Quotients, $\ell_\infty$ and abstract Ces\`aro spaces

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 Ces\`aro spaces $CX$.


Introduction and preliminaries
Let 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 ε, should we impose on the space X so that the implication (∆) X ∈ (P) + ε =⇒ CX ∈ (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 : f → C(f )(x) := 1 x x 0 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 (∆) 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 p < ∞ contains, for any ε > 0, an (1 + ε)-copy of ℓ p which is (1 + ε)-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 ) θ,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 G , of the space D E (S), where E is a Banach ideal space. Essentially, they were able to show that if E ∈ (G ) and an injective positive sublinear operator S has some special geometric properties corresponding to the property G , then also D E (S) ∈ (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 ℓ ∞ , as long as a rearrangement invariant space X contains such a copy as well.
A Banach space (X, · X ) is said to be a Banach lattice if there is a lattice order on X, say , such that for x, y ∈ X with |x| |y| we have x X y X . By X + we denote the positive cone of a Banach lattice X, that is, X + = {x ∈ X : x 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 . If, additionally, the operator norm of a lattice isomorphism T : X → Y is equal one, then we will emphasize this fact by calling T a lattice (or an order) isometry.
Let (Ω, Σ, µ) be a complete σ-finite measure space. Denote by L 0 (Ω) = L 0 (Ω, Σ, µ) the set of all (equivalence classes of) real-valued µ-measurable functions defined on a measure space (Ω, Σ, µ). A Banach ideal space X = (X, · X ) on (Ω, Σ, µ) is understood to be a Banach space X such that X is a linear subspace of L 0 (Ω, Σ, µ) satisfying the so-called ideal property, which means that if f, g ∈ L 0 (Ω, Σ, µ), |f (t)| |g(t)| µ-almost everywhere on Ω and g ∈ X, then f ∈ X and f X g X . If it is not stated otherwise we assume that a Banach ideal space X on (Ω, Σ, µ) contains a function which is positive µ-almost everywhere on Ω (such a function is called the weak unit in X). For a function f ∈ L 0 (Ω, Σ, µ) we define a support of f as a set supp(f ) := {x ∈ Ω : f (x) = 0}. Moreover, by a support supp(X) of the Banach ideal space X on (Ω, Σ, µ) we mean the smallest (in the sense of inclusion) µ-measurable subset, say A, of Ω such that f χ Ω\A = 0 for all f ∈ X. The existence of the weak unit is equivalent to the following condition that supp(X) = Ω. We say that a Banach ideal space X is non-trivial if X = {0}.
For two Banach ideal spaces X and Y on Ω the symbol X E ֒→ Y means that the embedding X ⊂ Y is continuous with the norm not bigger than E > 0, that is to say, f Y E f X for all f ∈ X. If the embedding X E ֒→ Y holds with some (maybe unknown) constant E > 0 we simply write X ֒→ Y . Moreover, the symbol X = Y (X ≡ 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 in X satisfying 0 x n |x| and x n ↓ 0 (that is, x n+1 x n and inf n∈N {x n } = 0) we have x n → 0 as n → ∞. 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 where X * a is the space of all order continuous functionals, that is to say, x * ∈ X * a if and only if x * (x n ) → 0 as n → ∞ for any sequnce {x n } ∞ n=1 ⊂ X + and x n ↓ 0, and X s is the space of all singular functionals, that is to say, x * ∈ X s if and only if x * (x) = 0 for any x ∈ X a . For an order continuous Banach lattice X the spaces X * a and X * coincide. If X is a Banach ideal space on (Ω, Σ, µ) such that supp(X a ) = supp(X) then the space X * a coincide with the Köthe dual space 2 (or associated space) X ′ Recall that X 1 ֒→ X ′′ and we have the equality X ≡ X ′′ if and only if the norm in X has the Fatou property, that is to say, if the conditions 0 f n ↑ f ∈ L 0 with {f n } ∞ n=1 ⊂ X and sup n∈N f n X < ∞ imply that f ∈ X and f n X ↑ 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 ∈ J, g ∈ X and |g| |f | then also g ∈ 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, x X/J = inf{ y X : Q(y) = Q(x)}, where Q : J → 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 ∈ L 0 (which means that they have the same distribution functions ) and if f ∈ X then g ∈ 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 (0, 1) or the semi-axis (0, ∞) with the Lebesgue measure m (in this case we will talk about function spaces) and the set of positive integers N with the counting measure # (and then we will refer to sequence spaces). Moreover, by a non-increasing rearrangement of a function f : Ω → R we mean the function f * : under the convention inf{∅} = ∞. In the sequence case, however, we need a little modification of the above definition, namely x * n = inf{λ > 0 : d x (λ) < n}, where x = {x n } ∞ n=1 . The fundamental function ϕ X of a rearrangement invariant space X on Ω is defined by the following formula ϕ X (t) = χ (0,t) X for t > 0, (with a fairly obvious modification when Ω = N, i.e., ϕ X (n) = n k=1 e k X for n ∈ N, where {e n } ∞ n=1 is the canonical basic sequence of X) where χ A , throughout, will denote the characteristic function of a set A. It is well-known that fundamental function is quasi-concave on Ω, that is: ϕ X (0) = 0; ϕ X is positive and non-decreasing; t → ϕ X (t)/t is non-increasing for t > 0 or, equivalently, ϕ X (t) max{1, t/s}ϕ X (s) for all s, t ∈ Ω.
