This paper explores the deep connections between Banach space theory and analytic P-ideals, revealing symmetries, new examples, and characterizations that advance understanding in both areas.
Contribution
It uncovers symmetries between Banach spaces with unconditional bases and analytic P-ideals, introduces new examples, and provides novel characterizations impacting the theory.
Findings
01
Identified symmetries between Banach spaces and analytic P-ideals
02
Constructed new examples of Banach spaces and P-ideals
03
Developed a new characterization of precompact families
Abstract
We study the interplay between Banach space theory and theory of analytic P-ideals. Applying the observation that, up to isomorphism, all Banach spaces with unconditional bases can be constructed in a way very similar to the construction of analytic P-ideals from submeasures, we point out numerous symmetries between the two theories. Also, we investigate a special case, the interactions between combinatorics of families of finite sets, topological properties of the "Schreier type" Banach spaces associated to these families, and the complexity of ideals generated by the canonical bases in these spaces. Among other results, we present some new examples of Banach spaces and analytic P-ideals, a new characterization of precompact families and its applications to enhance Pt\'ak's Lemma and Mazur's Lemma.
\|\mu\|=\mu^{+}(2^{\omega})+\mu^{-}(2^{\omega})=\\
\sup\big{\{}|\mu(C_{0})|+|\mu(C_{1})|\colon 2^{\omega}=C_{0}\cup C_{1}\;\text{is a decomposition into clopen sets}\big{\}}.
\|\mu\|=\mu^{+}(2^{\omega})+\mu^{-}(2^{\omega})=\\
\sup\big{\{}|\mu(C_{0})|+|\mu(C_{1})|\colon 2^{\omega}=C_{0}\cup C_{1}\;\text{is a decomposition into clopen sets}\big{\}}.
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Full text
Analytic P-ideals and Banach spaces
Piotr Borodulin–Nadzieja
Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
We study the interplay between Banach space theory and theory of analytic P-ideals.
Applying the observation that, up to isomorphism, all Banach spaces
with unconditional bases can be constructed in a way very similar to the
construction of analytic P-ideals from submeasures, we point out numerous
symmetries between the two theories. Also, we investigate a special case,
the interactions between combinatorics of families of finite sets, topological properties of the “Schreier type” Banach spaces associated to these families, and the complexity of ideals generated by the canonical bases in these
spaces. Among other results, we present some new examples of Banach spaces and analytic P-ideals, a new characterization of precompact families and its applications to enhance Pták’s Lemma and Mazur’s Lemma.
The first author was supported by
National Science Center project no. 2018/29/B/ST1/00223. The second author was supported by the Austrian Science Fund (FWF) project no. P29907.
1. Introduction
Certain aspects of the interplay between Banach space theory and the theory of analytic P-ideals have already been studied in the past two decades. For example, Veličković and Louveau ([LV99]) considered several ideals inspired by classical Banach spaces; Veličković
([Vel99]) and Farah ([Far99]) used the Tsirelson space, a classical construction in Banach space theory, to construct certain peculiar analytic P-ideals disproving a conjecture proposed by Kechris and Mazur; on the other hand, Drewnowski and Labuda ([DL10] and [DL17]) studied analytic P-ideals from the point of view of specialists in Banach space theory, obtaining certain general results.
In [BNFP15] we proposed a more systematic study of the connections between these ideals and Banach spaces. The main tool for our studies was representability of ideals in Banach spaces, a natural generalization of the concept of summable
ideals. We say that an ideal on the set ω of all natural numbers is representable in a Banach space X if it is of the form
[TABLE]
for some sequence (xn) from X. In Section 4 we give a brief overview of this notion.
In this article we will present a way to connect analytic P-ideals and Banach spaces in a tighter way. It is well-known that each analytic P-ideal is the exhaustive ideal Exh(φ)={A⊆ω:φ(A∖n)→0} of some lower-semicontinuous (lsc) submeasure φ on ω, and for Fσ P-ideals we
can find such submeasures for which Exh(φ) coincides with the “finite” ideal Fin(φ)={A⊆ω:φ(A)<∞} of φ (see Section 2 for a detailed introduction to analytic P-ideals).
Considering an extended norm Φ on Rω (that is, a “norm” which may attain infinite values) it makes sense to define the exhaustive spaceEXH(Φ)={x∈Rω:Φ(Pω∖n(x))→0} where
PA:Rω→Rω is the natural projection associated to A⊆ω, and the finite spaceFIN(Φ)={x∈Rω:Φ(x)<∞} of Φ in the same manner. Under some natural assumptions on Φ, both
EXH(Φ) and FIN(Φ) equipped with the restrictions of Φ are Banach spaces. Moreover, it turns out that each Banach space with unconditional basis is (up to isomorphism) an exhaustive space for some extended norm.
In Section 5, first of all we show that if an ideal is representable in a Banach space, then it is representable in EXH(Φ) for some Φ via its canonical (unconditional) basis (en). Pointing out the interactions between analytic P-ideals and these spaces, we present numerous statements equivalent to (i) EXH(Φ)=FIN(Φ) including the following: (ii) EXH(Φ) is an Fσ subset of Rω;
(iii) EXH(Φ) does not contain a copy of c0; (iv) FIN(Φ) is separable; (v) all ideals representable in EXH(Φ) are Fσ.
Most of the results in this section
turned out to be just reformulations of known, often even classical facts in Banach space theory. However, to the best of our knowledge the language of exhaustive and finite spaces was not present in the literature. This language seems to be quite convenient and reveals certain symmetries between the theory of Banach space and the theory of analytic P-ideals.
There is a natural way of associating an lsc submeasure φ to an extended norm Φ: if τ∈Rω, then φ(A)=Φ(PA(τ)) is an lsc submeasure. So, an extended norm induces the Banach spaces EXH(Φ) and FIN(Φ), also it generates (many) analytic P-ideals. It seems interesting to study how the properties of Banach spaces induced by Φ impact properties of the ideals induced by Φ and vice versa. This is partially done
in Section 5. Also, concrete examples of Banach spaces motivate definitions of extended norms and it is interesting to see what concrete analytic P-ideals they induce. And
vice versa, one can consider extended norms associated to classical analytic P-ideals and look at the Banach spaces which they generate.
Pretentiously, one can summarize the above results saying that Banach spaces (with unconditional bases) are linearized versions of (non-pathological) analytic P-ideals and, vice versa, (non-pathological) analytic P-ideals are discretized
incarnations of Banach spaces (with unconditional
bases). We will show that this analogy leads us to new characterizations of classical properties, as well as, provides us with new tools of constructing interesting examples of ideals and Banach spaces.
From Section 5 on we consider Banach spaces and analytic P-ideals induced by families of finite sets. If F⊆[ω]<ω={A⊆ω:∣A∣<ω} covers ω then ΦF(x)=sup{∑i∈F∣xi∣:F∈F} is
an extended norm. The main result, Theorem 6.3, of the paper points out a deeper interaction between combinatorial properties of F, topological properties of EXH(ΦF), and complexity of F-ideals, that is, ideals of the form Exh(φ) where φ is induced by Φ in the way mentioned above(we will see that most important analytic P-ideals are of this form). We prove that the following are equivalent:
(i) F is precompact, that is, F⊆[ω]<ω (see [LA15] and [LAT09] for applications of precompact families), (ii) EXH(ΦF) is c0-saturated, (iii) EXH(ΦF)
does not contain ℓ1, and (iv) nontrivial F-ideals are not Fσ.
We apply this result frequently in the next sections. In Section 7, we present a combinatorial application related to Fremlin’s DU problem; also we show that this application enables us to strengthen Pták’s Lemma, a classical theorem in combinatorics and Mazur’s Lemma, a basic tool in Banach space theory.
In Section 8 we briefly discuss properties of the Schreier idealsIα (α<ω1), ideals generated by the so called Schreier families Sα. We show that the following holds: (a) I1 is the ideal of asymptotic density zero sets; (b) Iα is summable-like and not Fσ for α>1 (a quite
rare combination of properties); and (c) that (Iα) forms a strictly ⊆-decreasing sequence, intertwined by a sequence of Fσ ideals.
In Section 9, we investigate Banach spaces induced by families which are far from being precompact. We show that the family associated with the so called Farah ideal induces a Banach space which possesses the Schur property but which is not isomorphic
to ℓ1. As a generalization of this example, we provide a sufficient condition for EXH(ΦF) to satisfy the Schur property.
Acknowledgement
The authors would like to thank Jordi Lopez-Abad for his useful remarks and fruitful discussions on the subject of this article.
2. Basics of analytic P-ideals
We start with basic definitions, examples, and facts concerning analytic P-ideals. For more we refer the reader e.g. to [Hru11]. Recall that I⊆P(Ω) is an ideal on an infinite set Ω if it contains all
finite subsets of Ω, it is hereditary (that is, A⊆B∈I implies A∈I), and it is closed under taking unions of finitely many elements from it. In order to simplify the list of conditions in certain results, we allow
Ω∈I, i.e. P(Ω) is considered to be an ideal; all other ideals are called proper. In our investigations Ω will always be countably infinite, and hence, without loss of generality, mostly we will work with Ω=ω.
We will use the notations I+=P(Ω)∖I and I↾X={A⊆X:A∈I}.
An ideal I on ω is Fσ* * (Borel, analytic, etc) if I is an Fσ (Borel, analytic, etc) subset of P(ω) identified with compact Polish space 2ω in the standard way. I is a P-ideal if for each countable C⊆I there is an
A∈I such that C⊆∗A for each C∈C (where C⊆∗A iff C∖A is finite). Finally, I is tall if each infinite subset of ω contains an infinite element of I.
For example, the ideal of finite sets Fin=[ω]<ω is an Fσ P-ideal which is not tall; the classical summable ideal
[TABLE]
is an Fσ tall P-ideal; and the density zero ideal
[TABLE]
is an Fσδ (not Fσ)
tall P-ideal. We will see more general examples later.
