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Advances in Mathematics
www.elsevier.com/locate/aim
On the permutative equivalence of squares of unconditional bases
✩Fernando Albiaca,∗,José L. Ansorenab
aDepartmentofMathematics,StatisticsandComputerSciences,andInaMat2, UniversidadPúblicadeNavarra,Pamplona31006,Spain
bDepartmentofMathematicsandComputerSciences,UniversidaddeLaRioja, Logroño26004, Spain
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received9July2020
Receivedinrevisedform13June 2022
Accepted5September2022 Availableonline26September2022 CommunicatedbyDanVoiculescu
MSC:
46B15 46B20 46B42 46B45 46A16 46A35 46A40 46A45
Keywords:
Uniquenessofunconditionalbasis
Weprove thatif thesquaresof twounconditionalbases are equivalentuptoapermutation,thenthebasesthemselvesare permutativelyequivalent. Thissettlesatwenty-fiveyear-old question raisedbyCasazza andKalton in[13]. Solvingthis problemprovidesanewparadigmtostudytheuniquenessof unconditionalbasisinthegeneralframeworkofquasi-Banach spaces.Multipleexamplesaregiventoillustratehowtoputin practicethistheoreticalscheme.Amongthemainapplications of thisprinciple we obtain the uniqueness of unconditional basis up to permutation of finite sums of spaces with this property,aswellasthefirstadditiontothescantlistofthe knownBanachspaceswithauniqueunconditionalbasesup topermutationsince[14].
©2022TheAuthor(s).PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).
✩ Both authors supported by the Spanish Ministry for Science, Innovation, and Universities, Grant PGC2018-095366-B-I00forAnálisisVectorial,MultilinealyApproximación.Thefirst-namedauthoralso acknowledgesthesupportfromSpanishMinistryforScienceandInnovation,GrantPID2019-107701GB-I00 forOperators,lattices,andstructureofBanachspaces.
* Correspondingauthor.
E-mailaddresses:[email protected](F. Albiac),[email protected] (J.L. Ansorena).
https://doi.org/10.1016/j.aim.2022.108695
0001-8708/©2022TheAuthor(s). PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Quasi-Banachspace Banachlattice
1. Introductionandbackground
Animportantlong-standingprobleminBanachspacetheory,eventuallysolvedinthe negative byGowersand Maureyin1997[18], askedwhetheranytwo BanachspacesX and Y suchthatX isisomorphictoacomplementedsubspaceofY andsuchthatY is isomorphic to acomplementedsubspaceof X areisomorphic (X Y , forshort). This is known, by analogy with asimilar result for cardinals inthe category of sets, as the Schröder-Bernstein problem forBanachspaces.
Pełczyński had noticed much earlier, back in 1969, that a little extra information about each space, namely being isomorphic to their squares, is all that is needed for theSchröder-BernsteinproblemforBanachspacestohaveapositiveoutcome[32].This observation,nowadaysknownasPełczyński’sdecompositionmethod,highlightedtherole played by the squares of the spaces, and the question arose whether any two Banach spacesX andY suchthatX2 Y2areisomorphic.Thisproblemwasalsosettledinthe aforementioned article byGowers and Maurey.Indeed, theauthors constructed in[18]
a BanachspaceX with X X3 butX X2. Then,if weputY = X2, we havethat X is isomorphicto acomplementedsubspaceofY ,thatY is isomorphicto asubspace of X, that X2 Y2, and that X Y . So, the pair of spaces X and Y serves as a counterexampleforbothquestions.
TheSchröder-Bernsteinproblem forBanachspacesisaverybasicandnaturalprop- erty that arises most of the time when one is trying to show that two Banach (or quasi-Banach) spaces are isomorphic. However, its practical implementation depends onknowingapriorilargeclassesofspaceswherethepropertyholds.Andthismightbe anintractable probleminalmostanygeneralsetting.
Wójtowicz [36] and Wojtaszczyk [35] discovered independently, with a lapse of 11 years,thefollowingbeautifulcriterioninthespiritoftheSchröder-Bernsteinproblemto check whethertwo unconditional bases (in possiblydifferent quasi-Banach spaces) are permutatively equivalent.
Theorem1.1(see[35,Proposition2.11] and[36,Corollary1]). Let(xn)∞n=1and(yn)∞n=1 be two unconditional basesof quasi-Banach spacesX and Y ,respectively. Supposethat (xn)∞n=1ispermutativelyequivalent toasubbasisof(yn)∞n=1andthat (yn)∞n=1ispermu- tativelyequivalenttoasubbasisof(xn)∞n=1.Then(xn)∞n=1and(yn)∞n=1arepermutatively equivalent. Inparticular, X Y .
ThevalidityoftheSchröder-Bernsteinprincipleforunconditionalbases hasaplayed acrucial role inthedevelopmentof thesubjectof uniquenessofunconditional basisin quasi-Banach spaces(see,e.g.,[5–9]).Casazzaand Kaltonbroughtthisprincipleto the reader’s awareness in [13] and used it to give new examples of Banach spaces with a
uniqueunconditional basis upto permutation. Thesimplifyingpower of theSchröder- Bernsteinprincipleforunconditionalbaseswouldhavemadelifemucheasieralsoforall theauthors whohad previouslyworked onthe problemof uniquenessof unconditional basisuptopermutationandwho,inordertoobtainthesameconclusions,hadtoimpose additional properties to the bases in relation to other general techniques such as the decomposition method (see e.g. [10, Proposition 7.7]). It is indeed remarkable that, althoughthe combinatorial arguments used byWojtaszczyk to prove Theorem 1.1 are somewhatstandard,theywentunnoticed untilclosetothe21stcentury!
