,n+1,...\\}}\rVert_{C(X_{F})}=1.$

Keeping in mind that the order continuous part of a rearrangement invariant
sequence space is always non-trivial we can complete the proof by referring
once again to Theorem 4.1. ∎

The assumption that appeared in Theorem 4.1 may at first glance look quite
wishfull, but it follows from the proof of the next result that rather natural
subclass of the class of rearrangement invariant spaces satisfies this
condition.

######  Proposition 4.4.

Let $X$ be a rearrangement invariant space on $(0,\infty)$ with the Fatou
property such that $L_{\infty}\hookrightarrow X$ and the Cesàro operator is
bounded on $X$. Suppose also that the space $X$ satisfies the condition
$({\mathscr{D}}_{\infty})$ from Corollary 3.8. Then the space $CX$ contains a
lattice isometric copy of $\ell_{\infty}$.

###### Proof.

To begin with, take $f=\chi_{(0,\infty)}/\varphi_{X}(\infty)$. The assumption
$L_{\infty}\hookrightarrow X$ implies that $\lVert f\rVert_{X}=1$. Moreover,

$\emph{d}_{X}(f,X_{a})=\lim_{n\rightarrow\infty}\lVert
f^{*}\chi_{(n,\infty)}\rVert_{X}=\lim_{n\rightarrow\infty}\lVert(f^{*}\chi_{(n,\infty)})^{*}\rVert_{X}=\lVert
f\rVert_{X}=1,$

where the first equality follows from Corollary 3.7 and the fact due to the
condition $({\mathscr{D}}_{\infty})$ that every function supported on the set
of finite measure is order continuous. Consequently, we have the equalities
$\lVert f\rVert_{X}=\emph{d}_{X}(f,X_{a})=1$. Again, due to the condition
$({\mathscr{D}}_{\infty})$, the ideal $X_{a}$ is non-trivial, so we can use
Theorem B to conclude that the space $X$ contains a lattice isometric copy of
$\ell_{\infty}$. On the other hand, for such $f$ it is not hard to see that

$\lVert f^{*}\chi_{(b,\infty)}\rVert_{CX}=1\text{ for every }b>0.$

In fact,

$\displaystyle\lVert
f^{*}\chi_{(b,\infty)}\rVert_{CX}=\frac{1}{\varphi_{X}(\infty)}\lVert
C(\chi_{(b,\infty)})\rVert_{X}=\frac{1}{\varphi_{X}(\infty)}\lVert(C(\chi_{(b,\infty)}))^{*}\rVert_{X}=\frac{1}{\varphi_{X}(\infty)}\lVert\chi_{(0,\infty)}\rVert_{X}=1.$

But this means that the condition in Theorem 4.1 is satisfied with $a=0$ and
$b>0$. We therefore conclude that the space $CX$ contains a lattice isometric
copy of $\ell_{\infty}$ and the proof is done. ∎

Some spaces, such as Orlicz spaces (and certain variants of Orlicz spaces like
Orlicz–Lorentz spaces or Musielak–Orlicz spaces; cf. [Hud98, Remarks on p.
526]), have this (from our perspective) pleasant property that if they contain
a lattice isomorphic copy of $\ell_{\infty}$, then they actually contain a
lattice isometric copy of $\ell_{\infty}$. This simple observation along with
the results from [KT17] leads us to the following

######  Theorem 4.5.

Let $X$ be an order continuous rearrangement invariant space with the Fatou
property and let $F$ be an Orlicz function. Suppose that the Cesàro operator
$C$ is bounded on the Calderón–Lozanovskiĭ space $X_{F}$. If the Cesàro space
$C(X_{F})$ contains a lattice isometric copy of $\ell_{\infty}$, then $X_{F}$
contains it also.

###### Proof.

Assume that $X_{F}$ does not contain a lattice isometric copy of
$\ell_{\infty}$. Then the space $X_{F}$ does not contain even an order
isomorphic copy of $\ell_{\infty}$. However, in view of the Lozanovskiĭ
theorem [Loz69], this means that $X_{F}$ is order continuous. At this stage,
applying [KT17, Theorem 3], we obtain that the space $CX$ is order continuous
as well. Consequently, the space $CX$ cannot contain a lattice isometric copy
of $\ell_{\infty}$ and we are done. ∎

As we already mentioned, if $X$ is a Banach function space with a trivial
ideal $X_{a}$, then the characterization given in Theorem B is no longer valid
in general. What seems to be quite interesting, the case of Cesàro spaces $CX$
(at least, when $X$ is a rearrangement invariant space) is different, because
a suppport of the ideal $(CX)_{a}$ is equal to a support of the space $X$ even
if the ideal $X_{a}$ is trivial (see [KT17, Lemma 8]). Therefore, we can still
use Theorem B and obtain the next

######  Theorem 4.6.

Let $X$ be a rearrangement invariant space with the Fatou property such that
the ideal $X_{a}$ is trivial. Suppose that there exists a function $f>0$ with

  * (i)

$C(f)(0^{+})=1$,

  * (ii)

$\lVert f\rVert_{CX}=\lVert\emph{id}\colon X\rightarrow L_{\infty}\rVert$.