A function φ:P(ω)→[0,∞] is a submeasure (on ω) if φ(∅)=0, φ(X)≤φ(X∪Y)≤φ(X)+φ(Y) whenever X,Y⊆ω, and φ({n})<∞ for n∈ω. A submeasure φ is
lower semicontinuous (lsc) if φ(X)=sup{φ(F):F∈[X]<ω}(=limn→∞φ(X∩n)) for each X⊆ω. Submeasures turn out to be the ultimate tool in studying analytic P-ideals. If φ is an lsc
submeasure on ω then for X⊆ω define tailφ(X)=inf{φ(X∖F):F∈[ω]<ω}(=limn→∞φ(X∖n)),111The original notation for tailφ(X) was ∥X∥φ, we decided to use a new notation to avoid any confusions when working with Banach spaces. and let
[TABLE]
It is easy to see that Fin(φ) is an Fσ ideal, and similarly Exh(φ) is an Fσδ P-ideal. Clearly, Exh(φ)⊆Fin(φ) always holds, and Exh(φ) is tall iff φ({n})→0.
Theorem 2.1**.**
([Maz91], [Sol96])* Let I be an ideal on ω. Then the following holds: (1) I is Fσ iff I=Fin(φ) for some lsc φ; (2) I is an analytic P-ideal iff I=Exh(φ) for some lsc φ; and (3) I is an Fσ P-ideal iff I=Fin(φ)=Exh(φ) for some lsc φ.*
In particular, analytic P-ideals are Fσδ. Let us overview some basic classes of analytic P-ideals.
Example 2.2**.**
Summable ideals and Farah ideals. Let h:ω→[0,∞) be a function.
The summable ideal generated by h is
[TABLE]
Summable ideals are Fσ P-ideals, and Ih is tall iff h(n)→0. Of course, I1/n is a summable ideal up to an bijection between ω and ω∖{0} (similarly, properties/classes of ideals on ω are extended to ideals on arbitrary countably infinite sets).
In [Far00, Example 1.11.1] Farah gave an example of an Fσ P-ideal which is not a summable ideal. Motivated by his example we define a class of ideals which we call Farah ideals and which contain all summable ideals: Let (Pn) be a partition of ω into finite nonempty sets, and let ϑ=(ϑn) be a sequence of submeasures, ϑn:P(Pn)→[0,∞).
The Farah ideal generated by ϑ is
[TABLE]
where ϑn(A)=ϑn(A∩Pn). All these ideals are Fσ P-ideals, and Jϑ is tall iff max{ϑn({i}):i∈Pn}n→∞0. Clearly, if all ϑn are measures then Jϑ is a summable ideal and of course all summable ideals are obviously Farah ideals.
Notice that φ(A)=∑n∈ωϑn(A) is an lsc submeasure and Jϑ=Fin(φ)=Exh(φ).
Farah’s original example on ω∖{0} is the special case when Pn=[2n,2n+1), ϑ0({1})=1, and ϑn=min{n,∣A∣}/n2 for A⊆Pn if n>0.
Example 2.3**.**
Density and generalized density ideals. Let (Pn) and ϑ=(ϑn) be as above.
The generalized density ideal generated by ϑ
is
[TABLE]
These ideals are Fσδ P-ideals, and Zϑ is tall iff max{ϑn({i}):i∈Pn}n→∞0.
Clearly Jϑ⊆Zϑ. An ideal I is a density ideal if I=Zϑ where all submeasures in the sequence are measures.
Clearly, ψ(A)=sup{ϑn(A):n∈ω} is an lsc submeasure and Zϑ=Exh(ψ).
For example, as
Z={A⊆ω∖{0}:∣A∩[2n,2n+1)∣/2n→0}, the density zero ideal is a density ideal.
Clearly, Fin, and in general, trivial modifications of Fin, that is, ideals of the form {A⊆ω:∣A∩X∣<ω} for some X⊆ω are both Farah ideals and a (generalized) density ideals. We will show that apart from these examples, these classes are disjoint.
Theorem 2.4**.**
(reformulation of [Sol96, Theorem 3.3]) An analytic P-ideal I is not Fσ iff there is a decreasing sequence (Xn) in I+ which has no I-positive pseudointersection (that is, if X⊆∗Xn for every n, then X∈I).
Fact 2.5**.**
(see [FL19, Fact 2.3])
Let I be an analytic P-ideal. Then the following are equivalent: (i) I is nowhere tall, that is, I↾X is not tall for any X∈I+. (ii) If I=Exh(φ) then I={A⊆ω:A is finite
or limn∈Aφ({n})=0}. (iii) I is a trivial modification of Fin, or I is isomorphic (via a bijection between ω and ω×ω) to the density ideal {∅}⊗Fin={A⊆ω×ω:∀n{k:(n,k)∈A} is finite}.
Corollary 2.6**.**
A generalized density ideal Zϑ is either a trivial modifications of Fin, or it is not Fσ (in particular, not a Farah ideal).
Proof.
Applying Theorem 2.4, the sequence Xn=(ω∖n)×ω witnesses that {∅}⊗Fin is not Fσ. Now if Zϑ is neither this ideal nor a trivial modification of Fin, then,
according to Fact 2.5, there is an X∈Zϑ+ such that Zϑ↾X is tall. Notice that Zϑ↾X is also a generalized density ideal: If ϑn:P(Pn)→[0,∞), Qn=X∩Pn, and ηn=ϑn↾P(Qn), then Zϑ↾X=Zη. Let ψ be the canonical lsc submeasure generating Zη, that is,
ψ(A)=sup{ηn(A):n∈ω}.
We show that if Y⊆X is Zη-positive and 0<ε<tailψ(Y), then there is a Z⊆Y such that tailψ(Z)=ε. This is enough because then we can construct a sequence Y0⊇Y1⊇⋯ in Zη+ such that tailψ(Yn)→0, in particular, all pseudointersections of this sequence belong to Zη. Applying Theorem 2.4, Zη is not Fσ, and hence neither is Zϑ.
If Y and ε are as above, then there is an E∈[ω]ω={A⊆ω:∣A∣=ω} such that ηn(Y)>ε for every n∈E. As Zη is tall, we know that max{ηn({i}):i∈Qn}→0, and hence we
can assume that ∀n∈E∀i∈Qnηn({i})<ε.
Now, for each n, fix a Zn⊆Y∩Qn such that ηn(Zn)≤ε but ηn(Zn∪{i})>ε for every i∈(Y∩Qn)∖Zn. Then clearly ηn(Zn)→ε and hence Z=⋃n∈EZn is as required.
∎
There are many analytic P-ideals which are neither Fσ nor generalized density ideals; one of the most
important example of such an ideal is the following:
Example 2.7**.**
Trace ideal of the null ideal. The Gδ-closure of a set A⊆2<ω={s:s is a function, dom(s)∈ω, and ran(s)⊆2} is defined as [A]δ={x∈2ω:∃∞nx↾n∈A} where ∃∞ stands for “there exists infinitely many” (and dually, ∀∞ means “for all but finitely many”). Then [A]δ is a Gδ subset of 2ω, and every Gδ subset of 2ω is of this form. Let N denote the σ-ideal of null subsets of 2ω with respect to the usual product probability measure. The trace of N is defined as follows:
[TABLE]
This is a Fσδ tall P-ideal. If φ(A)=∑{2−∣s∣:s∈A is ⊆-minimal}, then φ is an lsc submeasure and tr(N)=Exh(φ).
We know (see [HHH07]) that tr(N) is totally bounded, that is, whenever tr(N)=Exh(ψ) for some lsc ψ, then ψ(2<ω)<∞, in particular, applying Theorem 2.1 (1), tr(N) is not Fσ. Also, we know (see [BNFP15, Proposition 6.2]) that tr(N)summable-like (see the definition later), hence it is not a generalized density ideal either.
3. Basics of Banach spaces
In this section we recall some basic notions and results from the theory of Banach spaces, see any classical textbook on the subject for more details (e.g. Schlumprecht’s online available lecture notes [Sch18]). All Banach spaces we work with are considered over R. A sequence (en) in a Banach space X is basic if for every x∈[(en)]:=span({en:n∈ω}) there is a unique (αnx)∈Rω such that x=∑n∈ωαnxen. A basic sequence (en) in X is a (Schauder) basis of X if X=[(en)].
Two basic sequences (en) in X and (vn) in Y are (isometrically) equivalent if there is an (isometric) isomorphism T:[(en)]→[(vn)] such that T(en)=vn for every n.
Recall that if (xn) is sequence in X, then we say that ∑n∈ωxn is unconditionally convergent if
[TABLE]
Let us mention a couple of equivalent statements: (a) ∀ε>0∃N∈ω∀E∈[ω∖N]<ω∥∑n∈Exn∥<ε; (b) ∀ permutation π:ω→ω∑n∈ωxπ(n) is convergent; (c) ∀A⊆ω∑n∈Axn is convergent; and (d) ∀(εn)∈{±1}ω∑n∈ωεnxn is convergent.
The classical Dvoretzky-Rogers theorem (see [DR50]) says that in each infinite dimensional X there is a sequence (xn) such that ∑n∈ωxn is unconditionally convergent but ∑n∈ω∥xn∥=∞.
Now, a basic sequence (basis, resp.) (en) is unconditional if x=∑n∈ωαnxen converges unconditionally for every x∈[(en)]. For every unconditional basic sequence (basis, resp.) (en) there are C,K∈[1,∞) such that for every n∈ω, F⊆n, αk∈R, and εk=±1 (k<n) the following holds:
[TABLE]
Moreover, the following stronger form of the second inequality also holds:
[TABLE]
If K is such a constant, then we say that the base (en) is K-unconditional. If C and K are the smallest such that the above inequalities hold, then C≤K≤2C. Also, if any of these inequalities holds for a sequence (en) of non zero elements from X (with all possible parameters in the inequality and fixed C or K), then (en) is a (K-)unconditional basic sequence in X.
If (en) is a(n unconditional) basis in X then the associated coordinate functionalsen∗:X→R are determined by x=∑n∈ωen∗(x)en for every x. Then en∗∈X∗={bounded linear functionals on X} and (en∗) is a(n unconditional) basic sequence in X∗.