The state of art of the Schröder-Bernstein problem for Banach spaces in the pre- Gowers era was described by Casazza in [12]. His paper with Kalton [13] appeared just oneyear after Gowersand Maurey disproved the Schröder-Bernstein problem for BanachspacesandWojtaszczyk’sreinterpretedtheSchröder-Bernsteinprincipleforun- conditionalbases.Thus,itisnotsurprisingthatthefollowingquestionwastimelyraised in[13]:
Question 1.2. (See [13, Remarks following the proof of Theorem 5.7].) Suppose that (xn)∞n=1and(yn)∞n=1aretwounconditionalbaseswhosesquaresarepermutativelyequiv- alent.Doesitfollowthat(xn)∞n=1and(yn)∞n=1 arepermutativelyequivalent?
This problem was a driving force for the present investigation and we solve it in theaffirmative. Infactwe show thattheresult stillholds replacing theassumption on thesquare of thebases withthe weakerassumption thatsomepowersof thebases are permutativelyequivalent. Wewill dothatinSection2.
Answering Question 1.2 in the positive offers a new paradigm to tackle the prob- lem of uniqueness of unconditional basis up to permutation in the general setting of quasi-Banach spaces. The necessary ingredients and preparatory results leadingto the main theoreticaltool,namelyTheorem 3.9, arepresented inaself-contained fashionin Section3.
In Sections 4 and 5 we embark on acomprehensive survey of quasi-Banach spaces withaunique unconditionalbasisup topermutation towhich thescheme ofSection3 canbe applied.
InSection6wefurtherexploittheusefulnessofTheorem3.9toshowthattheunique- nessofunconditionalbasisispreservedwhenwetakefinitedirectsumsofawideclassof quasi-Banachspaceswiththisproperty.WhencombinedwiththespacesfromSections4 and 5weobtain amyriadof new examplesof spaceswith uniquenessof unconditional basisuptopermutation.Amongthem,wefindlocallyconvexquasi-Banachspaces,i.e., Banachspaces.Asfar aswe areaware,theseexamplesare thefirstcontributionto the theoryofuniquenessofunconditionalbasisofBanachspacessince[14].
Weusestandardterminologyand notationinBanachspacetheory ascanbe found, e.g., in [4]. Most of our results, however, will be established in the general setting of quasi-Banach spaces; the unfamiliar reader will find general information about quasi- Banach spaces in [24]. We next gather the notation that it is more heavily used. In
keepingwithcurrentusagewewillwritec00(J ) forthesetofall(aj)j∈J∈ FJ suchthat
|{j ∈ J : aj = 0}|<∞,where F couldbe therealor complexscalarfield.Givens∈ N weputN[s]={1,. . . ,s}.Givenaquasi-BanachspaceX ands∈ N wedenotebyκ[s,X]
thesmallestconstantC suchthatforallvectors(fj)sj=1∈ X wehave
s j=1
fj
≤ C
⎛
⎝s
j=1
fj
⎞
⎠ .
Note thatκ[2,X] isthesocalled modulusof concavityof thespace X (see[24]).IfX is ap-Banachspace,0< p≤ 1,thenκ[s,X]≤ s1/p−1.
TheclosedlinearspanofasubsetV ofX willbedenotedby[V ].Acountablefamily B = (xn)n∈N inX isanunconditionalbasicsequence ifforeveryf ∈ [xn: n∈ N ] thereis auniquefamily(an)n∈N inF suchthattheseries
n∈Nanxnconvergesunconditionally to f .IfB isanunconditionalbasicsequence,there isaconstantK≥ 1 suchthat
n∈N
anxn
≤K
n∈N
bnxn
for any finitely non-zero sequence of scalars (an)n∈N with |an| ≤ |bn| for all n ∈ N (see [2, Theorem 1.10]). If this inequality is satisfied for a given K we say that B is K-unconditional. If we additionallyhave [xn: n ∈ N ]= X then B is anunconditional basis of X.IfB isanunconditionalbasisofX,thenthemap
F : X → FN, f =
n∈N
anxn→ (x∗n(f ))n∈N = (an)n∈N
will be called thecoefficient transform withrespect to B, andthefunctionals (x∗n)n∈N
thecoordinatefunctionals ofB.
GivenacountablesetN ,wewriteEN := (en)n∈N forthecanonicalunitvectorsystem of FN, i.e., en = (δn,m)m∈N for eachn ∈ N , where δn,m = 1 if n = m and δn,m = 0 otherwise. A sequence space willbe aquasi-Banach space X ⊆ FN for which EN is a normalized1-unconditionalbasis.