Then the Cesàro space $CX$ contains a lattice isometric copy of
$\ell_{\infty}$.

###### Proof.

At the beginning observe that $X\hookrightarrow L_{\infty}$, because
$X_{a}=\\{0\\}$ (cf. [KT17, Theorem B]). Moreover, since we can always replace
the norm $\lVert\cdot\rVert_{X}$ by a constant multiple of itself, it follows
that without loss of generality we can assume that $\lVert\text{id}\colon
X\rightarrow L_{\infty}\rVert=1$.

Let $f$ be a function whose existence is guaranteed by our assumptions, that
is to say, $C(f)(0^{+})=1$ and $\lVert f\rVert_{CX}=\lVert\text{id}\colon
X\rightarrow L_{\infty}\rVert=1$. We will show that

($\ast$) $\emph{d}_{CX}(f,(CX)_{a})=1.$

Indeed, we have the following inequalities

$\displaystyle\emph{d}_{CX}(f,(CX)_{a})$ $\displaystyle=\inf\\{\lVert
f-g\rVert_{CX}\colon g\in(CX)_{a}\text{ and }0<g\leqslant f\\}$
$\displaystyle=\inf\\{\lVert C(f)-C(g)\rVert_{X}\colon g\in(CX)_{a}\text{ and
}0<g\leqslant f\\}$ $\displaystyle\quad\quad\quad(\text{because }f-g\geqslant
0)$ $\displaystyle\geqslant\inf\\{\lVert C(f)-C(g)\rVert_{L_{\infty}}\colon
g\in(CX)_{a}\text{ and }0<g\leqslant f\\}$
$\displaystyle\quad\quad\quad(\text{in view of the embedding
}X\xhookrightarrow{1}L_{\infty})$ $\displaystyle\geqslant\inf\\{\left\lvert
C(f)(0^{+})-C(g)(0^{+})\right\rvert\colon g\in(CX)_{a}\text{ and }0<g\leqslant
f\\}=1,$

where the first equality is due to Lemma 3.1 and the last equality is the
consequence of the fact that $C(g)(0^{+})=0$, whenever $g\in(CX)_{a}$ and the
ideal $X_{a}$ is trivial (see [KT17, Lemma 14]). In summary, we proved the
claim $(\ast)$, because we have shown that

$1\leqslant\emph{d}_{CX}(f,(CX)_{a})\leqslant\lVert f\rVert_{CX}\leqslant 1.$

Now, to complete the proof, it is enough to note that
$\operatorname{supp}((CX)_{a})=\operatorname{supp}(X)$ (see [KT17, Lemma 8])
and use once again Hudzik’s result from [Hud98] (cf. Theorem B). ∎

######  Remark 4.7.

Let us now make a few remarks regarding the above Theorem 4.6:

(a) The assumption about the symmetry of the space $X$ cannot in general be
omitted (because, for example, the space $L_{\infty}(w)$ with the weight $w$
defined as $w(x)=x$ (for $x>0$) contains a lattice isometric copy of
$\ell_{\infty}$, but $C(L_{\infty}(w))\equiv L_{1}$ and the space $L_{1}$ is
even separable).

(b) It may happen (consider, for example, the space $X=L_{1}\cap L_{\infty}$
on $(0,\infty)$) that the space $X$ contains a lattice isometric copy of
$\ell_{\infty}$ but the Cesàro space $CX$ is trivial.

(c) If $X$ is a rearrangement invariant space on $(0,1)$ with the Fatou
property such that the ideal $X_{a}$ is trivial, then the space $X$ is just
$L_{\infty}$ up to equivalence of norms (for the proof see, e.g. [KT17,
Theorem B]).

######  Corollary 4.8.

Let $X$ be a rearrangement invariant space on $(0,\infty)$ with the Fatou
property such that the Cesàro space $CX$ is non-trivial. Suppose that either
the ideal $X_{a}$ is non-trivial and the Cesàro operator $C$ is bounded on $X$
or $X_{a}=X$. Then the Cesàro space $C(X\cap L_{\infty})$ contains a lattice
isometric copy of $\ell_{\infty}$.