We will need two special types of bases:
We say that a basis (en) in X is shrinking if (en∗) is a basis in X∗ (i.e. [(en∗)]=X∗); equivalently, ∀x∗∈X∗∥x∗↾[(en)n≥N]∥N→∞0; equivalently, (en∗) is a boundedly complete basis (see below) in [(en∗)]. Furthermore, if (en) is an unconditional basis in X, then (en) is shrinking iff X does not contain a copy of ℓ1. For example, the standard basis of c0 is shrinking.
We say that a basis (en) in X is boundedly complete if ∑n∈ωanen is convergent whenever sup{∥∑n<Nanen∥:N∈ω}<∞; equivalently, the natural embedding X→[(en∗)]∗ is an isomorphism; equivalently, (en∗) is a shrinking basis in [(en∗)]. Furthermore, if (en) is an unconditional basis in X, then (en) is boundedly complete iff X does not contain a copy of c0. For example, the standard basis of ℓ1 is boundedly complete.
In general, characterisations of those spaces which contain copies of c0 or ℓ1 are among the most well-studied fundamental problems in the theory of Banach spaces. Let us here recall two related results we will need later. We say that a
sequence (xn) in X is perfectly bounded if sup{∥∑n∈Fxn∥:F∈[ω]<ω}<∞.
Theorem 3.1**.**
(Bessaga-Pełczyński c0 theorem, see e.g. [BP58])*
Let X be a Banach space. Then X does not contain a copy of c0 iff ∑n∈ωxn is unconditionally convergent for every perfectly bounded sequence (xn) from X.*
In the next theorem we need certain notions associated to the weak topologies of Banach spaces, we give rather direct definitions here without defining the weak topology itself: A sequence (an) in Xconverges weakly to a∈X if x∗(an)→x∗(a) for every x∗∈X∗; and similarly, (an) is weakly Cauchy if (x∗(an)) is Cauchy/convergent in R for every x∗∈X∗.
Theorem 3.2**.**
(Rosenthal ℓ1 theorem, see [Ros74]). Let X be a Banach space and (an) a bounded sequence in X. Then (an) has a subsequence (an)n∈E (E∈[ω]ω) such that (exactly) one of the following holds: (a) (an)n∈E is weakly Cauchy; (b) (an)n∈E is equivalent to the standard basis of ℓ1.
At the end of this section, as the first application of Banach space theory to analytic P-ideals, we will present a natural common generalization of Farah ideals and generalized density ideals (motivated by examples from [LV99]).
Example 3.3**.**
Common generalization of Farah and generalized density ideals. Let X⊆Rω be a linear space equipped with a norm ∥⋅∥ generating a complete topology such that the canonical algebraic basis
(en) of c00={x=(xn)∈Rω:∀∞nxn=0} is a 1-unconditional base of X (e.g. X=c0 or X=ℓp, 1≤p<∞). Let (Pn) and ϑ=(ϑn) be as in the definitions of
Jϑ and Zϑ.
We associate the following family to ϑ and X:
[TABLE]
where of course ϑn(A)=ϑn(A∩Pn). Applying that (en) is unconditional, it is easy to show that this family is an ideal. Now if ϑ and X are as above, then define φϑ,X:P(ω)→[0,∞] as follows:
[TABLE]
Applying 1-unconditonality of (en), it is easy to show that φϑ,X is subadditive, and of course, it is monotone and lsc by definition; furthermore, a straightforward argument shows that Jϑ,X=Exh(φϑ,X), hence Jϑ,X is an analytic P-ideal.
Notice that Jϑ=Jϑ,ℓ1 and Zϑ=Jϑ,c0.
Remark 3.4**.**
An easy application of the Bessaga-Pełczński c0 theorem shows that the ideal Jϑ,X is Fσ whenever X does not contain a copy of c0 .
4. Representing ideals in Banach spaces
In [BNFP15] the authors considered a natural generalization of summable ideals. Let G be a completely metrizable Abelian group. Unconditional convergence of infinite sums in G is defined just like in Banach spaces, and the appropriate translations of its equivalent formalizations hold as well. Now, fix a sequence h:ω→G and define
[TABLE]
Then IhG is an ideal on ω, and ideals of this form are called representable in G.
Example 4.1**.**
(see [BNFP15])
Summable ideals are exactly those represented in R. Similarly, an ideal is representable in Rω iff it is the intersection of countable many summable ideals (such an ideal is not necessarily Fσ). Finally, we show that Z is representable in c0 (over ω∖{0} for now): Z=Ihc0 where h(n)=2−kek if n∈[2k,2k+1) and (ek) is the standard basis of c0 (similarly, every non-pathological generalized density ideal is representable in c0).
One of the main theorems of [BNFP15] says that the class of ideals represented in Polish Abelian groups coincide with the class of analytic P-ideals, and the class of those represented in Banach spaces is still quite large containing all important examples. Allow us to first state the theorem, and give a detailed introduction to the notion we use in it afterwards:
Theorem 4.2**.**
([BNFP15]) An ideal is representable in a Polish Abelian group iff it is an analytic P-ideal. An ideal is representable in a Banach space iff it is a non-pathological analytic P-ideal.
If μ={μα:α<κ} is a family of measures on ω such that sup{μα({n}):α<κ}<∞ for every n, then their pointwise supremum φμ(A)=sup{μα(A):α<κ} is always an lsc
submeasure. Submeasures of this form are called non-pathological. Notice that if φ is non-pathological, then there is a countable family μ={μn:n∈ω} of measures such that φ(F)=φμ(F) for
every F∈[ω]<ω, and hence (by lsc) φ=φμ. Furthermore, if we replace μn with {μn↾E:E∈[ω]<ω}, then φμ does not change, and hence we can assume that
supp(μn)={k∈ω:μn({k})=∅} is finite for every n. Actually, a trivial compactness argument shows that if φ is non-pathological, then for every F∈[ω]<ω there is a measure μF such that
supp(μF)⊆F, μF≤φ, and μF(F)=φ(F); in particular φ=sup{μF:F∈[ω]<ω}. Moreover, applying the Banach-Alaoglu theorem, one can show that this hold for every F∈Fin(φ).
Also, by extending μ with countable many measures of finite support, we can always assume (without changing Exh(φμ)) that ⋃n∈ωsupp(μn)=ω, i.e. that φμ({n})>0 for every
n. From now on when using a family μ of measures, we will always assume that μ={μn:n∈ω} is countable, supp(μn) is finite for every n, and 0<sup{μn({k}):n∈ω}<∞ for every k.
We say that an analytic P-ideal I is non-pathological if there is a non-pathological φ such that I=Exh(φ) (it is trivial to construct pathological submeasures, for examples of pathological ideals see
[Far00, Theorem 1.9.5]). For example, if ϑn is non-pathological for every n, then the Farah ideal Jϑ and the generalized density ideal Zϑ are non-pathological. Trace of null is also
non-pathological: Let F={F⊆2<ω:F is a finite antichain}, for every F∈F define the measure μF as μF(A)=∑s∈A∩F2−∣s∣, and let μ={μF:F∈F}. Then tr(N)=Exh(φμ).
Let G and h be as above and fix an invariant compatible (hence complete) metric d on G. Then φhG(A)=sup{d(0,∑n∈Fh(n)):F∈[A]<ω} is an lsc submeasure and IhG=Exh(φhG).
If I is represented in a Banach space X, then it is is represented in ℓ∞, I=Ihℓ∞, h(n)=(h(n)0,h(n)1,…). Switching to h′(n)=(∣h(n)0∣,∣h(n)1∣,…) does not change the ideal, and if we define
the measures μk(A)=∑n∈A∣h(n)k∣ then φh′ℓ∞=sup{μk:k∈ω} is non-pathological. If we do not wish to embed X in ℓ∞, then the above construction pulled back to X gives us the
non-pathological submeasure
[TABLE]
and IhX=Exh(ψhX).
∎
There is an important special class of non-pathological submeasures: for every cover F⊆[ω]<ω of ω and sequence τ=(τn)∈(0,∞)ω, define μ(F,τ) as follows: μ∈μ(F,τ) iff
μ is a measure on ω, supp(μ)∈F, and μ({n})=τn for n∈supp(μ); then μ(F,τ) is a family of measures as above. Let φF,τ=φμ(F,τ) and
IF,τ=Exh(φF,τ). Most of our main examples of analytic P-ideals can be easily written of this form: in the case of Ih let F=[ω]<ω and τn=h(n). In the case of Farah’s example let
F={F∈[ω∖{0}]<ω:∀n>0∣F∩[2n,2n+1)∣=n}, τ1=1, and τk=1/n2 for k∈[2n,2n+1) and n>0. Finally, it is easy to present density ideals and tr(N) in this form.
As we will see later, combinatorics of ideals of this simple form is strongly related to properties of certain easily definable but interesting Banach spaces.
Problem 4.3**.**
Can all non-pathological analytic P-ideals be written of the form IF,τ?
We close this section with some general results on representability of ideals in Banach spaces. In [BNFP15] the authors considered questions of representability of ideals in classical Banach spaces. The case of c0 seems to be particularly interesting. As summable ideals can be represented in R, they are representable in every Banach space. The next theorem says that apart from these trivial representations no Fσ P-ideals can be represented in c0, whereas, as we saw above, c0 is a “natural”
Banach space for representing density ideals.
Theorem 4.4**.**
Among Fσ tall P-ideals only summable ideals are representable in c0.
Representability of ideals was considered independently by Drewnowski and Labuda (in a bit different setting). They proved another theorem revealing some connections between c0 and representability of Fσ ideals.
Theorem 4.5**.**
(see [DL10] and [DL17])* If X is an infinite dimensional Banach space, then there is an ideal representable in X which is not a summable ideal; furthermore X does not contain a copy of c0 iff IhX is Fσ for every h:ω→X.*
Proof.