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 ϕ on Ω generates the Lorentz function space Λ ϕ on Ω endowed with the norm Recall also that for a quasi-concave function ϕ the Marcinkiewicz function space M ϕ on Ω is defined in the following way 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 ϕ (note that every such a function is equivalent to a concave function) we have the following embeddings Let F be an non-decreasing convex function on [0, ∞) such that F (0) = 0 and let X be a Banach ideal space on Ω. The Calderón-Lozanovskiȋ space X F on Ω is defined as the space of all measurable functions f : Ω → R for which the followng norm, the so-called Luxemburg-Nakano norm, is finite The spaces X F were introduced by Calderón [Cal64, p. 122] and Lozanovskiȋ [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 Λ ϕ , then X F is the corresponding Orlicz-Lorentz space Λ F,ϕ . It is also clear that in the special situation, when F (t) = t, the Orlicz-Lorentz space Λ F,ϕ coincide, up to equality of norms, with the Lorentz space 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 Λ ϕ on N as λ ϕ and the Orlicz space L F on N as ℓ F , etc. We will use the notation A B or B A to denote an estimate of the form A CB for some constant C > 0 depending on the involved parameters only. We also write A ≍ B for A B 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 ℓ ∞ and complete one of them (Theorem 2.1). This connection between, on the one hand, the existence of a lattice isometric copy of ℓ ∞ 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 ℓ ∞ 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]. 4 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 ℓ ∞ one more time Lozanowskiȋ'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 ℓ ∞ if and only if the space X is not order continuous. Of course, every isomorphic copy of ℓ ∞ is automatically complemented in the ambient space due to the fact that the space ℓ ∞ is isometrically injective. Note also that a Banach lattice X contains a (lattice) almost isometric copy of ℓ ∞ whenever it contains a (lattice) isomorphic copy of ℓ ∞ (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 ℓ 1 . Moreover, if a Banach space X contains an asymptotically isometric copy of ℓ ∞ , then it contains even an isometric copy of ℓ ∞ (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 ℓ ∞ . Indeed, let X and Y be two Banach function spaces on Ω with non-trivial intersection X ∩ Y such that the first one is not order continuous and the second one is strictly convex (for example, we can take X = L ∞ and Y = L 2 ). Then the space is strictly convex and is not order continuous. Consequently, in view of the Lozanowskiȋ result [Loz69], it contains a lattice isomorphic copy of ℓ ∞ but clearly cannot contain an isometric copy of ℓ ∞ . 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 ℓ ∞ . 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, · ), then for any 0 < ε < 1 there exists x ∈ S(X) such that d X (x, Y ) 1 − ε.
It can be demonstrated that the above result is in general not true for ε = 0. Actually, for a Banach space X to have the property that given a proper closed linear subspace Y of X there exists x ∈ S(X) with 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 ϕ, in S(X * ) which does not achieve its norm. Taking Y = ker(ϕ) we see immediately that Y is a proper closed linear subspace of X and there is no such element x ∈ S(X) with 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 ψ ∈ S(X * ) such that Y ⊂ ker(ψ) and x ∈ S(X) with ψ(x) = 1. Consequently, we have the following inequalities: 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 ℓ ∞ . Anyhow, for X = ℓ ∞ and f = χ N we have f X = d X (χ 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 σ-Dedekind complete Banach lattice with the semi-Fatou property such that (X d a ) d = X * . If we can find an element f ∈ X such that f X = d X (f, X a ) = 1, then X contains a lattice isometric copy of ℓ ∞ .