###### Proof.

This fact follows (more or less) directly from Theorem 4.6. We just need to
check a few details. Moreover, we will give the proof only in the case when
the ideal $X_{a}$ is non-trivial and the operator $C$ is bounded on $X$,
because the remaining case is much easier.

First of all, let us note that

$X\cap L_{\infty}\overset{1}{\hookrightarrow}L_{\infty}\text{ (so, also
}(X\cap L_{\infty})_{a}=\\{0\\})\text{ and }C(X\cap L_{\infty})\neq\\{0\\}.$

Put $f_{\varepsilon}=\chi_{(0,\varepsilon)}$, where $\varepsilon>0$. We claim
that

($\star$) $\text{there exists }\varepsilon_{0}>0\text{ such that }\lVert
f_{\varepsilon_{0}}\rVert_{CX}\leqslant 1.$

To show $(\star)$, it is enough to observe that $\varphi_{X}(0^{+})=0$ (this
is actually equivalent to the fact that the ideal $X_{a}$ is non-trivial, see
[BS88, Theorem 5.5 (a), pp. 67–68]; cf. [KT17, Theorem B]) and the Cesàro
operator $C$ is bounded on $X$, hence

$\lVert f_{\varepsilon}\rVert_{CX}=\lVert
C(\chi_{(0,\varepsilon)})\rVert_{X}\leqslant\lVert C\rVert_{X\rightarrow
X}\varphi_{X}(\varepsilon)\rightarrow 0\text{ as }\varepsilon\rightarrow 0.$

Consequently, we have the following equalities

$\lVert f_{\varepsilon_{0}}\rVert_{C(X\cap L_{\infty})}=\max\\{\lVert
f_{\varepsilon_{0}}\rVert_{CX},\lVert
f_{\varepsilon_{0}}\rVert_{Ces_{\infty}}\\}=\max\\{\lVert
f_{\varepsilon_{0}}\rVert_{CX},1\\}=1.$

This means that we have defined the function $f_{\varepsilon_{0}}$ with

$C(f_{\varepsilon_{0}})(0^{+})=1\text{ and }\lVert
f_{\varepsilon_{0}}\rVert_{C(X\cap L_{\infty})}=1=\lVert\text{id}\colon X\cap
L_{\infty}\rightarrow L_{\infty}\rVert.$

Now, because the ideal $(X\cap L_{\infty})_{a}$ is trivial, we can use Theorem
4.6 and end the proof. ∎

######  Remark 4.9.

(a) Let $X=L_{p}+L_{\infty}$ on $(0,\infty)$, where $1<p<\infty$. It is clear
that in this case the ideal $X_{a}$ is non-trivial******Precisely, it can be
shown that $(L_{p}+L_{\infty})_{a}=\\{f\in L_{p}+L_{\infty}\colon
f^{*}(\infty)=0\\}$ (remembering that $\lVert
f\rVert_{L_{p}+L_{\infty}}\asymp(\int_{0}^{1}f^{*}(t)^{p}dt)^{1/p}$ (see
[BL76, p. 109]) one can, for example, modify the proof of an analogous fact
for the case when $p=1$ included in [KT17]; cf. [KPS82, pp. 115–116]). and the
Cesàro operator is bounded on $X$. However, it is also clear††††††Just note
that $L_{\infty}\overset{1}{\hookrightarrow}L_{p}+L_{\infty}$, so $\lVert
f\rVert_{L_{\infty}}\leqslant\max\\{\lVert f\rVert_{L_{p}+L_{\infty}},\lVert
f\rVert_{L_{\infty}}\\}\leqslant\lVert f\rVert_{L_{\infty}}$. that $X\cap
L_{\infty}\equiv L_{\infty}$. Therefore, using Theorem 4.6 (via Corollary 4.8)
we re-prove Proposition 4.9 from [KKM21] which states that the space
$Ces_{\infty}$ contains, as one would expect, a lattice isometric copy of
$\ell_{\infty}$.

(b) Let $X=L_{F}$ be an Orlicz space generated by an Orlicz function $F$ such
that $b_{F}\coloneqq\sup\\{x>0\colon F(x)<\infty\\}=1$, $F(1)\leqslant 1$ and
the left derivative of $F$ at $x=1$ is finite. Then there exists an Orlicz
function, say $G$, with $b_{G}=\infty$ such that $L_{F}\equiv L_{G}\cap
L_{\infty}$. The proof of this fact is analogous to the proof of [Mal89,
Theorem 12.1 (a), p. 99]. However, to get an isometry instead of isomorphism,
we need to notice two things: (1) due to the assumption regarding the
derivative of $F$, we can always extend $F$ (but not necessarily in a uniqe
way!) to an Orlicz function, say $\widetilde{F}$, with
$b_{\widetilde{F}}=\infty$; (2) the function $(\widetilde{F}\vee
F_{\infty})(x)\coloneqq\max\\{\widetilde{F}(x),F_{\infty}(x)\\}$, where