To prove the first statement, fix a sequence (xn) such that ∑n∈ωxn is unconditionally convergent but ∑n∈ω∥xn∥=∞. We can assume that 0<∥xn∥<1 for every n. Fix a partition (Pn) of ω into finite sets such that 1≤∣Pn∣⋅∥xn∥<2 for every n and define h:ω→X as h(k)=xn for every k∈Pn. We claim that that IhX is not a summable ideal. Assume on the contrary that IhX=If for some f:ω→[0,∞). Then liminfn∈ω∑k∈Pnf(k)=ε>0 because ∥∑k∈Pnh(k)∥=∣Pn∣⋅∥xn∥≥1 and hence ⋃n∈EPn∈If+ for every infinite E⊆ω. Now if kn∈Pn such that f(kn)=max{f(k):k∈Pn}, then on the one hand, as ∑n∈ωh(kn)=∑n∈ωxn is unconditionally convergent, ∑n∈ωf(kn)<∞ follows, but on the other hand f(kn)≥ε/∣Pn∣>ε∥xn∥/2 and hence ∑n∈ωf(kn)≥∑n∈ωε∥xn∥/2=∞, a contradiction.
Second part of the statement: Z is representable in c0 and we know that Z is not Fσ.
Conversely, recall that φ(A)=φhX(A)=sup{∥∑k∈Fh(k)∥:F∈[A]<ω} is an lsc submeasure and IhX=Exh(φ) (see the remarks on the proof of Theorem 4.2). Now, by Theorem 3.1, if X does not contain a copy of c0, then A∈IhX iff ∑n∈Ah(n) is unconditionally convergent iff h↾A is perfectly bounded iff φ(A)<∞. Therefore, IhX=Fin(φ) is Fσ.
∎
5. The Banach spaces FIN(Φ) and EXH(Φ)
The notion of representability of ideals in Banach spaces connects the theories of Banach space and of analytic P-ideals. One can consider questions of the form: Which ideals are representable in a Banach spaces satisfying certain properties or vice versa, what do we know about Banach spaces in which an ideal or family of ideals is represented?
On the other hand, an ideal can be represented in a Banach space in many different ways and sometimes the fact that an ideal can
be represented in a Banach space does not provide much information if we do not assume something more about the way it is represented. The extreme example illustrating this remark is the case of summable ideals: they are
represented in every Banach space.
In this section we will present a more canonical way of representing ideals in Banach spaces which enables us to avoid this issue. Through a general construction of Banach spaces, analogous to the way we associate ideals to
submeasures, we show that all non-pathological analytic P-ideals have representations via 1-unconditional basic sequences. As a byproduct of this approach we obtain some structure theorems for Banach spaces (most of them well-known, though).
An extended norm is a function Φ:Rω→[0,∞] satisfying all properties of a norm but possibly taking infinite values (where we let 0⋅∞=0 at the axiom Φ(αx)=∣α∣Φ(x)). To avoid strange examples of extended norms, more precisely, to deepen the analogy with lsc submeasures, we will assume that extended norms satisfy some natural additional conditions, we say that Φ is nice if the following holds:
(a)
∀x∈c00={(xn)∈Rω:∀∞nxn=0}Φ(x)<∞;
(b)
(monotonicity) If x,y∈Rω and ∣xn∣≤∣yn∣ for every n, then Φ(x)≤Φ(y);
where PA:Rω→Rω is the projection associated to A⊆ω, that is, PA(x)(n)=x(n) if n∈A and [math] otherwise. Now
fix a nice extended norm Φ, define tailΦ(x)=inf{Φ(Pω∖F(x):F∈[ω]<ω)(=limn→∞Φ(Pω∖n(x))), and consider the following linear spaces:
[TABLE]
When working with these spaces, (en) denotes the standard (algebraic) basis of c00.
Proposition 5.1**.**
Let Φ be a nice extended norm. Then EXH(Φ)⊆FIN(Φ) equipped with Φ are Banach spaces, EXH(Φ) is the completion of c00, and (en) is a 1-unconditional basis in EXH(Φ). Furthermore, every 1-unconditional basis (bn) in a Banach space X is isometrically equivalent to (en) in EXH(Φ) for some nice Φ. In particular, every unconditional basis is equivalent to (en) in
EXH(Φ) for some nice Φ.
Proof.
As Φ is finite on c00 and Φ(x)≤Φ(Pn(x))+Φ(Pω∖n(x)), EXH(Φ)⊆FIN(Φ) follows.
FIN(Φ) is complete: Let (xn) be a Cauchy sequence in FIN(Φ). Applying monotonicity, Φ(P{k}(xn−xm))≤Φ(xn−xm) for every k,n,m, and hence (P{k}(xn))k∈ω is Cauchy in the kth 1-dimensional
coordinate space of Rω (a Banach space, as Φ is finite on c00), P{k}(xn)n→∞yk, let y=(yk). We will first show that y∈FIN(Φ). The sequence {xn:n∈ω} is bounded, let
say Φ(xn)≤B for every n. We show that Φ(y)≤2B, i.e. (by lsc of Φ) Φ(PM(y))≤2B for every M∈ω. Fix an M>0. If n is large enough, say n≥n0, then Φ(P{k}(y−xn))≤B/M for every k<M and hence
[TABLE]
Now we will prove that xn→y. If not, then there are ε>0 and n0<n1<⋯<nj<… such that Φ(xnj−y)>ε, that is, Φ(PMj(xnj−y))>ε for some Mj∈ω∖{0} for every j. Pick j0 such that Φ(xnj0−xn)<ε/2 for every n≥nj0 and then pick j1>j0 such that Φ(P{k}(xnj1−y))≤ε/(2Mj0) for every k<Mj0. Then
[TABLE]
a contradiction.
EXH(Φ)=c00: c00 is dense in EXH(Φ) because Φ(x−Pn(x))=Φ(Pω∖n(x))n→∞0 for every x∈EXH(Φ). We have to show that EXH(Φ) is closed. Let x∈FIN(Φ) be an accumulation point of EXH(Φ). For any ε>0 we can find y∈EXH(Φ) such that Φ(x−y)<ε, and then n0 such that Φ(Pω∖n(y))<ε for every n≥n0. If n≥n0 then Φ(Pω∖n(x))≤Φ(Pω∖n(x−y))+Φ(Pω∖n(y))<2ε.
(en) is a 1-unconditional basis in EXH(Φ): This follows from monotonicity of Φ (see the characterizations of unconditional bases in Section 2).
Suppose (bn) is a 1-unconditional basis in a Banach space X. In particular, for each x∈X there is a unique (xn)∈Rω such that x=∑n∈ωxnbn, and hence we can assume that X⊆Rω. For
x=(xn)∈Rω define Φ(x)=sup{∥∑k<nxkbk∥:n∈ω}. Applying 1-unconditionality of (bn), Φ is a nice norm, and of course Φ(x)=∥x∥ for every x∈X. Now, x∈EXH(Φ) iff
Φ(Pω∖n(x))n→∞0 iff ∑n∈ωxnbn is Cauchy/convergent iff x∈X.
∎
Remark 5.2**.**
Notice that tailΦ(x)=tailΦ(y) whenever x−y∈EXH(Φ), in particular, we can consider tailΦ:FIN(Φ)/EXH(Φ)→[0,∞), and it is straightforward to check that this map is the canonical quotient norm on FIN(Φ)/EXH(Φ).
It is not a coincidence that when introducing the notions of FIN(Φ) and EXH(Φ) we mimicked the notations from the theory of analytic P-ideals. Every nice extended norm Φ gives rise in a natural way to many non-pathological submeasures and vice versa:
Proposition 5.3**.**
Let Φ be a nice extended norm and
σ=(σn)∈Rω. Then the function φ(A)=Φ(PA(σ))
is a non-pathological submeasure. Conversely, every non-pathological submeasure is of this form for some nice Φ and σ=1=(1,1,…). In particular, every non-pathological analytic P-ideal I is represented in EXH(Φ) for
some nice Φ by the unconditional basis (en), i.e. I=I(en)EXH(Φ).
Proof.
Lower semicontinuity of Φ and 1-unconditionality of (en) imply that φ is an lsc submeasure. To show that it is non-pathological, notice that φ=φ(σnen)EXH(Φ)=ψ(σnen)EXH(Φ)
(see the proof of Theorem 4.2, in particular Exh(φ)=I(σnen)EXH(Φ)).
Now let μ={μn:n∈ω} be a family of measures and φμ(A)=sup{μn(A):n∈ω} be the associated non-pathological submeasure. Define Φμ:Rω→[0,∞] as follows:
[TABLE]
Then Φμ is a nice extended norm and Φμ(PA(1))=sup{μn(A):n∈ω}=φμ(A), and hence Exh(φμ)=I(en)EXH(Φμ). Let us point out that we do not need to use φμ=ψ(en)EXH(Φμ) to see that Exh(φμ)=I(en)EXH(Φμ). Indeed, A∈Exh(φμ) iff φμ(A∖n)=Φμ(PA∖n(1))→0 iff PA(1)∈EXH(Φμ) iff ∑i∈Aei is unconditionally convergent.
∎
Also, notice that (σn) and (∣σn∣) generate the same submeasure, and we can assume, without changing the ideal, that σn>0 for every n (i.e. that σ∈(0,∞)ω just like τ in the definition of μ(F,τ)).
The next result was motivated by [Din17, Theorem 3.2] and, in principle, it gathers some known results (Ding [Din17], Drewnowski and Labuda [DL10], Bessaga-Pełczyński [BP58]) and put them in the framework of the notions and notations we introduced. For example, notice that (1)↔(2) below can be seen as a natural counterpart of Theorem 2.1 (3).
Theorem 5.4**.**
Let Φ be a nice extended norm. Then the following are equivalent:
(1)
FIN(Φ)=EXH(Φ).
2. (2)
EXH(Φ)* is Fσ in Rω.*
3. (3)
EXH(Φ)* does not contain c0, i.e. (en) is a boundedly complete basis.*
4. (4)
FIN(Φ)* is separable.*
5. (5)
Rω/EXH(Φ)* is Borel reducible to Rω/ℓ∞.*
6. (6)
Only Fσ ideals can be represented in EXH(Φ).
7. (7)
Z* is not represented in EXH(Φ).*
8. (8)
If σ∈Rω and φ(A)=Φ(PA(σ)) then Exh(φ)=Fin(φ).
Proof.