Let's us consider the space X = (L ∞ , · X ), where · X is an equivalent norm on L ∞ given by the formula: Then the subspace X a is trivial and However, the space X cannot contain a lattice isometric copy of ℓ ∞ as we have already explained above. Therefore the assumption (X d a ) 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 σ-finite measure space with the semi-Fatou property such that supp(X a ) = supp(X). The space X contains a lattice isometric copy of ℓ ∞ if and only if there exists an element f ∈ X with f X = d X (f, X a ) = 1.
Proof. Using Theorem 1 from [Hud98] we can find a sequence Reffering to Theorem 2.1, if a Banach ideal space X contains a lattice isometric copy of ℓ ∞ , then we can find f ∈ X with f X = d X (f, X a ) = 1. So, looking from a slightly different perspective, this means that f X/Xa = 1 and there exist a singular functional S ∈ S(X * ) with S(f ) = 1 (note only that (X/X a ) * ≈ X ⊥ a = X s , where X s is the space of all singular functionals, that is to say, those S ∈ X * for which S(x) = 0 for all x ∈ 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 where the last inequality follows from the fact that if f X = 1, then d X (f, X a ) 1. Therefore, we only need to prove the reverse inequality. To do this, take f ∈ X \ X a and let {f n } ∞ n=1 ⊂ X a * In less general situation, e.g. when X is a Banach ideal space, this condition simply means that supp(Xa) = supp(X) (cf. [Hud98, p. 523]).
6 be a sequence that realizes the distance of the function f from the ideal X a , that is to say, Thus, we obtain that 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 ) = S X * for some f ∈ B(X), i.e., d X (f, X a ) = 1 (cf. Lemma 2.2) and the space X contains a lattice isometric copy of ℓ ∞ 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 σ-Dedekind complete Banach lattice with the semi-Fatou property such that (X d a ) d = X. Then X contains a lattice isometric copy of ℓ ∞ 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 ℓ ∞ /c 0 † considering a more general construction ℓ F /(ℓ F ) a , where ℓ F is the Orlicz sequence space (in fact, they even allowed for ℓ 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 Proposition 2.4. Let X be a Banach sequence space such that X ֒→ ℓ ∞ , the ideal X a is nontrivial and X a = X. Then X/X a is not a dual space.
Proof. It is well-known that the space ℓ ∞ /c 0 contains an isomorphic copy of c 0 (Γ), where card(Γ) = c and c denotes the cardinal of the continuum, that is, c = 2 ℵ 0 . Let us briefly remind that we can proceed as follows: (1) for each irrational number r, take a sequence {q r n } ∞ n=1 of distinct rational numbers converging to r and put F r = {q r n : n ∈ N}; (2) the sets {F r } r∈R form an uncountable family of infinite sets such that F r ∩ F r ′ is finite whenever r = r ′ ; (3) observe that ℓ ∞ ≡ ℓ ∞ (Q); (4) the closed linear span of {π(χ Fr ) : r ∈ R}, where π : ℓ ∞ → ℓ ∞ /c 0 is the cannonical quotient map, is isometric to c 0 (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 ℓ ∞ (Γ). But this is impossible, because where the second inequality follows from the fact that X ֒→ ℓ ∞ . † As mentioned in [LW83]: The Banach space ℓ∞/c0 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(βN \ N) (just to be clear βN, as usual, is the Čech-Stone compactification of N endowed with the discrete topology, that is, βN is the unique compact Hausdorff space containing N as a dense subspace so that every bounded continuous function on N extends to a continuous function on βN); we also recommend taking a look at [DR91].
The above result is a direct generalization of [GH96, Proposition 6.2] which was proved in the case when X = ℓ F .