$\displaystyle F_{\infty}(x)=\begin{cases}0&\text{for }0\leqslant x\leqslant
1\\\ \infty&\text{for }x>1\end{cases},$

is equal to $F$. Now, the embedding
$L_{F}\overset{1}{\hookrightarrow}L_{\widetilde{F}}\cap L_{\infty}$ follows
directly from [Mal89, Theorem 12.1]. Therefore, it remains to show that also

($\sharp$) $L_{\widetilde{F}}\cap
L_{\infty}\overset{1}{\hookrightarrow}L_{F}.$

Take $f\in L_{\widetilde{F}}\cap L_{\infty}$ with $\max\\{\lVert
f\rVert_{L_{\widetilde{F}}},\lVert f\rVert_{L_{\infty}}\\}=1$. Let us consider
two cases:

$1^{\circ}$. $\lVert f\rVert_{L_{\infty}}=1$. Then $\left\lvert
f(x)\right\rvert\leqslant 1=b_{F}$, so $\int_{0}^{\infty}F(\left\lvert
f(x)\right\rvert)dx=\int_{0}^{\infty}\widetilde{F}(\left\lvert
f(x)\right\rvert)dx\leqslant 1$. Moreover, since $\lVert
f\rVert_{L_{\infty}}=1=b_{F}$, it follow that $\int_{0}^{\infty}F(\left\lvert
f(x)\right\rvert/\lambda)dx=\infty$ for every $\lambda<1$. Therefore, $\lVert
f\rVert_{L_{F}}=1$ as well.

$2^{\circ}$. $\lVert f\rVert_{L_{\widetilde{F}}}=1$ and $\lVert
f\rVert_{L_{\infty}}<1$. Then $\int_{0}^{\infty}F(\left\lvert
f(x)\right\rvert)dx=\int_{0}^{\infty}\widetilde{F}(\left\lvert
f(x)\right\rvert)dx\leqslant 1$, so $\lVert f\rVert_{L_{F}}\leqslant 1$. In
order to obtain a contradiction, suppose that $\lVert f\rVert_{L_{F}}<1$. Then
$\int_{0}^{\infty}F(\left\lvert f(x)\right\rvert/\lambda_{0})dx\leqslant 1$
for some $\lambda_{0}<1$. Hence, $\left\lvert
f\right\rvert/\lambda_{0}\leqslant 1=b_{F}$ and, consequently, also
$\int_{0}^{\infty}\widetilde{F}(\left\lvert
f(x)\right\rvert/\lambda_{0})dx\leqslant 1$. However, this implies that
$\lVert f\rVert_{L_{\widetilde{F}}}<1$, which is impossible. The proof of
$(\sharp)$ has been completed.

In particular, if $1\leqslant p<\infty$ and the Orlicz function $F_{p,\infty}$
is given in the following way

$\displaystyle F_{p,\infty}(x)=\begin{cases}x^{p}&\text{for }0\leqslant
x\leqslant 1\\\ \infty&\text{for }x>1\end{cases},$

then $L_{F_{p,\infty}}\equiv L_{p}\cap L_{\infty}$.

The above spaces provided some natural examples of spaces which satisfy the
assumptions of Corollary 4.8 (of course, in order to use Corollary 4.8, the
functions $F$ and $G$ cannot be completely arbitrary; however, translating the
assumptions of Corollary 4.8 into the language of the Orlicz spaces - the
$\Delta_{2}$-condition and the Matuszewska–Orlicz indices (cf. [LT79,
Proposition 2.b.5, p. 139]) will surely appear - does not present much
difficulty).

(c) Let $\varphi$ be an increasing concave function with $\varphi(0^{+})=1$
(this normalization is basically inessential). Since the norm in the Lorentz
space $\Lambda_{\varphi}$ is given by the formula

$\lVert f\rVert_{\Lambda_{\varphi}}=\lVert
f\rVert_{L_{\infty}}+\int_{0}^{\infty}f^{*}(t)\varphi^{\prime}(t)dt=\lVert\left(\lVert
f\rVert_{\Lambda_{\int_{0}^{t}\varphi^{\prime}(s)ds}},\lVert
f\rVert_{L_{\infty}}\right)\rVert_{\ell_{1}^{2}},$

it follows that

$\Lambda_{\varphi}\equiv\Lambda_{\int_{0}^{t}\varphi^{\prime}(s)ds}\cap
L_{\infty}\overset{1}{\hookrightarrow}L_{\infty}\text{ and
}\left(\Lambda_{\int_{0}^{t}\varphi^{\prime}(s)ds}\right)_{a}\neq\\{0\\},$

where, of course, the intersection space is considered with the equivalent
two-dimensional $\ell_{1}$-norm given above (after all, all norms on
$\mathbb{R}^{2}$ are equivalent). However, in this case the proof of Corollary
4.8 does not work. On the other hand, if we consider the general family of
equivalent norms on the space $X\cap L_{\infty}$ of the form