(1)→(2): We show that {x∈Rω:Φ(x)≤M} is closed, and hence FIN(Φ) is Fσ in Rω. Indeed, assume that Φ(x)>M+ε for some ε>0. By lsc, there is N∈ω such that Φ(PN(x))>M+ε. As
Φ is a norm on c00, we can find δ>0 such that Φ(PN(y))<ε whenever ∣y(n)∣<δ for each n∈N. For such y, Φ(x+y)≥Φ(PN(x+y))>M. In other words, {x∈Rω:Φ(x)>M} is open. (Similarly, EXH(Φ) is Fσδ.)
(2)→(1): Let EXH(Φ)=⋃n∈ωFn, where each Fn is closed in the product topology. As norms on finite dimensional linear space are equivalent, the restriction of every basic open subset of Rω is open in
(EXH(Φ),Φ) as well, and hence Fn is closed in (EXH(Φ),Φ) for every n. Applying the Baire Category Theorem, there are n∈ω, c∈EXH(Φ), and ε>0 such that Bc(ε)={x∈EXH(Φ):Φ(x−c)<ε}⊆Fn. Now if y∈FIN(Φ)∖{0}, then Pk(y)∈EXH(Φ) and 2Φ(Pk(y))εPk(y)+c∈Bc(ε)⊆Fn for every large enough k. Clearly, Pk(y)→y in the product topology, hence 2Φ(y)εy+c∈Fn⊆EXH(Φ), and so y∈EXH(Φ).
(3)↔(1): FIN(Φ)=EXH(Φ) says exactly that (en) is boundedly complete.
(1)→(4): Trivial.
(4)→(1): If x∈FIN(Φ)∖EXH(Φ), Φ(x)≥inf{Φ(Pω∖n(x)):n∈ω}>ε>0, then there are pairwise disjoint finite sets Fn⊆ω such that Φ(PFn(x))>ε for every n. For A⊆ω let FA=⋃n∈AFn and yA=PFA(x)∈FIN(Φ). If A,B⊆ω and n∈A△B then Φ(yA−yB)≥Φ(PFn(x))>ε, and therefore no set of size less than continuum can be dense in FIN(Φ).
(8)↔(1): If there is a σ∈FIN(Φ)∖EXH(Φ) and φ(A)=Φ(PA(σ)), then φ(ω)=Φ(σ)<∞ and so Fin(φ)=P(ω) but ω∈/Exh(φ). Conversely, if σ and φ are as in (8) and A∈Fin(φ)∖Exh(φ), then PA(σ)∈FIN(Φ)∖EXH(Φ).
∎
The above theorem raises several natural questions. One of them is whether EXH(Φ) not containing ℓ1 can be characterized in a way similar to the theorem above. Another is the question about the general relationship between EXH(Φ)
and FIN(Φ). The following proposition partially solves both of these questions.
Proposition 5.5**.**
Let Φ be a nice extended norm. Then EXH(Φ) does not contain ℓ1 (i.e. (en) is a shrinking basis) iff
FIN(Φ) is isomorphic to EXH(Φ)∗∗.
Proof.
We know (see e.g. [Sch18, Proposition 3.3.6]) that is (en) is a shrinking basis in X then X∗∗ is isomorphic to the space {α=(αn)∈Rω:Ψ(α):=sup{∥∑k<nαkek∥:n∈ω}<∞}=FIN(Ψ), and notice that Ψ=Φ in the case of X=EXH(Φ).
Conversely, if FIN(Φ) is isomorphic to EXH(Φ)∗∗, then ∣EXH(Φ)∗∗∣≤c, and by [OR75, Main theorem], EXH(Φ) does not contain ℓ1.
∎
It is however unclear for us if EXH(Φ) not containing ℓ1 could be characterized in terms of ideals represented in EXH(Φ) like in the case of c0 in Theorem 5.4. Although for nice norms induced by
families of finite sets (see below) we have such characterization: EXH(Φ) does not contain a copy of ℓ1 if and only if no nontrivial Fσ ideal can be represented in EXH(Φ) by its unconditional basis, see Theorem
6.3.
In the rest of this section, we will present numerous examples of nice extended norms together with Banach spaces and analytic P-ideals induced by them. In fact all these examples will be of a very special form, generated by families of finite
subsets of ω (see also remarks after Theorem 4.2). The
study of these families and the norms they induce have become an important section of Banach space theory in the past decades (see e.g. [AD92], [Cas93], [LAM08], [LA15], and [LAT09]).
Let F⊆[ω]<ω be a cover of ω containing all singletons. Define an extended norm ΦF by
[TABLE]
It is plain to check that it is a nice norm. Thus, EXH(ΦF) and FIN(ΦF) are Banach spaces. From now on, we will write XF=EXH(ΦF). Notice that we can always assume that F is hereditary (i.e. closed for taking subsets) because the hereditary closure F↓ of F generates the same extended norm.
Now fix a sequence τ∈(0,∞)ω and consider a submeasure φF,τ defined from ΦF as before, i.e.
[TABLE]
Ideals of the form IF,τ will be called F-ideals. Notice that trivial modifications of Fin are F-ideals for every F. When we wish to exclude them we will talk about non-trivialF-ideals.
When studying classical examples, we will typically use one particular weight sequence. Let λ=(λk)k≥1, λk=2−n if k∈[2n,2n+1), that is,
[TABLE]
Depending on the underlying set, we will modify λ accordingly, on ω∖n we consider the appropriate restriction of λ, and in the case of 2<ω we consider (λt)t∈2<ω, λt=2−∣t∣; these
variants of λ will also be denoted by λ.
Example 5.6**.**
If F=[ω]≤1 then FIN(ΦF)=ℓ∞, EXH(ΦF)=c0, IF,τ={A⊆ω:A is finite or (τn)n∈A→0}, and hence, by Fact 2.5, up to isomorphism the only nontrivial F-ideal is {∅}⊗Fin.
Example 5.7**.**
If F=[ω∖{0}]<ω then
FIN(ΦF)=EXH(ΦF)=ℓ1 and, as IF,τ=Iτ (the summable ideal generated by τ), the family of F-ideals coincides with the family of all summable ideals. Notice that there are many
other families such that I1/n is of the form IE,τ, for example consider E={E∈[ω∖{0}]<ω:∑n∈Eλn≤1}, then I1/n=IE,λ. At the same time,
EXH(ΦE) contains a copy of c0, e.g. because λ witnesses that FIN(ΦE)=EXH(ΦE).
Example 5.8**.**
If P=⋃{P([2n,2n+1)):n∈ω} then FIN(ΦP) is isometrically isomorphic to the ℓ∞-product of (ℓ1(2n):n∈ω), similarly EXH(ΦP) is isometrically isomorphic to c0-product of this sequence. All P-ideals are density ideals including the density zero ideal Z=IP,λ.
Example 5.9**.**
Let A={finite antichains in 2<ω}. Then IA,λ=tr(N) (see the discussion on non-pathological ideals after Theorem 4.2). EXH(ΦA) contains both copies of c0 and ℓ1. Actually these copies are visible
to the naked eye: copies of c0 live on branches and copies of ℓ1 live on antichains, more precisely, {ef↾n:n∈ω} is equivalent to the standard basis of c0 for each f∈2ω and {es:s∈A} is equivalent to the
standard basis of ℓ1 for each infinite antichain A⊆2<ω.
Also, the norm ΦA is in fact a well-known norm on a certain subspace of FIN(ΦA). Let M(2ω) be the space of signed Radon measures of bounded variation on 2ω equipped with the variation-norm, that is,
[TABLE]
We claim that M(2ω) can be isometrically embedded in FIN(ΦA): For every μ∈M(2ω) let xμ:2<ω→R, xμ(t)=μ([t]) where [t]={f∈2ω:t⊆f} is the basic clopen set generated by t. Then the function μ↦xμ is linear and injective (because μ is uniquely determined by its values on basic clopen sets). We show that ∥μ∥=ΦA(xμ), and hence this is an isometric embedding M(2ω)→FIN(ΦA).
Let μ∈M(2ω), ε>0, and C0, C1 be disjoint clopen sets such that 2ω=C0∪C1 and ∣μ(C0)∣+∣μ(C1)∣>∥μ∥−ε. There are A0,A1∈A such that Ci=⋃t∈Ai[t], notice that
A0 and A1 must be disjoint and A:=A0∪A1 is also an antichain. Now, by definition ΦA(xμ)≥∑t∈A∣xμ(t)∣=∑t∈A0∣μ([t])∣+∑t∈A1∣μ([t])∣≥∣μ(C0)∣+∣μ(C1)∣>∥μ∥−ε. The same argument works in the other direction as well.
We do not know if EXH(ΦA) (or FIN(ΦA)) is isomorphic to a known Banach space.
6. Spaces and ideals generated by compact families
In the rest of the paper we will mostly discuss properties of XF and F-ideals. First of all, in this section, we will focus on precompact families of finite sets. We say that an F⊆[ω]<ω is precompact if
F⊆[ω]<ω, or equivalently, if every sequence (Fn) in F has a subsequence which forms a Δ-system (for detailed studies on precompact families, see e.g. [LA15] and [LAT09]). The name is motivated by the fact that an F⊆[ω]<ω is compact iff every sequence (Fn)
in F has a subsequence which forms a Δ-system with root from F. We will leave the following simple fact without proof.
Fact 6.1**.**
If H⊆P(ω), then H↓=H↓. In particular, (a) if H is closed then so is H↓; and (b) if F⊆[ω]<ω then
F⊆[ω]<ω iff F↓ does not contain infinite chains.
Notice that if F⊆[ω]<ω, then F, F↓, F∩[ω]<ω and F↓∩[ω]<ω=F↓∩[ω]<ω generate the same space XF.
Example 6.2**.**
Probably the most obvious examples of compact families of finite sets are the locally finite ones, that is, when {F∈F:n∈F} is finite for every n (and non-empty, as we always work with covers). It follows from [BNFP15, Proposition 6.4] that in this case all F-ideals are generalized density ideals.