TheBanachenvelope ofaquasi-BanachspaceX consistsofaBanachspace xX together with a linear contraction JX: X → X satisfyingx the following universal property:for every BanachspaceY andeverylinearcontraction T : X → Y thereis aunique linear contraction pT :xX → Y such that pT ◦ JX = T . We say thata Banach space Y is the BanachenvelopeofX viaJ : X→ Y iftheassociatedmap pJ :xX→ Y isanisomorphism.
Other morespecific terminologywillbeintroducedincontextwhenneeded.
2. Permutativeequivalenceofpowersofunconditionalbases
Suppose thatBx = (xn)n∈N and Bu = (un)n∈N are (countable)families ofvectors inquasi-BanachspacesX andY ,respectively.WesaythatBx= (xn)n∈N C-dominates
Bu= (un)n∈N ifthereisalinearmapT fromtheclosedsubspaceofX spannedbyBxinto Y withT (xn)= un foralln∈ N suchthatT ≤ C.IfT isanisomorphicembedding, BxandBuaresaidtobeequivalent.WesaythatBxispermutativelyequivalent toafamily By= (ym)n∈MinY ,andwewriteBx∼ By,ifthereisabijectionπ :N → M suchthat Bx and(yπ(n))n∈N are equivalent. Asubbasis of anunconditionalbasis Bx= (xn)n∈N
isafamily(xn)n∈M forsomesubsetM ofN .
Let (Xi)i∈F be a finite collection of (possibly repeated) quasi-Banach spaces. The Cartesianproduct
i∈FXi equippedwiththequasi-norm
(xi)i∈F = sup
i∈Fxi, xi∈ Xi
isa quasi-Banachspace. Suppose thatBi = (xi,n)n∈Ni is anunconditional basis ofXi
foreachi∈ F .Set
N =
i∈F
{i} × Ni. (2.1)
Then the countable sequence
i∈FBi := (xi,n)(i,n)∈N given by xi,n = (xi,n,j)j∈F, where
xi,n,j =
xi,n if i = j,
0 otherwise, isanunconditionalbasisof
i∈FXi.IfF = N[s] andXi = X foralli∈ F ,theresulting directsumiscalledthes-foldproduct ofX andwesimplywriteXs.Similarly,ifBi =B for all i ∈ F = N[s], we put Bs and say that Bs is the s-fold product of B. We will refer to the 2-fold product of a basis as to thesquare of that basis. We start with an elementarylemma.
Lemma2.1.LetB = (xn)n∈N beanunconditionalbasisofaquasi-BanachspaceX.For agivens∈ N,considerthes-foldproductBs= (xi,n)(i,n)∈N[s]×N.Thenforanyfunction α : N → N[s],the basic sequence (xα(n),n)n∈N (which is permutatively equivalent to a subbasisof Bs) isequivalent toB.
Proof. SupposethatB isK-unconditional.IfweputNi= α−1(i) fori∈ N[s] then
n∈N
anxα(n),n
= sup
i∈N[s]
n∈Ni
anxn
,
forall(an)∞n=1∈ c00.Hence, 1
κ[s, X]
n∈N
anxn
≤
n∈N
anxα(n),n
≤K
n∈N
anxn
.
ThefollowingversionoftheHall-KönigLemma(alsoknownasMarriageLemma)for infinitefamiliesoffinitesets isessentialintheproofofTheorem 2.4.
Theorem 2.2 (see [19, Theorem 1]).Let N be a set and (Ni)i∈I be a family of finite subsets of N .Supposethat
|F | ≤
i∈F
Ni
for every F ⊆ I finite. Then there is a one-to-one map φ : I → N with φ(i) ∈ Ni for every i∈ I.
Theorem 2.3. LetBx andBy be twounconditional bases of quasi-BanachspacesX and Y ,respectively.Supposethat Bsxispermutativelyequivalent toasubbasisofBsy forsome s≥ 2.ThenBx ispermutativelyequivalent toasubbasisof By.
Proof. Put Bx = (xn)n∈N, By = (yn)n∈M, Bsx = (xi,n)(i,n)∈N[s]×N and Bsy = (yi,n)(i,n)∈N[s]×M.Byhypothesisthereisaone-to-onemap
π = (π1, π2) : N[s]× N → N[s] × M
suchthattheunconditionalbasesBxsand(yπ(i,n))(i,n)∈N[s]×N areequivalent.Forn∈ N set Mn={π2(i,n) : i∈ N[s]}.IfF isafinite subsetofN wehave
π(N[s]× F ) ⊆ N[s] ×
n∈F
Mn,
and sinceπ is one-to-one,
s|F | ≤ s
n∈F
Mn
.
Hence, |F | ≤ |∪n∈F Mn|. We also have |Mn| ≤ s for all n ∈ N .Therefore, by The- orem 2.2, there exist aone-to-one map φ :N → M, amap α : N → N[s], and amap β : M→ N[s] suchthat
π(α(n), n) = (β(n), φ(n)), n∈ N ,
from where it follows that the unconditional basic sequences Bx = (xα(n),n)n∈N and By = (yβ(n),φ(n))n∈N are equivalent. Since, on the other hand, by Lemma 2.1, Bx is equivalent to B and By is permutatively equivalent to (ym)m∈M, where M = φ(N ), we aredone.