Let us now recall the well-known Phillips-Sobczyk theorem, which states that c 0 is not complemented in ℓ ∞ (to be precise, what Phillips proved was that c is not complemented in ℓ ∞ ; look at [CSCY00], cf. [AK06,). 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. Lozanovskiȋ, 1973). Let X be a Banach ideal space on a complete σ-finite measure space (Ω, Σ, µ) such that X a = 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 where T is a lattice isomorphism and Q is the canonical quotient map. It follows that T (c 0 ) ⊂ X a (see Proposition 1 and Theorem 1 in [Wój05]; cf. [PW07]). Thus, if q : ℓ ∞ → ℓ ∞ /c 0 is a quotient mapping, then there exist a unique mapping U : ℓ ∞ /c 0 → X/X a defined in the following way U : x + c 0 → T (x) + X a . It is not hard to see that QT = U q. In short, the following diagram commutes and the space X/X a contains a lattice isomorphic copy of ℓ ∞ /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 : X ∋ x → xy ∈ L 1 (µ), where y ∈ X ′ is chosen so that y X ′ = 1 and y(t) > 0 for µ-a.e. t ∈ Ω. Then we can define the functional ♯ · ♯ on X as follows Now, since the space ℓ ∞ 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 (µ) is already strictly convex; another words, we can always equip the space L 1 (µ) with a new norm, say |·|, such that the space (L 1 (µ), |·|) is strictly convex. But this gives that ♯ · ♯ is an equivalent norm on X and that the space (X, ♯ · ♯) is strictly convex as well (this follows from the well-known fact that if T : X → 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 ℓ ∞ /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 ℓ ∞ /c 0 , is not isomorphic to a subspace of X and this implies that X a is not complemented in X.
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 ℓ ∞ (cf. [GH96,Proposition 5.1] and [Wój05, Theorem 2]).

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 · X/M defined as x + M X/M := inf{ y X : y ∈ x + M }.
In simple words, this norm measures the distance from a coset to the origin of the space X/M , that is Therefore, the study of the function x → 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 Λ ϕ ; 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, 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 ℓ ∞ . But let's start with some auxiliary lemmas.
Lemma 3.1. Let X be a Banach ideal space on (Ω, Σ, µ) and let J be an order ideal of X. If f ∈ X then we have the following formula Proof. Without loss of generality we can assume that f > 0. Of course, to show the above equality we need only to prove that f n = f n χ Ωn + f χ Ω\Ωn for n ∈ N.
Observe that f n f for every n ∈ N. Consequently, Taking limits on both sides of the above inequality, we see immediately that Moreover, since J is an order ideal of X, it follows that { f n } ∞ n=1 ⊂ 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 (Ω, Σ, µ) and let J be an order ideal of X.
Proof. Take 0 < g f ∈ X. Let {f n } ∞ n=1 ⊂ J be a sequence that realizes the distance, that is to say, lim n→∞ f − f n = d X (f, J). It follows from Lemma 3.1 that we can additionally assume that f n f for every n ∈ N. Put g n = min{g, f n } for n ∈ N.
Since J is an order ideal of X, it is clear that {g n } ∞ n=1 ⊂ J. Observe also that 0 < g − g n f − f n for all n ∈ N, so due to the ideal property of X we get that But this simply means that d X (g, J) d X (f, J).
Lemma 3.3. Let X be a Banach ideal space on (Ω, Σ, µ) and let J be an order ideal of X. If f ∈ X then d X (f, J) = d X (|f | , J).
Proof. To show one of the inequalities let's take f ∈ X and notice that On the other hand, for each 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.
To begin with, due to the fact that the sequence { f * χ (0,1/n)∪(n,∞) X } ∞ n=1 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 = X a , because otherwise there is nothing to prove. Denote First of all, it is not hard to see that or, equivalently, f * − g X L for each g ∈ X a . Indeed, where the equality follows from the fact that g ∈ X a . On the other hand, our assumption that X a = {0} implies that 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 Ω, where Ω = (0, 1) or Ω = (0, ∞), with the Fatou property. Then for f ∈ X we have the following equality Proof. We can assume that X a = {0}, X a = X and f / ∈ 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 is true even without additional assumptions on the function f ∈ X. In fact, we have 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 • . Suppose that f * (∞) = 0. Then, as a consequence of Ryff's theorem (see [BS88, Corollary 7.6, p. 83]), there is a measure preserving transformation σ from the support of f onto the support of f * such that f * • σ = f m-almost everywhere on the support of f . If we were able to show that for each g ∈ X a there exists h g ∈ X a such that f − h g X f * − g X , the proof of this part would be complete. For this, let's take g ∈ X a and put h g = g • σ. Due to [CKP14, Lemma 2.6] and the fact that h * g = (g • σ) * , we conclude that h g ∈ X a . Now, since we have the equalities Combining this inequality with the inequality (♥), we see that we have just finished the proof in the first case.