Before the main result of this section, we recall some notions and results from the theory of Banach spaces and also from general topology:
If X,Y are Banach spaces then we say that X is Y-saturated if every infinite dimensional closed subspace of X contains a copy of Y.
If X is topological space, then by recursion on α we define the Cantor-Bendixson (CB) derivatives of X as follows: X(0)=X, X′={accumulation points of X}, X(α+1)=(X(α))′, and if γ is limit then
X(γ)=⋂α<γX(α). Let
[TABLE]
denote the CB-rank of X. If X is countable and compact, then rk(X) is always a successor ordinal, and if rk(X)=α+1, then
X(α) is (nonempty) finite (and so X(α+1)=∅). The classical theorem of Mazurkiewicz and Sierpiński says (see [MS20]) that in this case, if ∣X(α)∣=M, then X is homeomorphic to the ordinal ωαM+1 (equipped with the topology generated by its natural well-order).
We will use the following easy observation: If X and Y are compact metric spaces and f:X→Y is a continuous open surjection with finite fibers (i.e. ∣f−1({y})∣<ω for every y∈Y), then X(α)=f−1[Y(α)] for every α. In particular, in this case rk(X)=rk(Y).
Theorem 6.3**.**
Let F⊆[ω]<ω be a cover of ω. Then the following are equivalent:
(1)
F* is precompact.*
(2)
XF* is c0-saturated.*
(3)
XF* does not contain ℓ1, i.e. (en) is a shrinking basis.*
(4)
Non-trivial F-ideals are not Fσ.
Proof.
(1)→(2): As XF=XF, we may assume that F is compact. We know (see [CG91]) that if F is compact then XF can be embedded isometrically in a C(K) space, more precisely, in
C(KF) where
[TABLE]
It is trivial to check that KF is compact, and that the map XF→C(KF), x=(xn)↦fx, fx(y)=∑n∈supp(y)xnyn is an isometry.
Since Banach spaces of the form C(K), where K is countable and compact, are c0-saturated (see [PS59]), XF is also c0-saturated.
(2)→(3): Trivial.
(3)→(1) and (4)→(1): Suppose now that F is not precompact. Then, since we can assume that F is hereditary (because XF=XF↓), there is an infinite A⊆ω such that A∩n∈F for each n. As ΦF(PA(x))=sup{∑i∈A∩n∣xi∣:n∈ω}=∑i∈A∣xi∣, the sequence (ei)i∈A is isometrically equivalent to the standard basis of ℓ1. In particular, if τ=(τn)∈(0,∞)ω such that τn→0, ∑n∈Aτn=∞, and ∑n∈/Aτn<∞, then IF,τ is a tall summable ideal.
(1)→(4): As above, we may assume that F is compact. Now, suppose on the contrary that there is an Fσ P-ideal I which is not a trivial modification of Fin but I=IF,τ for some compact F.
Applying Fact 2.5 and Corollary 2.6, there is a D∈I+ such that I↾D is tall. Now, F[D]={D∩F:F∈F} is compact in P(D), it covers D, and I↾D=IF[D],τ↾D. Therefore, we can assume that our counterexample is tall (from now on D=ω, of course): I=Exh(φ)=Fin(φ) for some lsc φ such that φ({n})→0.
Let α be the minimal ordinal such that there are a compact F⊆[ω]<ω of rank α+1 covering ω and a τ∈(0,∞)ω such that I=IF,τ. Fix F and τ witnessing this
definition.
Claim A**.**
We may assume that F(α)={∅}.
Proof of the claim.
We know that F(α) is finite. Let N>max{max(F):F∈F(α)} and F[ω∖N]={F∖N:F∈F}. Since F→F[ω∖N], F↦F∖N is continuous, open, and has finite fibers, rk(F[ω∖N])=rk(F). Define the family G=F[ω∖N]∪{{n}:n≤N}. Then rk(G)=rk(F), G(α)={∅}, and IF,τ=IG,τ, hence we can work with G instead of F.
∎
Clearly, α≥1. In fact α≥2. If α=1 then F is locally finite, hence I is a tall generalized density ideal (see Example 6.2), which cannot be an Fσ ideal.
The map KF→F, y↦supp(y) is a continuous open surjection with finite fibers, hence KF is also of rank α+1, KF(α)={constant [math] sequence}, and so there is a homeomorphism ωα+1→KF, β↦yβ such that
yωα is the constant [math] sequence.
Let fn∈C(KF) be the image of en∈XF under the isometric embedding XF→C(KF), that is, fn(y)=yn.
As fn is continuous, ran(fn)⊆{−1,0,1}, and fn(yωα)=0, we can fix γn<ωα such that fn(yδ)=0 for every δ∈(γn,ωα]. We can assume that γ0<γ1<⋯ tends to ωα.
For n∈ω let
[TABLE]
a compact subset of KF.
Clearly ⋃n∈ωKn=KF. Define the lsc submeasures ψ and ψn for n∈ω on P(ω) as follows:
[TABLE]
where τifi↾Kn∈C(Kn) and ∥⋅∥ denotes the sup-norm on these spaces. Then Exh(ψ)=I and hence we can assume that ψ≤φ on P(ω) (otherwise we can work with φ′=ψ+φ because I=Exh(φ′)=Fin(φ′)). Let
[TABLE]
Notice that (In)n∈ω forms a non-increasing sequence of ideals containing I.
Claim B**.**
I=In* for every n∈ω.*
Proof of the claim.
Let G={supp(y):y∈Kn}. Then G is a continuous open image of Kn with finite fibers, and hence it is compact and rk(G)=rk(Kn)<α+1. Notice that Kn is a subset
of KG={y∈{−1,0,1}ω:supp(y)∈G} and typically it is a proper subset. The family G does not necessarily cover ω, so define G=⋃G and H=G∪[ω∖G]≤1. Then H is compact and rk(G)≤rk(H)≤max{rk(G),2}<α+1 (because α≥2).
We show that In=IH,σ where σ↾G=τ↾G and σi=2−i for i∈ω∖G, and hence In=I by the definition of α.
Kn⊆KH and if gi is the image of ei∈XH→C(KH), then gi↾Kn=fi↾Kn, in particular, if ∑i∈Aσigi is unconditionally convergent in C(KH), then ∑i∈Aσifi↾Kn is unconditionally convergent in C(Kn). As fi(y)=0 for every i∈(ω∖G) and y∈Kn, ∑i∈Aτifi↾Kn is also unconditionally convergent, therefore IH,σ⊆In.
Conversely, assume that ∑i∈Aτifi↾Kn is unconditionally Cauchy, that is,
[TABLE]
We know that ∥∑i∈Bτifi↾Kn∥<ε holds iff ∣∑i∈Bτifi(y)∣=∣∑i∈Bτiyi∣<ε for every y∈Kn. We would like to show that ∑i∈Aσigi is also unconditionally Cauchy in KH. Fix ε>0, let N≥Nε/2 such that ∑i∈ω∖(G∪N)2−i<ε, finally let B∈[A∖N]<ω and y∈KH be arbitrary.
If supp(y)∈/G then
[TABLE]
Now assume that supp(y)∈G but y∈/Kn (for y∈Kn we are done) and denote B′=B∩supp(y)⊆G. Then ∣∑i∈Bσiyi∣=∣∑i∈B′σiyi∣=∣∑i∈B′τiyi∣. We know that there is y′∈Kn with the same support. Now if we partition B′=P∪Q where P={i:yiyi′=1} and Q={i:yiyi′=−1}, then
[TABLE]
If E,F∈[ω]<ω are non-empty then we write E≤F if max(E)≤min(F), and similarly, E<F if max(E)<min(F).
Claim C**.**
There is a sequence (An)n∈ω such that (i) An∈In∖I for every n, (ii) ψn(An)<2−n for every n, and (iii) An∩Ak=∅ if n=k.
Proof of the Claim.
Fix an arbitrary sequence (In)n∈ω such that (i) holds. Let (Jk) be a sequence such that Jk=In for each k and some n, and Hn={k∈ω:Jk=In} is infinite for every n∈ω. By recursion on k we can construct
a sequence B0<B1<⋯ of finite sets such that Bk⊆Jk and φ(Bk)>k (because Jk∈/I=Fin(φ)). Now the sequence An=⋃k∈HnBk satisfies (i) and (iii). By throwing out finitely many points from each An we obtain the desired family.
∎
In particular, since ψn≤ψk for n≤k, we know that A≥n:=⋃k≥nAk∈In for every n, and by (ii) that ψn(A≥n)<2−n+1.
We can easily construct a sequence X0′<X1′<⋯ of non-empty finite subsets ω such that
φ(Xn′)≈1, X0′⊆A0, and Xn+1′⊆Amn where mn=max(Xn′)+1 for every n.
Claim D**.**
There is a set X⊆⋃nXn′ such that φ(X∩Xn′)→0 and φ(X)=∞.
Proof fo the claim.
Let X′=⋃n∈ωXn′. Consider the ideals I′=I↾X′ and Zϑ, where ϑ=(ϑn), ϑn=φ↾Xn′. Clearly I′⊆Zϑ but since I′ is an Fσ ideal and Zϑ is a tall generalized density ideal, they are not equal and every X∈Zϑ∖I′ is
as desired.
∎
Finally, we show that X∈Exh(ψ), and hence Exh(ψ)=Fin(φ), a contradiction.
Let Xn=X∩Xn′. Notice that X∖mn⊆A≥mn because Xn+1⊆Amn, Xn+2⊆Amn+1 etc. Now ψ=sup{ψy:y∈KF} where
[TABLE]
is an lsc submeasure on ω. Fix such y and the smallest m such that y∈Km.
If m≤mn then ψy(X∖mn)≤ψmn(X∖mn)≤ψmn(A≥mn)<2−mn+1≤2−n.
If mn≤mn′≤m<mn′+1 for some n′≥n then
[TABLE]
where ψy(X∩[mn,mn′))=0 because y=yδ for some δ>γm−1≥γmn′−1 and so fi(y)=0 for every i<mn′; as Xn′+1=X∩[mn′,mn′+1) we know that
[TABLE]
and ψy(X∖mn′+1)<2−n′−1+1≤2−n just like in the first case.