Theorem2.4. LetBx and By be twounconditional bases of quasi-Banachspaces X and Y .Supposethat Bsx∼ Bsy forsome s≥ 2.ThenBx∼ By.
Proof. Applying Theorem 2.3 yields thatBx is permutatively equivalent to asubbasis of By, and switching the roles of the basis also the other way around. An appeal to Theorem1.1finishestheproof.
Corollary2.5.LetB be anunconditional basisofaquasi-Banachspace. SupposethatBt ispermutativelyequivalent toasubbasisof Bs forsome t> s≥ 1. ThenB2∼ B.
Proof. Sincet≥ s+1,Bs+1ispermutativelyequivalenttoasubbasisofBs.Byinduction we deduce thatBu+1 is permutatively equivalent to asubbasis of Bu forevery u≥ s, and so by transitivity, Bu is permutatively equivalent to a subbasis of Bs for every u≥ s.Inparticular, B2s ispermutatively equivalentto asubbasis ofBs.Therefore, by Theorem2.3,B2ispermutativelyequivalenttoasubbasisofB.SinceB ispermutatively equivalenttoasubbasisofB2, applyingTheorem1.1we aredone.
3. Anewtheoretical approachtotheuniquenessofunconditionalbasisin quasi-Banachspaces
From astructural point of view,it is usefulto know if agiven space hasan uncon- ditional basisand, ifthe answer is yes, whetherthis is theunique unconditional basis ofthespace. Recallthataquasi-BanachspaceX with anunconditionalbasisB issaid tohaveauniqueunconditionalbasis,ifeverysemi-normalizedunconditionalbasisofX isequivalent to B. Forconvenience, from nowon allbases willbe assumed to besemi- normalized.Notethat,ifB = (xn)n∈N isasemi-normalizedunconditionalbasisthenit isequivalenttothenormalizedbasis(xn/xn)n∈N.
For a Banach space with a symmetric basis it is rather unusual to have a unique unconditionalbasis. Itiswell-known that2 hasauniqueunconditional basis[26], and aclassical resultof Lindenstrauss and Pełczyński [28] assertsthat1 and c0 also have aunique unconditional basis. Lindenstrauss and Zippin [29] completed the picture by showing that those three are theonly Banach spaces in which allunconditional bases areequivalent.
OncewehavedeterminedthataBanachspacedoesnothaveasymmetricbasis(atask thatcanbefarfromtrivial)wemustrethinktheproblemofuniquenessofunconditional basis.Infact,anunconditionalnon-symmetricbasisadmitsacontinuumofnonequivalent permutations(cf.[20,Theorem2.1]).HenceforBanachspaceswithoutsymmetricbases itismorenaturalto considerinsteadthequestionofuniquenessofunconditionalbases upto(equivalenceand)apermutation,(UTAP)forshort.WesaythatX hasa(UTAP) unconditional basis B if everyunconditional basis in X is permutatively equivalent to B. Thefirstmoversinthis directionwereEdelsteinand Wojtaszczyk,whoprovedthat finitedirectsumsofc0,1and2havea(UTAP)unconditionalbasis[16].Bourgainetal.
embarkedonacomprehensivestudyaimedatclassifyingthoseBanachspaceswithunique unconditional basis up to permutation, which culminated in 1985 with their Memoir [10].Theyshowed thatthe spacesc0(1),c0(2), 1(c0),1(2) andtheir complemented subspaceswith unconditionalbasisallhavea(UTAP)unconditional basis,while2(1) and 2(c0) do not. However, the hopes of attaining a satisfactory classification were shattered when they found a nonclassical Banach space, namely the 2-convexification T(2) of Tsirelson’s space having a (UTAP) unconditional basis. Their work also left manyopen questions,mostofwhichremainunsolvedasoftoday.
Inturn,inthecontextofquasi-BanachspacesthatarenotBanachspaces,theunique- nessofunconditionalbasisseemstobethenormratherthananexception.Forinstance, it was shownin[21] that awide class of nonlocallyconvex Orlicz sequence spaces, in- cludingthep spacesfor0< p< 1,haveauniqueunconditional basis.Thesameistrue in manynonlocallyconvex Lorentz sequence spaces([6,23])and (UTAP) inthe Hardy spacesHp(T ) for0< p< 1 ([35]).
This section is gearedtowards Theorem 3.9, which tells us that, underthree man- ageable conditions regarding a space and a basis, the unconditional bases of a space areallpermutativelyequivalent. Thetechniques usedintheproofofthistheoremarea developmentofthemethodsintroducedbyCasazzaandKaltonin[13,14] toinvestigate theproblem ofuniquenessofunconditionalbasisinaclassofBanachlatticesthatthey called anti-Euclidean. Thesubtle butcrucial role playedby thelatticestructure of the space inthe proof ofTheorem 3.9 hasto be seeninthatit will permitto simplify the untangled way in which thevectors of one basiscan be writtenin termsof the other.
These techniques have been extended to the nonlocally convex setting and efficiently used intheliteratureto establishtheuniquenessofunconditionalbasisuptopermuta- tionofthespacesp(q) forp∈ (0,1]∪ {∞} andq∈ (0,1]∪ {2,∞} (see[5–9]),withthe convention that∞ heremeansc0.