We claim that d f χ Ω > −hg = d f −hg . Indeed, for all 0 x < f * (∞) we have and the claim follows. In consequence, we have Case 2: Suppose now that m(Ω > ) < ∞ and f χ Ω> ∈ X a . Put It follows directly from the above definition that f χ Ω> ∈ Γ f and Γ f ⊂ X a . We claim that To see this, for each g ∈ X a let us define a function g f in the following way the opposite inequality is evident and the equality (♠) is proved. Consequently, since g ∈ X a , so On the other hand, since m(Ω > ) < ∞ and f χ Ω> ∈ X a , applying Theorem 3.4, we conclude that Case 3: Let m(Ω > ) < ∞ and f χ Ω> ∈ X \ X a . Observe that f * (0 + ) = ∞, because otherwise, keeping in mind that m(Ω > ) < ∞, we would get f χ Ω> ∈ X a which is undoubtedly a contradiction. We will show that Take n 0 ∈ N satisfying f * (1/n 0 ) > f * (∞) and note that this limit exists because the sequence is non-increasing for n n 0 and bounded from below. Now, there is a measure preserving transformation τ : Ω > → (0, m(Ω > )) such that f * • τ = f . Put T n = τ −1 ((0, 1/n)) for n = n 0 , n 0 + 1, ... and denote It is clear that f n ∈ X a for n n 0 . Moreover, since f * (0 + ) = ∞, so f * (∞)χ (0, 1 n ) for n ∈ N big enough, and, consequently, we have On the other hand, we get L d X (f, X a ) = inf{ f − g X : g ∈ X a and |g| |f |} inf{ f * − g * X : g ∈ X a and |g| f } inf{ f * − g * X : g ∈ X a and 0 g * 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 (♦), it is enough to show that To see this, take ε > 0 and h ∈ X a as above. We can find a number, say m ∈ N, such that h * χ (0, 1 m )∪(m,∞) < ǫ, 12 because h * ∈ X a (see [CKP14, Lemma 2.6]). Hence, and (♣) follows. But this means that the proof of (♦) is finished as well. However, using Theorem 3.4 again, we see that . 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 N with the Fatou property. Then for f ∈ X we have the following equality 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.
Corollary 3.7. Let X be a rearrangement invariant space on Ω, where Ω = (0, 1), Ω = (0, ∞) or Ω = N, such that the ideal X a is non-trivial. Then for f ∈ X we have the following equality where Ω n = Ω ∩ (0, 1 n ) ∪ (n, ∞) . The above corollary can be interpreted in the following way: the function f ∈ X can be approximated by a bounded function supported in set of finite measure to within d X (f, X a ) + ε, where ε > 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, ∞) with the Fatou property such that L ∞ ֒→ X. Suppose that Then for all f ∈ X we have the following formula Before we provide the proof, let us note that the above condition (D ∞ ) itself plays in rearrangement invariant spaces an analogous role to the ∆ 2 (∞)-condition in the case of Orlicz spaces. In fact, if X is an Orlicz space L F on (0, ∞) generated by the Orlicz function F which vanishes outside zero and F ∈ ∆ 2 (∞), then L ∞ ֒→ X and X ∈ (D ∞ ). However, the assumptions of the above corollary are true also in the case of Lorentz spaces Λ ϕ provided ϕ(0 + ) = 0 and ϕ(∞) < ∞.
Proof. Suppose that X is as above and observe that the sequence (f * χ (n,∞) ) * ∞ n=1 is bounded and converges uniformly to a bounded function f * (∞)χ (0,∞) , so in view of the embedding L ∞ ֒→ X we get 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 ϕ be a quasi-concave function such that lim t→0 + t/ϕ(t) = 0. Then the Marcinkiewicz space M ϕ contains a lattice isometric copy of ℓ ∞ .
Proof. Let us consider the function ψ defined in the following way for t > 0.