$\lVert f\rVert_{X\oplus_{F}L_{\infty}}=\lVert(\lVert f\rVert_{X},\lVert
f\rVert_{L_{\infty}})\rVert_{F}\text{ for }f\in X\cap L_{\infty},$

where $\lVert\cdot\rVert_{F}$ is a norm on $\mathbb{R}^{2}$ with the ideal
property, then for our proof to work, it is enough to require that

$\lVert(x_{0},1)\rVert_{F}=\lVert(0,1)\rVert_{F}=1\text{ for some }x_{0}>0,$

which means that the unit sphere in $(\mathbb{R}^{2},\lVert\cdot\rVert_{F})$
contains an order interval beginning at point $(0,1)$. For example, the
“hexagonal” norm

$\lVert(x,y)\rVert_{hex}=\max\left\\{\left\lvert
x+\frac{\sqrt{3}}{3}y\right\rvert,\left\lvert
x-\frac{\sqrt{3}}{3}y\right\rvert,\frac{2\sqrt{3}}{3}\left\lvert
y\right\rvert\right\\}\text{ for }x,y\in\mathbb{R},$

seems to be a good candidate for such a norm. But we will stop here.

(d) The Cesàro operator $C$ is not bounded on the Zygmund space $L\log L$ but
the Cesàro space $C(L\log L)$ is non-trivial (cf. [KKM21]) and the space
$L\log L$ is order continuous. On the other hand, the Cesàro operator is
bounded on $L_{\infty}$ but the ideal $(L_{\infty})_{a}$ is trivial.

##  5\. Appendix. Local approach to order continuity in abstract Cesàro
sequence spaces

We are going to present here a result characterizing the ideal of order
continuous elements in the Cesàro sequence spaces. This result, as well as
it’s proof, is analogous to Theorem 7 from [KT17] (to be fair, due to the fact
that $X_{a}$ is always non-trivial subspace of $X$, whenever $X$ is a Banach
sequence space, it is much simpler than its function counterpart). However, in
order to relieve the reader from the tedious obligation to check all the
details of this proof on his own, we rather prefer to provide a brief sketch
of the argument.

######  Proposition 5.1.

Let $X$ be a rearrangement invariant sequence space such that the Cesàro
operator $C$ is bounded on $X$. Then $(CX)_{a}=C(X_{a})$.

###### Proof.

To prove the equality $(CX)_{a}=C(X_{a})$ we will show the following
inclusions

$C(X_{a})\subset(CX)_{a}\subset(CX)_{b}\subset C(X_{a}).$

The first inclusion $C(X_{a})\subset(CX)_{a}$ holds true even for Banach
sequence spaces and its proof is mutatis mutandis the same as Lemma 11 in
[KT17], while the second inclusion $(CX)_{a}\subset(CX)_{b}$ follows directly
from [BS88, Theorem 3.11, p. 18]. Consequently, it remains only to show that
$(CX)_{b}\subset C(X_{a})$. We will do this in two steps.

$1^{\circ}$. Take $\chi_{A}$, where $A\subset\mathbb{N}$ and $\max(A)\coloneqq
n_{0}<\infty$. Let $n>n_{0}$. Since $p_{X}>1$‡‡‡‡‡‡Symbol $p_{X}$ denote here
the lower Boyd index. We will only remaind that the classical Hardy operator
$C$ is bounded in a rearrangement invariant space $X$ if and only if $p_{X}>1$
(for details we refer, for example, to [KPS82, Theorem 6.6, p. 138], [LT79,
pp. 129–131]) or [KMP07, pp. 126–129] and references given there)., so
$\ell_{p}\hookrightarrow X$ for all $1<p<p_{X}$ (this fact follows from
Proposition 2.b.3 in [LT79], which is admittedly formulated for rearrangement
invariant function spaces, but exactly as noted at the top of the page 132,
after some notation changes due to the use of discrete versions of Boyd’s
indices, his proof remains essentially unchanged in the comparison to the
proof of the afromentioned Proposition 2.b.3) and we have the following
inequalities

$\lVert
C(\chi_{A})\chi_{\\{n,n+1,\dots\\}}\rVert_{X}\leqslant\lVert\frac{1}{k}\chi_{\\{n,n+1,\dots\\}}(k)\rVert_{X}\preccurlyeq\lVert\frac{1}{k}\chi_{\\{n,n+1,\dots\\}}(k)\rVert_{\ell_{p}}\rightarrow
0,$

as $n\rightarrow\infty$.