Beforemovingon,recallthatthesupport ofavectorf ∈ X relativetoanunconditional basisB = (xn)n∈N withcoordinatefunctionals(x∗n)n∈N ∈ X∗is theset
supp(f ) ={n ∈ N : x∗n(f )= 0}, while thesupportofafunctionalf∗∈ X∗ istheset
supp(f∗) ={n ∈ N : f∗(xn)= 0}.
Although the dualof aquasi-Banach space X couldbe trivial,the existenceof abasis forX guaranteestheexistenceofarichdualspace.Thiswillallowustousethesupport of vectors andfunctionals on X inadecisiveway. To thataim weneed to introducea few moredefinitions.
Anunconditional basicsequence Bu = (um)m∈M inaquasi-Banachspace X issaid to be complemented if its closed linear span U = [Bu] is a complemented subspaceof X, i.e., there is a bounded linear map P : X → U with P|U = IdU. Notice that the
unconditionalbasicsequenceBu= (um)m∈MiscomplementedinX ifandonlyifthere exists asequence (u∗m)m∈M inX∗ such thatu∗m(un)= δm,n for all (m,n) ∈ M2 and thereisaboundedlinearmapPu: X→ X givenby
Pu(f ) =
m∈M
u∗m(f ) um, f ∈ X. (3.1)
Wewillreferto(u∗m)m∈M asasequenceofprojecting functionals forBu.AfamilyBu= (um)m∈M in X with mutually disjoint supportswith respect to agiven unconditional basisB isanunconditionalbasicsequence.Inthecasewhen,moreover,supp(um) isfinite for everym ∈ M we say thatBu is a block basic sequence (with respectto B). Notice that, usually, by a block basic sequence of abasis B = (xn)∞n=1 we mean a sequence (um)∞m=1 ofnonzerovectorswith
max(supp(um)) < min(supp(um+1)), m∈ N.
However,whendealingwithanunconditionalbasisB = (xn)n∈N indexedbyacountable setN where theorderisunimportant,itismoreconvenientto useouralternative(yet valid)definitionofblockbasicsequence.
We saythat ablock basic sequence Bu is well complemented (with respect to B) if we canchoose asequence of projecting functionals Bu∗ = (u∗m)m∈M with supp(u∗m) ⊆ supp(um) for all m ∈ M. In this case, B∗u is said to be a sequence of good projecting functionals forBu.
Thefollowing definition identifiesand gives relevanceto an unstatedfeatureshared by some unconditional bases. Examples of such bases can be found, e.g., in [6,13,21], where the property naturally arises in connection with the problem of uniqueness of unconditionalbasis.
Definition3.1.AnunconditionalbasisB = (xn)n∈N ofaquasi-Banachspacewillbesaid tobeuniversalforwellcomplementedblock basic sequences ifforeverysemi-normalized wellcomplementedblockbasicsequenceBu= (um)m∈MofB thereisamapπ :M→ N suchthat π(m)∈ supp(um) for every m∈ M, and Bu is equivalent to the rearranged subbasis(xπ(m))m∈M ofB.
Theideasinthefollowingdefinitionandpropositionareimplicitin[21].
Definition3.2. Anunconditional basisB = (xn)n∈N ofaquasi-Banach spaceX willbe saidtohavethepeakingproperty ifeverysemi-normalizedwellcomplementedblockbasic sequenceBu= (um)m∈M withrespecttoB satisfies
minf∈Msup
n∈N|u∗m(xn)| |x∗n(um)| > 0 (3.2) forsomesequence (u∗m)m∈M ofgoodprojectingfunctionalsforBu.
Proposition3.3.SupposeB = (xn)n∈N isanunconditionalbasisofaquasi-Banachspace X. If B hasthe peakingproperty then it isuniversal forwell complemented block basic sequences.
Proof. Let Bu = (um)m∈M be a semi-normalized well complemented block basic se- quence and Bu∗ = (u∗m)m∈M be a sequence of good projecting functionals for Bu such that(3.2) holds.There isπ :M→ N one-to-onewith
m∈Minf |x∗π(m)(um)| |x∗π(m)(um)| > 0.
Form∈ M letusput
λm= x∗π(m)(um), μm= xπ(m)(u∗m), and set
vm= λmxπ(m), v∗m= μmx∗π(m).
By [1, Lemma 3.1], Bv = (vm)m∈M is equivalent to Bu. In particular, Bv is semi- normalized so thatinfmλm > 0 andsupmλm<∞. It follows thatBv is equivalent to (xπ(m))m∈M.
Thelastingredientinthedeconstructionprocesswearecarryingoutisthefollowing featureaboutthelatticestructureofaquasi-Banachspace.
Definition 3.4.A quasi-Banachspace(respectively,aquasi-Banachlattice)X issaidto be sufficiently Euclidean if2 iscrudely finitely representable inX as acomplemented subspace (respectively, complemented sublattice), i.e., there is a positive constant C suchthatforeveryn∈ N thereareboundedlinearmaps(respectively,latticehomomor- phisms) In: n2 → X andPn: X → n2 withPn◦ In = Idn2 andInPn≤ C.Wesay thatX isanti-Euclidean (resp.latticeanti-Euclidean)ifitisnotsufficientlyEuclidean.
