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Banach spaces without local unconditional

arXiv:math/9306211v1 [math.FA] 21 Jun 1993Banach spaces without local unconditionalstructureRyszard A. KomorowskiNicole Tomczak-JaegermannAbstractFor a large class of Banach spaces, a general construction of sub-spaces without local unconditional structure is presented. As an ap-plication it is shown that every Banach space of finite cotype containseither l2 or a subspace without unconditional basis, which admits aSchauder basis.

Some other interesting applications and corollariesfollow.0IntroductionIn this paper we present, for a large class of Banach spaces, a general con-struction of subspaces with a basis which have no local unconditional struc-ture. The method works for a direct sum of several Banach spaces with baseswhich have certain unconditional properties.

It is then applied to Banachspaces with unconditional basis, to show that if such a space X is of finitecotype and it does not contain an isomorphic copy of l2, then X contains asubspace with a basis and without local unconditional structure. As an im-mediate consequence we get that if all subspaces of a Banach space X haveunconditional basis then X is l2 saturated (i.e., every infinite-dimensionalsubspace of X contains a copy of l2).

In particular, if X is a homogeneousBanach space non-isomorphic to a Hilbert space (i.e., X is isomorphic to itsevery infinite-dimensional subspace) then X must not have an unconditionalbasic sequence.1

We also discuss several other situations. Let us only mention here thatour method provides a uniform construction of subspaces without local un-conditional structure which still have Gordon–Lewis property in all Lp spacesfor 1 ≤p < ∞, p ̸= 2, and in all p-convexified Tsirelson spaces and theirduals 1 ≤p < ∞.The technique developed here is based on the approach first introducedby W. B. Johnson, J. Lindenstrauss and G. Schechtman in [J-L-S] for in-vestigating the Kalton–Peck space, which was the first example of a Banachspace which admits 2-dimensional unconditional decomposition but has nounconditional basis.

This approach was refined by T. Ketonen in [Ke] andsubsequently generalized by A. Borzyszkowski in [B], for subspaces of Lp,with 1 ≤p < 2.The essential idea of the approach from [J-L-S], [Ke] and [B] is summa-rized (and slightly generalized for our purpose) in Section 1. In the samesection we also introduce all definitions and notations.Our general con-struction is presented in Section 2.

The additional ingredient which appearshere consists of an ordered sequence of partitions of natural numbers, whichallows to replace some “global” arguments used before by “local” analogues.In Section 3 we prove the main application on subspaces of spaces with anunconditional basis. Other applications and corollaries are discussed in Sec-tion 4.After this paper was sent for publication we learnt about a spectacularstructural theorem just proved by W. T. Gowers.

This theorem combinedwith our Theorem 4.2 and a result from [G-M] shows that a homogeneousBanach space is isomorphic to a Hilbert space, thus solving in the positivethe so-called homogeneous space problem. More details can be found in thepaper by Gowers [G].The contribution of the first named author is a part of his Ph.

D. Thesiswritten at the University of Alberta under a supervision of the second namedauthor.During the final work on the paper the first named author wassupported by KBN.1Notation and preliminariesWe use the standard notation from the Banach space theory, which can befound e.g., in [L-T.1], [L-T.2] and [T], together with all terminology not2

explained here. In particular, the fundamantal concepts of a basis and aSchauder decomposition can be found in [L-T.1], 1.a.1 and 1.g.1, respectively.Let us only recall fundamental notions related to unconditionality.A basis {ej}j in a Banach space X is called unconditional, if there is aconstant C such that for every x = Pj tjej ∈X one has ∥Pj εjtjej∥≤C∥x∥,for all εj = ±1 for j = 1, 2, .

. ..

The infimum of constants C is denoted byunc ({ej}). The basis is called 1-unconditional, if unc ({ej}) = 1.A Schauder decomposition {Zk}k of a Banach space X is called C-uncon-ditional, for some constant C, if for all finite sequences {zk} with zk ∈Zk forall k, one has ∥Pk εkzk∥≤C∥Pk zk∥.

For a subset K ⊂IN, by YK denotespan [Zk]k∈K.A Banach space X has local unconditional structure if there is C ≥1 suchfor every finite-dimensional subspace X0 ⊂X there exist a Banach space Fwith a 1-unconditional basis and operators u0 : X0 →F and w0 : F →Xsuch that the natural embedding j : X0 →X admits a factorization j = w0 u0and ∥u0∥∥w0∥≤C. The infimum of constants C is denoted by l.u.st (X).We will also use several more specific notation.