$2^{\circ}$. Take $a=\\{a_{n}\\}_{n=1}^{\infty}\in(CX)_{b}$. Let
$\\{s_{n}\\}_{n=1}^{\infty}$ be a sequence of sequences such that $s_{n}$
converges to $a$ in the norm of $CX$ and
$\\#(\operatorname{supp}(s_{n}))<\infty$ for all $n\in\mathbb{N}$. It follows
from the previous step that $\\{s_{n}\\}_{n=1}^{\infty}\subset C(X_{a})$. Now,
using the reverse triangle inequality, we get

$\lVert C(\left\lvert a\right\rvert)-C(\left\lvert
s_{n}\right\rvert)\rVert_{X}\leqslant\lVert C(\left\lvert
a-s_{n}\right\rvert)\rVert_{X}=\lVert a-s_{n}\rVert_{CX}\rightarrow 0,$

as $n\rightarrow\infty$. Since $X_{a}$ is a closed ideal of $X$, so
$C(\left\lvert a\right\rvert)\in X_{a}$ and the proof is completed. ∎

The immediate consequence of the above result is the following fact (which, in
the case of the abstract Cesàro function spaces, was the main result of
[KT17]).

######  Corollary 5.2.

Let $X$ be a rearrangement invariant sequence space such that the Cesàro
operator $C$ is bounded on $X$. Then $CX$ is order continuous if and only if
$X$ is order continuous as well.