Any (semi-normalized) unconditional basis ofa quasi-Banach space X is equivalent to the unitvector systemof asequence space and so it induces alattice structure on X. In general,we will say thatan unconditional basishas aproperty about lattices if its associatedsequence spacehasit. Andtheother way around,i.e., wewill saythata sequencespaceenjoysacertainpropertyrelevanttobasesifitsunitvectorsystemdoes.
A quasi-Banach latticeX is saidto be L-convex (or lattice-convex)ifthere isε> 0 so that whenever f and (fi)ki=1 in X satisfy 0 ≤ fi ≤ f for everyi = 1, . . . , k, and (1− ε)kf≤k
i=1fi we haveεf ≤ max1≤i≤kfi. Kalton [22] showed that a quasi- Banachlattice isL-convex ifand onlyifit isp-convexfor somep> 0. So,most quasi- Banachlattices(andunconditionalbases)occurring naturallyinanalysisareL-convex.
The space 1 is the simplest and most important example of anti-Euclidean space (seee.g.[1,CommentsprevioustoRemark2.9]).So,itishelpfultobeabletocounton conditionsthatguaranteethattheBanachenvelopeofagivenquasi-Banachspaceis1. Lemma3.5 (see [1,Proposition 2.10]).SupposeX is aquasi-Banachspacewith an un- conditionalbasisB thatdominatestheunitvectorbasisof1.ThentheBanachenvelope ofX is 1 viathecoefficient transform.
Thefollowing lemmais usefulwhendealingwith unconditionalbases thatdominate thecanonicalbasisof1.
GivenanunconditionalbasisB = (xn)n∈N withcoordinatefunctionals(x∗n)n∈N and A⊆ N finitewewillput
1A[B] =
n∈A
xn and 1∗A[B] =
n∈A
x∗n.
IfB isclearfrom contextwesimplywrite 1A=1A[B] and 1∗A=1∗A[B].
Lemma3.6(cf. [5,Lemma4.1]). LetB = (xn)n∈N beanunconditionalbasisofaquasi- BanachspaceX. Supposethat B dominatesthecanonicalbasisof 1.Thenevery semi- normalizedwell complementedblock basic sequence of X withrespect toB isequivalent toa well complementedblock basic sequence (um)m∈M for which(1∗supp(u
m))m∈M is a sequenceofgood projecting functionals.
Proof. LetC1be suchthat
n∈N|x∗n(f )|≤ C1f forallf ∈ X.Set C2= sup
m∈Mum, C3= sup
m∈Mu∗m, and C4= sup
n∈Nxn.
Fixm∈ M andput
Am=
n∈ N : |u∗m(xn)| > 1 2C1C2
.
Wehave
n∈N \Am
|x∗n(um) u∗m(xn)| ≤ 1 2C1C2
n∈N \Am
|x∗n(um)| ≤ 1 2. Hence,
λm: =
n∈Am
|x∗n(um) u∗m(xn)|
≥ −1 2+
n∈N
|x∗n(um) u∗m(xn)|
≥ −1
2+ u∗m(um) =1 2. Let
vm= λ−1m
n∈Am
|x∗n(um) u∗m(xn)| xn
and v∗m=1∗Am.Foreveryn∈ N wehave
v∗m(vm) = 1, λ−1m|u∗m(xn)| ≤ 2C3C4, and foreveryn∈ Am,
1≤ 2C1C2|u∗m(xn)|.
Hence,theresultfollowsfrom [1,Lemma3.1].
Wewillusethefullforceofthelatticestructureinducedbythebasisinthefollowing reductionlemma.
Lemma 3.7. LetX be a quasi-Banach space whose Banach envelope is anti-Euclidean.
Suppose that B is an L-convex, unconditional basis of X which is universal for well complemented block basic sequences. Then, if Bu is another unconditional basis of X, there arepositiveintegers s andt suchthat Bu ispermutativelyequivalent toasubbasis of Bs andB is permutativelyequivalent toasubbasisof Btu.
Proof. SinceBuislatticeanti-Euclidean,[5,Theorem3.4] yieldsthatBuispermutatively equivalent to awell complemented blockbasic sequence of Bs for somes ∈ N. By[1, Proposition 3.4], Bs is universal for well complemented block basic sequences so that Bu ispermutativelyequivalenttoasubbasis ofBs.SinceBsinheritstheconvexityfrom B, thebasisBuis L-convexanduniversalforwellcomplementedblockbasicsequences.
SwitchingtherolesofB and Buyields theconclusionofthelemma.
Remark 3.8.A remark onthe inherited order structure inaquasi-Banach latticeis in order here.Kaltonshowedin[22,Theorem4.2] thateveryunconditionalbasicsequence B0 of aquasi-Banach space with an L-convex unconditional basis B is L-convex.This argumentwouldhave,indeed,simplifiedtheproofofLemma3.7.However,herewewant tomakethepointthatthevalidityofthelemmadoesnotdependonsuchadeeptheorem as Kalton’s.
Weareready toprovethemain resultofthissection.
Theorem 3.9.LetX be aquasi-BanachspacewhoseBanach envelopeisanti-Euclidean.