It follows from [BS88, Theorem 5.2, p. 66] that ψ is also a quasi-concave function as a fundamental function of the space X ′ . In addition, due to the condition lim t→0 + t/ϕ(t) = 0, the function ψ is continuous (cf. [BS88, Corollary 5.3, p. 67]). Therefore, we can find 0 < a < 1 such that the function ψ is strictly increasing on (0, a). Consequently, the function ψ ′ is non-increasing and positive m-almost everywhere on (0, a). Put g = ψ ′ χ (0,a) . Then and, because the function t → ϕ(t)/t is non-increasing, we also have that 1 for t a.
This means that g Mϕ = 1. Moreover, we can repeat the above argument with the same result for the function ψ ε = ψχ (0,ε) , where 0 < ε < a, in place of the function ψ. Therefore, using Corollary 3.7, we get for a properly choosen subsequence {ε n } ∞ n=1 . Lemma 3.10. Let X be a rearrangement invariant space on (0, ∞). If L ∞ ֒→ X, then X contains a lattice isometric copy of ℓ ∞ .
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 ℓ ∞ , but it is clear that L ∞ ֒→ e L .
Proof. Take f = χ (0,∞) / χ (0,∞) X and let {N i } ∞ i=1 be a pairwise disjoint family of an infinite subsets of N such that ∞ i=1 N i = N. Moreover, let {ω j } ∞ j=1 be a sequence of real numbers such that 0 = ω 1 < ω 2 < ... and lim j→∞ ω j = ∞. Put Ω i j = (ω j , ω j+1 ) for i ∈ N and j ∈ N i . Consider the following sequence f χ . It is clear that this sequence is for every i ∈ N and f X = 1. Consequently, in order to complete the proof we can refer to [Hud98, Theorem 1].
Proof. It follows from [KPS82, Lemma 5.1] that the space Λ ϕ is order continuous if ϕ(∞) = ∞. Therefore, due to the well-known Lozanovskiȋ result, it does not contain even an order isomorphic copy of ℓ ∞ and proves the necessity. Now, suppose that ϕ(∞) < ∞. But this means that L ∞ ֒→ X and it is enough to use Lemma 3.10.
Let us add that it may happen that the Lorentz space Λ ϕ contains an order isomorphic copy of ℓ ∞ but not an order isometric copy of ℓ ∞ . Indeed, if ϕ(0 + ) > 0 and ϕ(∞) = ∞, then the space Λ ϕ is not order continuous but is strictly monotone.

On isometric copies of ℓ ∞ in abstract Cesàro spaces
For a Banach function space on (0, 1) or (0, ∞) we define the abstract Cesàro function space CX as a set where C denotes the continuous Cesàro operator defined as However, in the sequence case, by the discrete Cesàro operator we will understand the operator C d defined as , 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 such that C d (x) ∈ X for which the following norm 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 ֒→ CX and, consequently, the space CX is non-trivial.
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, ∞) or 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 ∈ X and 0 a < b ∞ such that f X = d X (f, X a ) = 1 and f * χ (0,a)∪(b,∞) CX = 1. Then CX contains a lattice isometric copy of ℓ ∞ .

15
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 ∈ X be as above. Put A = (0, a) ∪ (b, ∞) and A c = (a, b). In view of Theorem 3.5, we have the equality d X (f * , X a ) = d X (f, X a ) = 1. Note also that f * χ A c ∈ (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 * χ A , we have d CX (h, (CX) a ) = inf{ h − g CX : g ∈ (CX) a and 0 < g h} = inf{ Ch − Cg X : g ∈ C(X a ) and 0 < g h} (because h − g 0 and CX a = C(X a ), see [KT17, Theorem 16]) inf{ C(f * ) − g ′′ X : g ′′ ∈ X a } (we take the infimum on a larger set) where the last inequality follows from Corollary 3.2 and the fact that f * C(f * ). Now, since supp((CX) a ) = 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 ℓ ∞ in Cesàro spaces CX. To be more precise, Kamińska and Kubiak [KamK09] considered this problem for X = ℓ 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, in the sequence case and the modular-type inequality of the form in the function case (which looks a little like the boundedness ‡ 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 ℓ 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-Lozanovskiȋ spaces X F (recall, that the spaces (L 1 ) F and (ℓ 1 ) F coincide, up to the equality of norms, with the Orlicz spaces L F and ℓ F , respectively).