Let F be a Banach spacewith a basis {fl}l. For a subset A ⊂IN, by F |A we denote span [fl]l∈A. IfF ′ is another space with a basis {f ′l}l, by I : F →F ′ we denote the formalidentity operator, i.e., I(x) = Pl tlf ′l, for x = Pl tlfl ∈F.

With some abuseof notation, we will occasionally write ∥I : F →F ′∥= ∞when this operatoris not bounded.We say that a basis {fl}l dominates (resp. is dominated by) {f ′l}l, if theoperator I : F →F ′ (resp.

I : F ′ →F) is bounded. If the bases in F andF ′ are fixed and they are equivalent, by de(F, F ′) we denote the equivalenceconstant,de(F, F ′) = ∥I : F →F ′∥∥I : F ′ →F∥;(1.1)and we set de(F, F ′) = ∞if the bases are not equivalent.By D(F ⊕F ′) we denote the diagonal subspace of F ⊕F ′, i.e., the subspacewith the basis {(fj +f ′j)/∥fj +f ′j∥}j; an analogous notation will be also usedfor a larger (but finite) number of terms.The following proposition is a version of a fundamental criterium due toKetonen [Ke] and Borzyszkowski [B].

Since a modification of original argu-ments would be rather messy, we provide a shorter direct proof.3

Proposition 1.1 Let Y be a Banach space of cotype r, for some r < ∞,which has a Schauder decomposition {Zk}k, with dim Zk = 2, for k = 1, 2, . .

..If Y has local unconditional structure then there exists a linear, not necces-sarily bounded, operator T : span [Zk]k →span [Zk]k such that(i) T(Zk) ⊂Zk for k = 1, 2, . .

. ;(ii) If, for some K ⊂IN and some C ≥1, the decomposition {Zk}k∈K of YKis C-unconditional, then∥T |YK : YK →YK∥≤C2ψ l.u.st (Y ),(1.2)where ψ = ψ(r, Cr(Y )) depends on r and the cotype r constant Cr(Y )of Y only;(iii) infλ ∥T |Zk −λIZk∥≥1/8, for k = 1, 2, .

. ..The proof requires a fact already used in a more general form in [B].

Forsake of completeness and clarity of the exposition, we sketch the proof here.Lemma 1.2 Let Y be a Banach space of cotype r which has local uncondi-tional structure, and let q > r. For every ε > 0 and every finite-dimensionalsubspace Y0 ⊂Y there exist a Banach space E with a 1-unconditional ba-sis which is q-concave, and operators u : Y0 →E and w : E →Y suchthat the natural embedding j : Y0 →Y admits a factorization j = w u and∥u∥∥w∥≤(1 + ε) l.u.st (Y ). Moreover, the q-concavity constant of E satis-fies M(q)(E) ≤φ where φ = φ(r, q, Cr(Y )) depends on r, q and the cotype rconstant of Y only.ProofGiven ε > 0 and Y0, let F be a space with a 1-unconditional basis{fi}i and let u0 : Y0 →F and w0 : F →Y be such that j = w0u0 and∥w0∥∥u0∥≤(1 + ε) l.u.st (Y ).

It can be clearly assumed that F is finite-dimensional, say dim F = N. Let {f ∗i }i be the biorthogonal functionals.We let E to be IR N with the norm ∥· ∥E defined by∥(ti)i∥E = supεi=±1 ∥w0Xiεitifi∥for (ti) ∈IR N.We also set, u(x) =f ∗i (u0x)i, for x ∈Y0 and w(ti)i=Pi tiw0fi, for(ti) ∈E.4

It is easy to check that wu(x) = x, for x ∈Y0 and that ∥u∥≤∥w0∥∥u0∥and ∥w∥= 1. Clearly, the standard unit vector basis is 1-unconditional inE.

Using the cotype r of Y , it can be checked that E satisfies a lower restimate with the constant Cr(Y ). Thus E is q-concave for every q > r withthe q-concavity constant M(q)(E) depending on q, r and Cr(Y ).

(cf. [L-T.2]1.f.7).✷Proof of Proposition 1.1Assume that Y has the local unconditionalstructure.

It is enough to construct a sequence of operators Tn : span [Zk]k →span [Zk]k, such that for every n, the operator Tn satisfies (i), (ii) and(iii’) infλ ∥Tn |Zk −λIZk∥≥1/8, for k = 1, 2, . .

. , n.Then the existence of the operator T will follow by Cantor’s diagonal proce-dure and Banach–Steinhaus theorem.Fix n and ε > 0, set q = 2r.