Suppose B isan unconditionalbasisforX suchthat:
(i) B isL-convex, i.e., thelatticestructureinduced byB in X isL-convex;
(ii) B isuniversal forwellcomplementedblock basic sequences;and (iii) B ∼ B2.
ThenX has auniqueunconditional basisuptopermutation.
Proof. Let Bu be another unconditional basis of X. Since Br ∼ B for every r ∈ N, applying Lemma3.7 yieldsthatBu ispermutatively equivalent to asubbasis of B and that Bt is permutatively equivalent to a subbasis of But for some t ∈ N. Combining Theorem2.3with Theorem1.1yieldsBu∼ B.
Theorem2.3becomesinstrumentalinreachingtheconclusionoftheprevioustheorem.
Indeed, withoutit, and under thesame hypotheses as inTheorem 3.9, we wouldhave only been able to guarantee that given another unconditional basis Bu of X, Bu is permutativelyequivalenttoasubbasisofB andthatB ispermutativelyequivalenttoa subbasis of somes-foldproduct of Bu. Thanks to Theorem 2.3we canclose the“gap”
betweenB andBuandarriveatthepermutativeequivalenceofthetwobases.Although thisgap mightseemsmall, wewouldlike toemphasizethatinthelackofTheorem 3.9 thespecialistswereforcedtouseadditionalpropertiesofB toinferthatB istheunique unconditional basis of X. For instance, in the proof that 1(p) for 0 < p < 1 has a uniqueunconditionalbasisuptopermutation,theauthorsusedthatallsubbasesofthe canonicalbasisof1(p) arepermutatively equivalenttotheirsquare(see [5]).
4. Applicabilityof ourschemetoanti-Euclidean spaces
Most anti-Euclidean spaces scattered through the literature with a unique uncon- ditional basis (up to permutation) fulfill the hypotheses of Theorem 3.9. This can be checkedbylookingupthecorrespondingreferencescontainedherein.Still,withtheaim tobe asself-containedas possibleand fortheconvenienceofthereaderwenextsurvey how to verify the hypotheses of Theorem 3.9 in all known spaces (Banach and non- Banach)with auniqueunconditional basisaswell asinothernewcases.Thespacesin this section and thenext will be the protagonists of Section6, where we will combine themtogettheuniquenessofunconditionalbasisuptopermutationoftheirfinitedirect sums.
Inwhatfollows, thesymbol αi βi fori∈ I meansthatthefamiliesofpositivereal numbers(αi)i∈I and (βi)i∈I verifysupi∈Iαi/βi <∞.If αi βi andβi αi for i∈ I wesay(αi)i∈I are(βi)i∈I areequivalent,andwewrite αi ≈ βi fori∈ I.
4.1. The space1
As we mentioned above, 1 is anti-Euclidean. Since its canonical basis is perfectly homogeneous (see, e.g., [4, Section 9.1]), it is universal for well complemented block
basicsequences. Finally,sincethecanonicalbasisof1is symmetric,itis equivalentto itssquare.
4.2. Orliczsequencespaces
AnOrlicz function willbe aright-continuousincreasingfunctionϕ : [0,∞)→ [0,∞) such ϕ(0) = 0, ϕ(1) = 1 and ϕ(s+ t) ≤ C(ϕ(s)+ ϕ(t)) for some constantC and all s, t ≥ 0. The Orlicz space ϕ is the space associated to the Luxembourg quasi-norm definedfromthemodular(an)∞n=1→∞
n=1ϕ(|an|).Ourassumptionsonϕ yieldthatϕ is asymmetricsequencespace. Kaltonprovedin[21] thatifϕ satisfies
t ϕ(t), 0 ≤ t ≤ 1, (4.1)
and
Λϕ:= lim
ε→0+ inf
0<s<1
−1 log ε
1
ε
ϕ(sx)
sx2 dx =∞, (4.2)
thenϕhasauniqueunconditionalbasisuptopermutation.Itiseasytoshowthat(4.1) impliesthattheBanachenvelopeofϕisanti-Euclidean,anditisimplicitin[21] thatif (4.1) and(4.2) hold,thentheunitvectorsystemofϕisuniversalforwellcomplemented blockbasicsequences.Forthesakeofcompletenessandfurtherreference,werecordthese resultsandsketchaproof ofthem.
Proposition4.1(cf.[21]).Letϕ beanOrliczfunctionsuchthatboth(4.1) and(4.2) hold.
Then:
(i) TheBanachenvelope of ϕ is1 via theinclusionmap.
(ii) The unitvector systemof ϕ has thepeakingproperty.
Proof. Since 1 is the Orlicz sequence spaceassociated to thefunction t→ t, we have
ϕ⊆ 1. Then,(i)followsfromLemma3.5.
Assume by contradiction that Bu = (um)m∈M is a well complemented block basic sequence of ϕ, that(u∗m)m∈M isafamilyof well complementedprojectingfunctionals forBu,butthat
minf∈Msup
n∈N|u∗m(en)| |e∗n(um)| = 0.
Then,by[21,Theorem6.5],ϕhasa complementedbasicsequenceBysuchthatY = [By] is locally convex. Using (i) and [1, Lemma 2.1], it follows that the restriction of the inclusion mapof ϕ in1 to Y isanisomorphism. Therefore,by [21,Theorem 5.3],we reachtheabsurditythatΛϕ<∞.