Therefore, ψy(X∖mn)≤φ(Xn(y))+2−n for every y with some n(y)>n. As φ(Xn)n→∞0, we obtain that ψ(X∖mn)n→∞0 as well.
∎
Remark 6.4**.**
In the same way as above one can prove a strengthening of Theorem 4.4. Namely, if I is represented in C(γ+1), for some countable ordinal γ, then I is either a summable ideal or not Fσ: let γ be minimal such that there is an Fσ ideal I which is not summable but represented in C(γ+1). Since I is not summable, γ has to be infinite and then it is easy to see that it has to be a limit ordinal.
Since C(γ+1) is isomorphic to C0(γ+1)={f∈C(α+1):f(γ)=0} we may assume that I is represented in C0(γ+1). If (fn) is a sequence in C0(γ+1) representing I, then we may assume that there is an
increasing sequence γn of ordinals converging to γ such that fn(δ)=0 for δ>γn. For each n define In as the ideal represented by the sequence (fn↾(γn+1)) in C(γn+1) and notice that
(In) is non-increasing, every In contains I, and I=In because of the choice of γ. From this point on we can continue the proof as above, starting with Claim C.
Since every space of the form XF, for a precompact F, can be embedded in C(γ+1) for some γ, as a corollary we obtain that among Fσ ideals only summable ideals can be represented in XF for a
precompact F.
Problem 6.5**.**
Is tr(N) an F-ideal for some (pre)compact F?
7. Applications to Pták’s Lemma and Mazur’s Lemma
In this section we take a short digression to show that Theorem 6.3 has nice applications in combinatorics. Consider the following problem.
Problem 7.1**.**
Fix a measure μ:P(ω)→[0,∞]. Is there a compact hereditary family F of finite subsets of ω such that for every E∈[ω]<ω there is F∈F such that
F⊆E and μ(F)≥μ(E)/2?
This is a version of of Fremlin’s DU Problem (see [Fre], although Fremlin’s note contains many versions of DU problem but this one). Notice that if μ is a counting measure, then the answer is positive, witnessed by the Schreier
family. However, if μ is a measure generating a tall proper ideal, that is, we assume that μ(ω)=∞ and that μ({n})→0, then the
answer seems to be difficult to guess. Theorem 6.3 implies that in this situation it is negative.
Theorem 7.2**.**
Let μ:P(ω)→[0,∞) be a measure such that μ(ω)=∞ and μ({n})→0, F be a hereditary family of finite subsets of ω, and fix ε>0. If for every E∈[ω]<ω there is F∈F such that F⊆E and μ(F)≥εμ(E), then F is not compact.
Proof.
Assume F is as in the theorem, consider the lsc submeasure defined by
[TABLE]
let I=Exh(φ) and J=Exh(μ). As φ≤μ, we know that J⊆I. On the other hand, φ(A∖n)≥εμ(A∖n) for every A⊆ω and n∈ω, hence I⊆J also holds, and so I=J. But J is a tall Fσ ideal and I is and F-ideal, a contradiction with Theorem 6.3.
∎
This proof seems to be rather indirect. However, it is unclear for us how to prove it more directly. Note that most of the proofs of the negative answers to variants of the DU Problem uses precalibers of measures.
Theorem 7.2 seems to be surprising if compared to Pták’s Lemma (see [Pta63]). Denote by M the set of finitely supported probability measures on ω.
Theorem 7.3** (Pták’s Lemma).**
Let F be a hereditary family of finite subsets of ω, and fix ε>0. If for each ν∈M there is F∈F such that ν(F)≥ε, then
F is not compact.
Using Theorem 7.2 we can see that instead of testing F against every measure in M, we may focus on finitely supported measures generated by just one measure (see also [AT05, Remark II.3.33]). For a measure μ on ω denote Mμ the following family of finitely
supported probability measures on ω: ν∈Mμ iff supp(ν)∈[ω]<ω, μ(supp(ν))>0, and
ν(A)=μ(A∩supp(ν))/μ(supp(ν)) for A⊆ω.
Corollary 7.4**.**
Let μ be a measure on ω such that μ(ω)=∞ and μ({n})→0, F be a hereditary family of finite subsets of ω, and fix ε>0. If for each ν∈Mμ there is an F∈F such that ν(F)≥ε, then F is not compact.
Let X be a Banach space and let (xn) be a bounded weakly null sequence in X. Then for each ε>0 there is a finite convex combination y=∑iαixi such that ∥y∥<ε.
Mazur’s Lemma can be derived directly from Pták’s Lemma (see [AT05, Corollary II.3.35] for the detailed proof): For x∗∈X∗ define Fx∗={n∈ω:∣x∗(xn)∣≥ε/2}, and let F be the hereditary closure of {Fx∗:x∗∈X∗}. Notice that F is compact. Fix ε>0. Using Pták’s Lemma, one can find ν∈M such that ν(F)<ε for some F∈F, and then y=∑i∈supp(ν)ν({i})xi is as desired.
Using Corollary 7.4 instead of Pták’s Lemma in the above proof, we may specify the form of the convex combination in Mazur’s Lemma:
Corollary 7.6**.**
Let X be a Banach space, (xn) be a bounded weakly null sequence in X, and let μ be a measure on ω such that μ(ω)=∞ and μ({n})→0. Then for each ε>0 there is a
finite G⊆ω and a convex combination y=∑i∈Gαixi where αi=μ({i})/μ(G), such that ∥y∥<ε.
8. Schreier ideals
In Example 5.9 we already saw an example of a family of finite sets which induces a well-known analytic P-ideal, and at the same time generates a Banach space whose properties are quite unclear and which may be, at least hypothetically, a new and interesting example. This
section is devoted to an opposite situation. We will define a family of analytic P-ideals, which, to the best of our knowledge, were not described before in the literature and which are induced by the so called Schreier families, well-known in the theory
of Banach spaces (see e.g. [CG91], [GL00]).
We begin with recalling the definition of Schreier families (introduced in [AA92]). Fix a ladder system
[TABLE]
on ω1, that is,
ξ1α<ξ2α<⋯ tends to α for every limit α; and by recursion on α<ω1 define Sα=Sαξ⊆[ω∖{0}]<ω as follows: S0=[ω∖{0}]≤1,
[TABLE]
and for limit α let
[TABLE]
where if F⊆[ω∖{∅}]<ω and A⊆ω then F↾A={F∈F:F⊆A}.
Notice that the classical Schreier family S=S1={∅}∪{F⊆ω∖{0}:∣F∣≤min(F)}. For each α<ω1 the family Sα is hereditary, compact in P(ω), and spreading, that is, whenever n∈ω∖{0}, {a1<a2<⋯<an}∈Sα, b1<b2<⋯<bn, and ai≤bi for every i, then
{b1<b2<⋯<bn}∈Sα as well (see [AA92, Proposition 4.2]).
For α<ω1 let Xα=EXH(ΦSα) be the αth Schreier space and Iα=ISα,λ the αth Schreier ideal in the sense of Section 5, that is,
Iα=Exh(φα), where φα(A)=φSα,λ(A)=ΦSα(PA(λ))=sup{∑i∈A∩Fλi:F∈Sα}. For each α<ω1, rk(Sα)=ωα+1, and hence Sα is homeomorphic to ωωα+1, furthermore Xα can be embedded in C(Sα) (see [AA92, Propositions 4.9 and 4.10]).
Clearly, X0=c0 and I0=P(ω).
Proposition 8.1**.**
I1=Z.
Proof.
Let Pn=[2n,2n+1), n∈ω. Since P={Pn:n∈ω}⊆S1, φ1≥φP,λ and hence I1⊆Z=Exh(φP,λ) follows. Conversely, let A∈Z and denote
an=∣A∩Pn∣/2n. Fix an ε>0, assume that an<ε for every n≥N for some N, and also fix an F∈S. We show that s:=∑{λk:k∈F∩A∖2N}<(1−log2ε)ε and hence φ1(A∖2N)≤(1−log2ε)εε→00.
When estimating s from above we can assume that F⊆A∖2N (otherwise we can switch to a “better” F), in particular (a) min(F)∈Pn0 for some n0≥N, (b) ∣F∩Pn∣<ε2n for every n, and (c) s=∑k∈Fλk. If we consider only (a) and (b) above and that F∈S1 i.e. ∣F∣≤min(F), then s is maximal if F contains the maximal amount of points from consecutive Pns starting with Pn0. For such an F we have s<(n1−n0)ε where n1 is the first n such that F∩Pn=∅, this holds e.g. if n1 satisfies ε(2n0+2n0+1+⋯+2n1)≥2n0+1>∣F∣. A straightforward calculation shows that n1=n0+1−⌊log2ε⌋ is large enough, and hence s<(1−log2ε)ε.
∎
Notice that if α<β, then essentiallySα⊆Sβ, i.e. there is n∈ω such that if S∈Sα and n<minS, then S∈Sβ (in notation Sα↾(ω∖n)⊆Sβ). Using this remark, it
is trivial to see that Iβ⊆Iα for α<β and so (Iα)α<ω1 forms a non-increasing sequence of ideals. Actually, we know more:
Proposition 8.2**.**
If 0<β then Iβ is not Fσ, and if α<β, then Iβ⊊Fin(φβ)⊊Iα (with the only exception Fin(φ1)=P(ω)=I0). In particular, if β is a limit ordinal, then Iβ⊊⋂α<βIα.
Proof.
As Sβ is compact and Iβ is a non-trivial Sβ-ideal, it is not Fσ by Theorem 6.3.
Now let α<β. Then Iβ=Exh(φβ)⊆Fin(φβ) and as Iβ is not Fσ, this inclusion is strict. Similarly, it is enough to show that Fin(φβ)⊆Iα because Iα is not Fσ for α>0. If A∈/Iα then there are an ε>0 and a sequence F0<F1<⋯ in Sα such that Fn⊆A and φα(Fn)>ε for every n. Now as n≤min(Fn), En=Fn∪Fn+1∪⋯F2n−1∈Sα+1, and since Sα+1 is essentially included in Sβ, we can assume that En∈Sβ for every n, therefore φβ(A)≥φβ(En)>nε for every n, and so A∈/Fin(φβ).