## References

  * [AK06] F. Albiac and N. J. Kalton, Topics in Banach space theory, Springer-Verlag, New York, 2006.
  * [AB85] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Pure and Applied Math., Academic Press Inc., 1985.
  * [Ast12] S. V. Astashkin, Geometrical properties of Banach spaces generated by sublinear operator, Positivity 17 (2012), no. 2, 223–234.
  * [ALM17] S. V. Astashkin, K. Leśnik and M. Maligranda, Isomorphic structure of Cesàro and Tandori spaces, Canad. J. Math. 71 (2019), no. 3, 501–532.
  * [AM09] S. V. Astashkin and L. Maligranda, Structure of Cesàro function spaces, Indag. Math. (N.S.) 20 (2009), no. 3, 329–379.
  * [AM14] S. V. Astashkin and L. Maligranda, Structure of Cesàro function spaces: a survey, Banach Center Publ. 102 (2014), 13–40.
  * [BS88] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988.
  * [Ben96] G. Bennett, Factorizing Classical Inequalities, Mem. Amer. Math. Soc. 120 (1996), no. 576, 138 pp.
  * [BL76] J. Bergh and J. Löfström, Interpolation spaces, Springer-Verlag, Berlin-New York, 1976.
  * [Boy69] D. W. Boyd, Indices of function spaces and their relationship to interpolation, Canad. J. Math. 21 (1969), 1245–1254.
  * [Bou80] J. Bourgain, $l_{\infty}/c_{0}$ has no equivalent strictly convex norm, Proc. Amer. Math. Soc. 78 (1980), 225–226.
  * [CSCY00] F. Cabello Sánchez, J. M. F. Castillo and D. Yost, Sobczyk’s Theorem from A to B, Extracta Math. 15 (2000), vol. 2, 391–420.
  * [Cal64] A. P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190.
  * [CCH03] S. Chen, Y. Cui and H. Hudzik, Isometric copies of $l^{1}$ and $l^{\infty}$ in Orlicz spaces equipped with the Orlicz norm, Proc. Amer. Math. Soc. 132 (2003), no. 2, 473–480.
  * [CKP14] M. Ciesielski, P. Kolwicz and A. Panfil, Local monotonicity structure of symmetric spaces with applications, J. Math. Anal. Appl. 409 (2014), 649–662.
  * [CHL17] Y. Cui, H. Hudzik and K. Leśnik, Order asymptotically isometric copies of $l^{\infty}$, $l^{1}$ and $c_{0}$ in Banach function spaces, J. Math. Anal. Appl. 455 (2017), 1139–1151.
  * [CR16] G. P. Curbera and W. J. Ricker, Abstract Cesàro spaces: Integral representation, J. Math. Anal. Appl. 441 (2016), no. 1, 25–44.
  * [CR06] G. P. Curbera and W. J. Ricker, Banach lattices with the Fatou property and optimal domains of kernel operators, Indag. Math. (N.S.) 17 (2006), no. 2, 187–204.
  * [DS07] O. Delgado and J. Soria, Optimal domain for the Hardy operator, J. Funct. Anal. 244 (2007), no. 1, 119–133.
  * [Dis76] J. Diestel, Geometry of Banach spaces. Selected topics, Lecture Notes in Math. 485, Springer-Verlag, Berlin and New York, 1976.
  * [Dis84] J. Diestel, Sequences and series in Banach spaces, Springer-Verlag, Berlin and New York, 1984.
  * [Dow00] P. N. Dowling, On $\ell^{\infty}$ subspaces of Banach spaces, Collect. Math. 51 (2000), no. 3, 255–260.
  * [DR91] L. Drewnowski and J. W. Roberts, On the primariness of the Banach space $l_{\infty}/c_{0}$, Proc. Amer. Math. Soc. 112 (1991), no. 4, 949–957.
  * [FH99] P. Foralewski and H. Hudzik, On some geometrical and topological properties of generalized Calderón–Lozanovskiĭ sequence spaces, Houston J. Math. 25 (1999), no. 3, 523–542.
  * [FH97] P. Foralewski and H. Hudzik, Some basic properties of generalized Calderón–Lozanovskiĭ spaces, Collectanea Math. 48 (1997), no. 4–6, 523–538.
  * [GH99] A. S. Granero and H. Hudzik, On some proximinal subspaces of modular spaces, Acta Math. Hungar. 85 (1999), no. 1–2, 59–79.
  * [GH96] A. S. Granero and H. Hudzik, The classical Banach spaces $\ell_{\varphi}/h_{\varphi}$, Proc. Amer. Math. Soc. 124 (1996), no. 12, 3777–3787.
  * [Hud98] H. Hudzik, Banach lattices with order isometric copies of $l^{\infty}$, Indag. Math. (N.S.) 9 (1998), no. 4, 521–527.
  * [HKM96] H. Hudzik, A. Kamińska and M. Mastyło, Geometric properties of some Calderón–Lozanovskiĭ spaces and Orlicz–Lorentz spaces, Houston J. Math. 22 (1996), no. 3, 639–663.
  * [HM93] H. Hudzik and M. Mastyło, Almost isometric copies of $l^{\infty}$ in some Banach spaces, Proc. Amer. Math. Soc. 119 (1993), no 1, 209–215.
  * [HN05] H. Hudzik and A. Narloch, Relationships between monotonicity and complexity rotundity properties with some consequences, Math. Scand. 96 (2005), no. 2, 289–306.
  * [HW06a] H. Hudzik and K. Wlaźlak, Rotundity properties in Banach spaces via sublinear operators, Nonlinear Anal. 64 (2006), no. 6, 1171–1188.
  * [HW06b] H. Hudzik and K. Wlaźlak, Monotonicity properties in Banach spaces via sublinear operators and corrigendum to “Rotundity properties in Banach spaces via sublinear operators, Nonlinear Analysis 64 (2006) 1171–1188”, Nonlinear Anal. 67 (2006), no. 7, 2208–2216.
  * [Jag74] A. A. Jagers, A note on Cesàro sequence spaces, Nieuw Arch. Wisk. 22 (1974), no. 3, 113–124.