Let us also remind that by the Calderón-Lozanovskiȋ class we understand the set {f ∈ X F : F (|f |) X < ∞} (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-Lozanovskiȋ class we have the following inequality: If F does not satisfy the ∆ X 2 -condition § , then the space C(X F ) contains a lattice isometric copy of ℓ ∞ .
Proof. Since F / ∈ ∆ X 2 , it follows from Theorem 1 and Theorem 2 in [HKM96] that the Calderón-Lozanovskiȋ space X F contains a lattice isometric copy of ℓ ∞ . Therefore, in view of Theorem 2.1 (let us note the fact that supp((X F ) a ) = supp(X F )) and Theorem 3.5, there exists a function f ∈ X F such that Because the space X F is rearrangement invariant, so f * X F = 1 and, consequently, F (f * ) X 1. Now, we divide the proof into two parts ¶ . 1 • . Suppose that f * χ (c,∞) ∈ (X F ) a for some (equivalently, for all) c > 0. Consequently, we have that where the second equality follows from Theorem 3.4. Since X ∈ (OC), so we can find b > 0 with where the constant M > 0 is from the condition (M ). Therefore, we have the following inequalities But this means that f * χ (0,b) C(X F ) 1. What's more, thanks to our assumptions, f * χ (b,∞) ∈ (X F ) a , whence that is, C(f * χ (0,b) ) X F = 1. Now we can apply Theorem 4.1 and finish the proof of this part. § Let's just mention that the ∆ X 2 -condition is an appropriate modification of the well-known condition ∆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-Lozanovskiȋ space XF . For a more precise definitions we refer, for example, to [HKM96,p. 642].
¶ It follows from Theorem 2.1 that a rearrangement invariant space X on (0, ∞) with supp(Xa) = (0, ∞) contains a lattice isometric copy of ℓ∞ if and only if there is a function, say f , such that f = dX (f, Xa) = 1. In many situations (for example, if X is a Calderón-Lozanovskiȋ space) the function f is given in a simple form, i.e., f = ∞ n=1 fn, where fn = anχΩ n , an ∈ R+, Ωn are a pairwise disjoint measurable subsets of (0, ∞) and fn ∈ S(X) for n ∈ 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 fn, Xa) = 1, that is to say, the subsequence {fn k } ∞ k=1 still builds a lattice isometric copy of ℓ∞. Therefore, if we can find a subsequence, say {fn k } ∞ k=1 , which is uniformly bounded, then it is enough to put f = ∞ k=1 fn k . Clearly, f ∈ L∞. On the other hand, if some subsequence {fn k } ∞ k=1 is unbounded, then since X ֒→ L1 + L∞, so (taking further subsequence, if needed) we see that ∞ k=1 m(Ωn k ) < ∞.
2 • . Assume that f * χ (c,∞) ∈ X F \ (X F ) a and f * χ (0,c) ∈ (X F ) a for some (equivalently, for all) c > 0. Referring once again to Theorem 3.4, we have the following equalities Just as before, since X ∈ (OC), so where M > 0 is from the condition (M ). Moreover, using the condition (M ), we get that is, f * χ (b,∞) C(X F ) 1. Now, we claim that To prove (=) it is enough to show that Take ε > 0. Note that In consequence, we have where the last inequality follows from the fact that f * χ (B,∞) X F = 1 (to see this just note that due to our assumptions f * χ (0,B) ∈ (X F ) a and then 1) and the definition of the Luxemburg-Nakano norm. But this proves the claim (=). 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-Lozanovskiȋ class is closed under the Cesàro operator C. If F vanishes only at zero and does not satisfy the ∆ 2 (0)-condition, then the space C(X F ) contains a lattice isometric copy of ℓ ∞ .
Proof. To begin with, since X = X a ֒→ c 0 ֒→ ℓ ∞ and the Orlicz function F vanishes only at 0, it follows that the condition ∆ 2 (0) is equivalent to the condition δ X 2 (see [HN05, Lemma 4]) ‖ . Due to the lack of the δ X 2 -condition and Lemma 2.4 from [FH99], the Calderón-Lozanovskiȋ space X F contains a lattice isometric copy of ℓ ∞ (note only that the results from [FH99] have been proven for a wider than X F class of spaces, the so-called generalized Calderón-Lozanovskiȋ 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 ∈ X F with ‖ We leave the definition of the condition δ X 2 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].