Let E with a 1-unconditional basis {ej}jand operators u : Y{1,...,n} →E and w : E →Y be given by Lemma 1.2, suchthat j = w u and ∥u∥∥w∥≤(1 + ε) l.u.st (Y ); moreover, E is 2r-concave.Let Pk : Y →Zk be the natural projection onto Zk, for k = 1, 2, . .

.. For asequence of signs Θ = {θj}, with θj = ±1 for j = 1, 2, . .

., define ΛΘ : E →Eby ΛΘ(y) = Pj θjtjej, for y = Pj tjej ∈E. Then ∥ΛΘ∥= 1.For every k = 1, 2, .

. .

pick a sequence of signs Θk such thatsupΘ infλ ∥PkwΛΘuPk −λIZk∥≤(4/3) infλ ∥PkwΛΘkuPk −λIZk∥.Define Tn : span [Zk]k →span [Zk]k byTn(y) =nXk=1PkwΛΘkuPk(y)for y =Xkzk ∈span [Zk]k.Clearly (i) follows just from the definition of Tn. To prove (ii), let Kn =K∩{1, .

. .

, n}. Let rk denote the Rademacher functions on [0, 1].

Since (E, ∥·∥) is a 2r-concave Banach lattice with the 2r-concavity constant depending onr and Cr(Y ), and also the decomposition {Zk}k∈K of YK is C-unconditional,by Khintchine–Maurey’s inequality (cf. e.g., [L-T.2], 1.d.6) we have, for y ∈YKn,∥Tn |YK(y)∥=nXk=1PkwΛΘkuPk(y) =Xk∈KnPkwΛΘkuPk(y)5

=Z 10 Xk∈Knrk(t)Pk Xk∈Knrk(t)wΛΘkuPk(y)dt≤sup0≤t≤1Xk∈Knrk(t)Pk∥w∥Z 10Xk∈Knrk(t)ΛΘkuPk(y)dt≤C M∥w∥ Xk∈Kn|ΛΘkuPk(y)|21/2=C M∥w∥ Xk∈Kn|uPk(y)|21/2≤C M2∥w∥Z 10Xk∈Knrk(t)uPk(y)dt≤C M2∥w∥∥u∥Z 10Xk∈Knrk(t)Pk(y)dt≤C2M2 (1 + ε) l.u.st (Y ) ∥y∥.The constant M depends on r and M(2r)(E), hence, implicitely, on r andCr(Y ); so the function ψ so obtained satisfies the requirements of (ii).To prove (iii’), fix an arbitrary k = 1, 2, . .

. , n. Consider the 4-dimensionalspace H of all linear operators on Zk and the subspace H0 = span [IZk]spanned by the identity operator on Zk.

Consider the quotient space H/H0and for R ∈H, let eR be the canonical image of R in H/H0.Denote the biorthogonal functionals to the basis {ej}j in E by {e∗j}j andconsider operators Rj = Pkw(e∗j ⊗ej)uPk on Zk. Since dim Rj(Zk) = 1 < 2,it is easy to see that for every j = 1, 2, .

. ., one has∥eRj∥= infλ ∥Rj −λIZk∥≥(1/2)∥Rj∥.Also recall that if F is an m-dimensional space then for any vectors {xj}jin F one hassupθj=±1Xjθjxj ≥(1/m)Xj∥xj∥.This is a restatement of the estimate for the 1-summing norm of the identityon F, π1(IF) ≤m, and it is a simple consequence of the Auerbach lemma(cf.

e.g., [T]).So by the definition of Tn and by the choice of Θk and the above estimateswe getinfλ ∥Tn |Zk −λIZk∥≥(3/4) supΘ infλ ∥PkwΛΘuPk −λIZk∥6

=(3/4) supθj=±1Xjθj eRj ≥(1/4)Xj∥eRj∥≥(1/8)Xj∥Rj∥≥(1/8)∥XjRj∥= (1/8)∥IZk∥= 1/8,completing the proof.✷Finally let us introduce notations connected with partitions of the set ofnatural numbers IN, which are essential in the sequel. A subset A ⊂IN iscalled an interval if it is of the form A = {i | k ≤i ≤n}.

Sets A1 andA2 are called consecutive intervals if max Ai < min Aj, for i, j = 1, 2 andi ̸= j. A family of mutually disjoint subsets ∆= {Am}m is a partition of IN,if Sm Am = IN.For a partition ∆= {Am}m of IN, by L(∆) we denote the familyL(∆) = {L ⊂IN | |L ∩Am| = 1 for m = 1, 2, .

. .

}. (1.3)If ∆′ = {A′m}m is another partition of IN, we say that ∆≻∆′, if there existsa partition J (∆′, ∆) = {Jm}m of IN such thatmin Jm < min Jm+1andA′m =[j∈JmAjfor m = 1, 2, .