∎
The ideals Iα for α>1 resemble the density zero ideal Z and may be considered as kind of density ideals of “higher order”. However, the following proposition indicates that they are, in a sense, far from being (generalized)
density ideals.
An lsc submeasure φ is summable-like if there is an ε>0 such that for every δ>0 there is a sequence An∈[ω]<ω of pairwise disjoint finite sets with φ(An)<δ and there is a k∈ω such that \varphi\big{(}\bigcup_{n\in Y}A_{n}\big{)}\geq\varepsilon for every Y∈[ω]k. An analytic P-ideal J is summable-like if there is a summable-like submeasure φ such that J=Exh(φ). For example, Farah ideals which are not trivial modifications of Fin are summable-like, and we already mentioned that so is tr(N). As far as we know, tr(N) is the only known “natural” non Fσ example of such an ideal.
Proposition 8.3**.**
Iα* is summable-like for α>1.*
Proof.
Fix N∈ω∖{0}. We will define a sequence F0<F1<⋯ such that φα(Fn)=2−N for every n and φα(⋃n∈HFn)=1 for every H∈[ω]2N. Let Fn⊆[2N+n,2N+n+1) be first 2n many points in this interval. Then Fn∈S1⊆Sα (it is easy to see that Sn↾(ω∖n)⊆Sα for every n∈ω and α∈[n,ω1)) and hence φα(Fn)=2−N. Now, if H∈[ω]2N then ⋃n∈HFn∈S2↾(ω∖2)⊆Sα (here we need that min(F0)≥2), and therefore φα(⋃n∈HFn)=1.
∎
Remark 8.4**.**
So far we know that if α>0, then Iα is not Fσ and if α>1 it is summable-like, and of course, all of them contain the summable ideal. Since tr(N)⊆{A⊆2<ω:∣A∩n2∣/2n→0}=Z, one may wonder if every Iα contains tr(N). This is not the case, tr(N)⊈I2: Here of course we identify ω∖{0} with 2<ω in the standard way. Let A⊆ω∖{0} such
that
[TABLE]
where n∈[2k,2k+1). Now, A∈tr(N) because A∩[2n,2n+1) is a maximal antichain in A∖2n, and hence φtrn(A∖2n)=2−k if n∈[2k,2k+1). On the other hand, if we fix k then
[TABLE]
because it is the union of 2k many consecutive elements of S1 and min(F)=22k, therefore φ2(A∖2n0)≥∑{2n−k2−n:n∈[2k,2k+1)}=1.
We know that Iα is representable (via an unconditional basic sequence) in Xα and so in C(Sα) where Sα is compact and of Cantor-Bendixson rank ωα+1.
Problem 8.5**.**
Is it true that Iα cannot be represented (at least via unconditional basic sequences) in any C(K) where K is countable, compact, and of Cantor-Bendixson rank ≤ωα?
9. Schur Property
Combining Theorem 5.4 and Theorem 6.3 we can characterize both those spaces XF which does not contain a copy of c0 and those which does not contain a copy of ℓ1. Both of the
characterizations are topological but of different nature: If F is a family of finite sets covering ω, then (1) XF does not contain a copy of c0 iff XF is Fσ in Rω; and (2) XF does not contain a copy of ℓ1 iff F is precompact.
Since XF do contain a copy of c0 if F is compact, it follows that every XF contains either c0 or ℓ1. Therefore, the spaces XF are in a sense alloys of c0 and ℓ1.
We already presented several examples of XF and F-ideals for F being compact. In this section we want to study the families which are far from being compact and which induce Banach spaces from the other extreme.
First, we will show that a family motivated by Farah’s example induces an ℓ1-saturated Banach space which is not isomorphic to ℓ1.
Example 9.1**.**
Let F be the family of those finite sets F⊆ω∖{0} for which ∣F∩[2n,2n+1)∣≤n−12n for every n>0. Then IF,λ is a variant of Farah example (see Example 2.2).
Concerning the generated Banach spaces it is not hard to see that XF=FIN(ΦF) and so XF does not contain a copy of c0. In fact, we can prove that XF has a slightly stronger property. Recall that a Banach space X satisfies the Schur property if every weakly null sequence in X converges to [math] in norm. The canonical space with Schur property is ℓ1 and Rosenthal ℓ1 theorem (see Theorem 3.2) implies that every space with Schur property is ℓ1-saturated.
We will show that XF enjoys the Schur property. Indeed, assume that (xn) is such that Φ(xn)>ε for some ε>0 and each n. We are going to show that (xn) is not weakly null. Since finitely dimensional
spaces enjoy Schur property, without loss of generality we may assume that (xn) is a block sequence, i.e. there is a sequence of finite sets (Gn) such that supp(xn)⊆Gn and Gn<Gn+1 for every n. Moreover, passing to a subsequence if needed, we may assume that ∣{k:[2n,2n+1)∩Gk=∅}∣≤1 for every n. Let An=supp(xk)∩[2n,2n+1) if [2n,2n+1)∩Gk=∅. If there is no such k, let An=∅. We may assume that ∣An∣<n−12n for every n (by shrinking supports of xk’s if needed, without decreasing their norms).
Now, notice that ⋃n∈ωAn is infinite and ⋃n∈ωAn∈F. This means that the function f:XF→R defined by f(x)=∑n∈ωχA(n)x(n) is a functional on XF of norm 1 such that f(xn)>ε for every n, and so (xn) is not weakly null.
On the other hand XF is not isomorphic to ℓ1. Otherwise, {en:n∈ω} would be equivalent to the standard base of ℓ1 (as all bounded unconditional bases of ℓ1 are equivalent, see [LPc68]). But there is no c>0 such that ΦF(∑n<Nen)>c⋅N=∥∑n<Nen∥1 for every N∈ω.
There are many examples having the Schur property which are not isomorphic to ℓ1. It would be more interesting to find a space which is ℓ1-saturated and which does not have the Schur property. There are many examples of them, too (the
first one was given by Bourgan, [Bou], see also the construction of Azimi and Hagler, [AH86]), but most of these examples are quite technical. However, it is not clear if there is a space XF which does not have a
copy of c0 and does not posses the Schur property.
Problem 9.2**.**
Is there a family F⊆[ω]<ω such that XF does not have the Schur property and which does not have a copy of c0?
The following remarks (motivated by the example above) may indicate how to look for such a family (or how to prove that they do not exist).
Proposition 9.3**.**
Let F be a hereditary family of finite sets covering ω. Assume that (xn) is a sequence in XF with pairwise disjoint supports and that there is A∈F such that limsupn→∞ΦF(PA(xn))>0. Then (xn) is not weakly null.
Proof.
Let (xn) and A⊆ω be as above. Passing to a subsequence, we can assume that there is an ε>0 such that ΦF(PA(xn))>ε for every n.
Let Fn=supp(xn), Fn+={k∈Fn:xn(k)>0}, and Fn−=Fn∖Fn+. Notice that either ΦF(PFn+(xn))>ε/2 or ΦF(PFn−(xn))>ε/2. Hence, shrinking A again if needed, we may assume that ∣∑k∈Fn∩Axn(k)∣>ε/2 for every n.
Now, let x∗:XF→R, x∗(x)=∑n∈Ax(n). Since A∈F, x∗∈XF∗ (in fact, x∗ is of norm 1) but ∣x∗(xn)∣=∣∑k∈Fn∩Axn(k)∣>ε/2 for every n, and hence (xn) is not weakly null.
∎
Corollary 9.4**.**
Let F be as above. If for every sequence (xn) with pairwise disjoint supports which is not (strongly) null, there is A∈F such that limsupn→∞ΦF(PA(xn))>0, then XF has the Schur property.
Remark 9.5**.**
If a sequence (xn) in XF has A∈F as above, then it has a subsequence which is equivalent to the standard basis of ℓ1.
Problem 9.6**.**
Let F be as above. Assume that there is a normalized sequence (xn) in XF such that for each limn→∞ΦF(PA(xn))=0 for each A∈F. Does it mean that XF contains a copy of c0? Or, at least, that it is not ℓ1-saturated?
Notice that if the answer to the above is positive, then, because of Corollary 9.4, Problem 9.2 has the negative answer.
Problem 9.7**.**
Assume F⊆[ω]<ω is a family such that XF does not have a copy of c0. Is XFℓ1-saturated?
This problem is natural in the light Theorem 6.3: We know that XF does not have a copy of ℓ1 if and only if it is c0-saturated. The above problem asks if we have a similar
equivalence for spaces which do not have a copy of c0. It seems likely that the answer is positive.
Bibliography34
The reference list from the paper itself. Each links out to its DOI / PubMed record.
1[AA 92] Dale E. Alspach and Spiros Argyros. Complexity of weakly null sequences. Dissertationes Math. (Rozprawy Mat.) , 321:44, 1992.
2[AD 92] Spiros A. Argyros and Irene Deliyanni. Banach spaces of the type of tsirelson. preprint , 1992.
3[AH 86] Parviz Azimi and James N. Hagler. Examples of hereditarily l 1 superscript 𝑙 1 l^{1} Banach spaces failing the Schur property. Pacific J. Math. , 122(2):287–297, 1986.
4[AT 05] Spiros A. Argyros and Stevo Todorcevic. Ramsey methods in analysis . Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2005.
5[BNFP 15] Piotr Borodulin-Nadzieja, Barnabás Farkas, and Grzegorz Plebanek. Representations of ideals in Polish groups and in Banach spaces. J. Symb. Log. , 80(4):1268–1289, 2015.
6[Bou] Jean Bourgain. ℓ 1 superscript ℓ 1 \ell^{1} subspaces of Banach spaces. Lecture Notes. Free University of Brussels.
7[BP 58] Czesław Bessaga and Aleksander Pełczyński. On bases and unconditional convergence of series in Banach spaces. Studia Math. , 17:151–164, 1958.
8[Cas 93] Jesús M. F. Castillo. A variation on Schreier’s space. Riv. Mat. Univ. Parma (5) , 2:319–324 (1994), 1993.