  * [Jam57] R. C. James, Reflexivity and supremum of linear functionals, Ann. of Math. 66 (1957), no. 2, 159–169.
  * [Jam64] R. C. James, Uniformly non-square Banach spaces, Ann. of Math. 80 (1964), no. 2, 542–550.
  * [deJ77] E. de Jonge, The semi-M property for normed Riesz spaces, Compositio Mathematica 34 (1977), no. 2, 147–172.
  * [KamK09] A. Kamińska and D. Kubiak, On isometric copies of $\ell^{\infty}$ and James constants in Cesàro–Orlicz sequence spaces, J. Math. Anal. Appl. 349 (2009), 574–584.
  * [KLT19] A. Kamińska, H. J. Lee and H.-J. Tag, M-ideal properties in Orlicz-Lorentz spaces, Houston J. Math. 45 (2019), no. 1, 213–232.
  * [KA82] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Nauka, Moscow 1977 (in Russian); English transl. in: Pergamon Press, Oxford-Elmsford, New York 1982.
  * [KT17] T. Kiwerski and J. Tomaszewski, Local approach to order continuity in Cesàro function spaces, J. Math. Anal. Appl. 455 (2017), no. 2, 1636–1654.
  * [KK18a] T. Kiwerski and P. Kolwicz, Rotundity and monotonicity properties of selected Cesàro function spaces, Positivity 22 (2018), no. 1, 357–377.
  * [KK18b] T. Kiwerski and P. Kolwicz, Isometric copies of of $l^{\infty}$ in Cesàro–Orlicz function spaces, Results Math. 73 (2018), no. 2, 1–21.
  * [KKM21] T. Kiwerski, P. Kolwicz and L. Maligranda, Isomorphic and isometric structure of the optimal domains for Hardy-type operators, Studia Math. 260 (2021), no. 1, 45–89.
  * [KR61] M. A. Krasnoselskii and Y. B. Rutickii, Convex functions and Orlicz spaces, Noordhoff, Groningen, 1961.
  * [KPS82] S. G. Kreĭn, Yu. I. Petunin and E. M. Semenov, Interpolation of Linear Operators, Nauka, Moscow 1978 (in Russian); English transl.: AMS, Providence 1982.
  * [KMP07] A. Kufner, L. Maligranda and L. E. Persson, The Hardy Inequality. About its History and Some Related Results, Vydavatelsky Servis, Plzeň 2007.
  * [LW83] I. E. Leonard and J. H. M. Whitfield, A classical Banach space: $l_{\infty}/c_{0}$, Rocky Mountain J. Math. 13 (1983), 531–539.
  * [LM15] K. Leśnik and M. Maligranda, On abstract Cesàro spaces. Duality, J. Math. Anal. Appl. 424 (2015), no. 2, 932–951.
  * [LT77] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Springer-Verlag, Berlin-New York 1977.
  * [LT79] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II. Function Spaces, Springer-Verlag, Berlin-New York 1979.
  * [Loz65] G. Ja. Lozanovskiĭ, On reflexive spaces generalizing the reflexive spaces of Orlicz, Dokl. Akad. Nauk SSSR 163 (1965), 573–576 (in Russian); English transl. in: Soviet Math. Dokl. 6 (1965), 986–971.
  * [Loz69] G. Ja. Lozanovskiĭ, On isomorphic Banach structures, Sibirsk. Math. J. 10 (1969) 93–98 (in Russian); English transl.: Siberian Math. J. 10 (1969), 64–68.
  * [Loz73] G. Ja. Lozanovskiĭ, The second Nakano-dual space to a Banach lattice, Collection of articles dedicated to B. Z. Vulikh on the ocassion of his sixtieth birthday, Optimizacija 12 (1973), no. 29, 90–92 (in Russian).
  * [Mal89] L. Maligranda, Orlicz Spaces and Interpolation, Sem. Mat. 5, Universidade Estadual de Campinas, Dep. Mat., Campinas 1989.
  * [Mas91] Mastyło, Banach spaces via sublinear operators, Math. Japonica 36 (1991), 85–92.
  * [Mas92] Mastyło, Type and cotype of some Banach spaces, Int. J. Math. Math. Sci. 15 (1992), 235–240.
  * [MN91] P. Meyer-Nieberg, Banach Lattices, Springer-Verlag, Berlin 1991.
  * [NP10] A. Nekvinda and L. Pick, Optimal estimates for the Hardy averaging operator, Math. Nachr. 283 (2010), 262–271.
  * [ORS08] S. Okada, W. J. Ricker and E. Sánchez Pérez, Optimal Domain and Integral Extension of Operators Acting in Function Spaces, Birkhäuser-Verlag, Basel 2008.
  * [Par80] J. R. Partington, Equivalent norms on spaces of bounded functions, Israel J. Math. 35 (1980), no. 3, 205–209.
  * [Par81] J. R. Partington, Subspaces of certain Banach sequence spaces, Bull. London Math. Soc. 13 (1981), 162–166.
  * [PW07] A. Plichko and M. Wójtowicz, Copies of $c_{0}(\Gamma)$ and $\ell_{\infty}(\Gamma)/c_{0}(\Gamma)$ in quotients of Banach spaces with applications to Orlicz and Marcinkiewicz spaces, Indag. Mathem. (N.S.) 18 (2007), no. 2, 251–268.
  * [Ros70] H. P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36.
  * [SS83] E. Saab and P. Saab, On the Banach space $X^{**}/X$, Bull. Sci. Math, 107 (1983), no. 2, 139-144.
  * [Wnu99] W. Wnuk, Banach Lattices with order continuous norms, Polish Scientific Publishers PWN, Warszawa 1999.
  * [Wnu84] W. Wnuk, On the order-topological properties of the quotient space $L/L_{A}$, Studia Math. 79 (1984), 139–149.
  * [Wój05] M. Wójtowicz, The lattice copies of $\ell_{\infty}(\Gamma)/c_{o}(\Gamma)$ in a quotient of Banach lattices, Indag. Mathem. (N.S.) 16 (2005), no. 1, 147–155.
  * [Zan83] A. C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam 1983.