Moreover, it is not hard to see that But this means that f * χ {n,n+1,...} C(X F ) 1. Note also that f * χ {1,2,...,n−1} ∈ (X F ) a , so 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 (=) from Corollary 4.2, we can show that 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.
(b) It may happen (consider, for example, the space X = L 1 ∩ L ∞ on (0, ∞)) that the space X contains a lattice isometric copy of ℓ ∞ 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 ∞ 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, ∞) 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 ∩ L ∞ ) contains a lattice isometric copy of ℓ ∞ .
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 Put f ε = χ (0,ε) , where ε > 0. We claim that To show (⋆), it is enough to observe that ϕ 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 Consequently, we have the following equalities This means that we have defined the function f ε 0 with C(f ε 0 )(0 + ) = 1 and f ε 0 C(X∩L∞) = 1 = id : X ∩ L ∞ → L ∞ . Now, because the ideal (X ∩ L ∞ ) a is trivial, we can use Theorem 4.6 and end the proof.
Remark 4.9. (a) Let X = L p + L ∞ on (0, ∞), where 1 < p < ∞. It is clear that in this case the ideal X a is non-trivial * * and the Cesàro operator is bounded on X. However, it is also clear † † that X ∩ L ∞ ≡ L ∞ . Therefore, using Theorem 4.6 (via Corollary 4.8) we re-prove Proposition 4.9 from [KKM21] which states that the space Ces ∞ contains, as one would expect, a lattice isometric copy of ℓ ∞ . (b) Let X = L F be an Orlicz space generated by an Orlicz function F such that b F := sup{x > 0 : F (x) < ∞} = 1, F (1) 1 and the left derivative of F at x = 1 is finite. Then there exists an Orlicz function, say G, with b G = ∞ such that L F ≡ L G ∩ L ∞ . 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 Take f ∈ L F ∩ L ∞ with max{ f L F , f L∞ } = 1. Let us consider two cases: 1 • . f L∞ = 1. Then |f (x)| 1 = b F , so ∞ 0 F (|f (x)|)dx = ∞ 0 F (|f (x)|)dx 1. Moreover, since f L∞ = 1 = b F , it follow that ∞ 0 F (|f (x)| /λ)dx = ∞ for every λ < 1. Therefore, f L F = 1 as well.
In particular, if 1 p < ∞ and the Orlicz function F p,∞ is given in the following way (c) Let ϕ be an increasing concave function with ϕ(0 + ) = 1 (this normalization is basically inessential). Since the norm in the Lorentz space Λ ϕ is given by the formula where, of course, the intersection space is considered with the equivalent two-dimensional ℓ 1norm given above (after all, all norms on 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 ∩ L ∞ of the form where · F is a norm on R 2 with the ideal property, then for our proof to work, it is enough to require that (x 0 , 1) F = (0, 1) F = 1 for some x 0 > 0, which means that the unit sphere in (R 2 , · F ) contains an order interval beginning at point (0, 1). For example, the "hexagonal" norm (x, y) hex = max x + √ 3 3 y , x − √ 3 3 y , 2 √ 3 3 |y| for x, y ∈ 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 ∞ but the ideal (L ∞ ) 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 ) ⊂ (CX) a ⊂ (CX) b ⊂ C(X a ).
The first inclusion C(X a ) ⊂ (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 ⊂ (CX) b follows directly from [BS88, Theorem 3.11, p. 18]. Consequently, it remains only to show that (CX) b ⊂ C(X a ). We will do this in two steps. 1 • . Take χ A , where A ⊂ N and max(A) := n 0 < ∞. Let n > n 0 . Since p X > 1 ‡ ‡ , so ℓ p ֒→ 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 C(χ A )χ {n,n+1,... } X 1 k χ {n,n+1,... } (k) X 1 k χ {n,n+1,... } (k) ℓp → 0, as n → ∞. 2 • . Take a = {a n } ∞ n=1 ∈ (CX) b . Let {s n } ∞ n=1 be a sequence of sequences such that s n converges to a in the norm of CX and #(supp(s n )) < ∞ for all n ∈ N. It follows from the previous step that {s n } ∞ n=1 ⊂ C(X a ). Now, using the reverse triangle inequality, we get C(|a|) − C(|s n |) X C(|a − s n |) X = a − s n CX → 0, as n → ∞. Since X a is a closed ideal of X, so C(|a|) ∈ 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.