4.3. Lorentz sequencespaces
Letw = (wn)∞n=1 beaweight,i.e.,asequenceofpositive scalars,and let0< p<∞.
Supposethatw decreases tozero.TheLorentz space d(w,p) isthequasi-Banach space consistingofallf = (an)∞n=1∈ FN suchthat
fd(w,p)= sup
π∈Π
∞
n=1
|aπ(n)|pwn
1/p
<∞,
whereΠ isthesetofallpermutationsofN.Theunitvectorsystemisasymmetricbasis ofd(w,p).Itwasprovedin[6] thatiftheweightfulfills the condition
k∈Ninf
k n=1
wn
kp > 0, (4.3)
thend(w,p) has auniqueunconditional basisuptopermutation.Next,we deducethis resultbycombining Theorem3.9withargumentsfrom [6].
Proposition 4.2(cf. [6]).Let 0< p< 1 andlet w = (wn)∞n=1 be aweight decreasing to zero.Thend(w,p)⊆ 1 if andonlyif (4.3) holds.Moreover, if(4.3) holds,then
(i) the Banachenvelope ofd(w,p) is 1 viatheinclusionmap, and (ii) the unitvectorsystemof d(w,p) has thepeakingproperty.
Proof. Fork∈ N writesk=k
n=1wn.Assumethatd(w,p)⊆ 1andletC bethenorm oftheinclusionmap.If|A|= k wehave
1A1= k, and 1Aw,p= s1/pk . Thusk≤ Cs1/pk foreveryk∈ N.
Wewillusetheweak-Lorentzspaced∞(u,p) associatedtoaweightu = (un)∞n=1and 0< p<∞, whichconsists ofallsequences f ∈ c0 whosenon-increasing rearrangement (a∗k)∞k=1satisfies
fd∞(u,p)= sup
k
k
n=1
un
1/p
a∗k <∞.
Wehaved∞(u,p)⊆ d(u,p) forevery0< p<∞ andeveryweightu.Ifup = (np− (n− 1)p)∞j=1therearrangementinequalityandthemeredefinitionofthespacesyields
[d(up, p)]p· [d∞(up, p)]1−p⊆ 1.
Wealsohavetheobviousinclusion
d(up, p)⊆ [d(up, p)]p· [d(up, p)]1−p. Summingup,weobtaind(up,p)⊆ 1.
Assumethatw fulfills (4.3).Wededucethatd(w,p)⊆ d(up,p).Therefore,d(w,p)⊆
1.Then,(i)followsfromLemma3.5.Toprove(ii),wepickasemi-normalizedwellcom- plemented blockbasicsequence (um)m∈M withgoodprojectingfunctionals(u∗m)m∈M. ByLemma3.6,wecansuppose thatu∗m=1∗supp(um)sothat
sup
n∈N|u∗m(en)| |e∗n(um)| = sup
n∈N|e∗n(um)|.
Finally,note thattheproofof[6,Theorem 2.4] gives
m∈Minf sup
n∈N|e∗n(um)| > 0. 4.4. Tsirelson’sspace
Casazza and Kalton established in [13] the uniquenessof unconditional basisup to permutationofTsirelson’s spaceT anditscomplementedsubspaceswithunconditional basis as a byproduct of their study of complemented basic sequences in lattice anti- EuclideanBanachspaces.TheirresultansweredaquestionbyBourgainetal.([10]),who hadprovedtheuniquenessofunconditionalbasisuptopermutationofthe2-convexifyed Tsirelson’sspaceT(2)ofT (seeExample5.10inSection5forthedefinition).UnlikeT(2), whichis“highly”Euclidean,thespaceT isanti-Euclidean.Toseethelatterrequiresthe notionofdominance,introduced in[13].
LetB = (xn)∞n=1bea(semi-normalized)unconditionalbasisofaquasi-Banachspace X.Given f ,g∈ X, wewritef ≺ g ifm< n forallm∈ supp(f) andn∈ supp(g).The basisB issaidtobeleft(resp.right)dominant ifthereisaconstantC suchthatwhenever (fi)Ni=1 and (gi)Ni=1 are disjointly supported families with fi ≺ gi (resp. gi ≺ fi) and
fi≤ gi foralli∈ N[N],thenN
i=1fi≤ CN
i=1gi.IfX isaBanachspacewith a left(resp. right) dominantunconditional basisB there is auniquer = r(B) ∈ [1,∞]
such thatr isfinitely blockrepresentable inX.In thecasewhen r(B)∈ {1,∞},X is anti-Euclidean(see[13,Proposition5.3]).
ThecanonicalbasisoftheTsirelsonspaceT isrightdominant[13,Proposition5.12], and r(T )= 1. Moreover,by[13, Proposition5.5] and[15,page 14],thecanonical basis (aswellaseachofitssubbases)isequivalenttoitssquare.Inourlanguage,[13,Theorem 5.6] says thateveryleft (resp.right) dominantunconditional basisis universalforwell complementedblockbasicsequences. Finally,sinceitislocally convex,T istriviallyan L-convex lattice.