. .

. (1.4)In such a situation, for m = 1, 2, .

. ., K(A′m, ∆) denotes the familyK(A′m, ∆) = {K ⊂A′m | |K ∩Aj| = 1 for j ∈Jm}.

(1.5)Finally, if ∆i = {Ai,m}m, for i = 1, 2, . .

., is a sequence of partitions of IN,with ∆1 ≻. .

. ≻∆i ≻.

. ., we set, for m = 1, 2, .

. .

and i = 2, 3, . .

.Ki,m = K(Ai,m, ∆i−1). (1.6)2General construction of subspaces withoutlocal unconditional structureWe will now present an abstract setting in which it is possible to con-struct spaces without local unconditional structure, but which still admit a7

Schauder basis. As it is quite natural, we work inside a direct sum of severalBanach spaces with bases, with each basis having certain unconditional prop-erty related to some partitions of IN.

The construction of a required subspacerelies on an interplay between a “good” behaviour of a basis on members ofthe corresponding partition and a “bad” behaviour on sets which select onepoint from each member of the partition. (Recall that the notation Ki,m usedbelow was introduced in (1.6).

)Theorem 2.1 Let X = F1 ⊕. .

. ⊕F4 be a direct sum of Banach spaces ofcotype r, for some r < ∞, and let {fi,l}l be a normalized monotone Schauderbasis in Fi, for i = 1, .

. .

, 4. Let ∆1 ≻.

. .

≻∆4 be partitions of IN, ∆i ={Ai,m}m for i = 1, . .

. , 4.

Assume that there is C ≥1 such that for everyK ∈Ki,m with i = 2, 3, 4 and m = 1, 2, . .

., the basis {fs,l}l∈K in Fs |K isC-unconditional, for s = 1, . .

. , 4; moreover, there iseC ≥1 such that fori = 1, 2, 3 and m = 1, 2, .

. .

we have∥I : Fi |Ai,m →Fi+1 |Ai,m∥≤eC. (2.1)Assume finally that one of the following conditions is satisfied:(i) there is a sequence 0 < δm < 1 with δm ↓0 such that for every i = 1, 2, 3and m = 1, 2, .

. .

and every K ∈Ki+1,m we have∥I : D(F1 ⊕. .

. ⊕Fi) |K →Fi+1 |K∥≥δ−1m ;(2.2)(ii) there is a sequence 0 < δm < 1 with δm ↓0 and Pm δ1/2m= γ < ∞suchthat for every i = 1, 2, 3 and m = 1, 2, .

. .

and every K ∈Ki+1,m wehave∥I : Fi+1 |K →Fi |K∥≥δ−1m . (2.3)Then there exists a subspace Y of X without local unconditional structure,but which still admits a Schauder basis.Remarks1.The space Y will be constructed to have a 2-dimensionalSchauder decomposition.If the bases {fi,l}l are unconditional, for i =1, .

. .

, 4, this decomposition will be unconditional.2. Recall that a space which admits a k-dimensional unconditional de-composition has the GL-property (cf.

[J-L-S]) (with the GL-constant depend-ing on k). Therefore the subspace Y discussed in Remark 1 above has theGL-property but fails having the local unconditional structure.8

ProofWe will define 2-dimensional subspaces Zk of X which will form aSchauder decomposition of Y = span [Zk]k. This decomposition will be C′-unconditional on subsets associated with the partitions ∆1, . .

. , ∆4, for someC′ depending on C. We shall use Proposition 1.1 to show that if Y hadthe local unconditional structure then, letting ψ = ψ(r, Cr(X)) to be thefunction defined in this proposition, we would havel.u.st (Y ) ≥κδ−αt(2.4)for an arbitrary t = 1, 2, .

. .

; in case (i) we have κ > (21433C4 eC2ψ)−1 andα = 1/3; in case (ii) we have κ > (213(1 + 4γ)C3 eC2ψ)−1 and α = 1/2. Thisis impossible, which will conclude the proof.For k = 1, 2, .

. ., vectors xk and yk spanning Zk will be of the formxk=α1,kf1,k + .

. .

+ α4,kf4,k,yk=α′1,kf1,k + . .

. + α′4,kf4,k,such that for k = 1, 2, .

. .

and any scalars s and t, we will have(1/2) max(|s|, |t|) ≤∥sxk + tyk∥≤4(|s| + |t|). (2.5)Set Zk = [xk, yk], for k = 1, 2, .

. ..Clearly, {Zk}k is a 2-dimensionalSchauder decomposition for Y , in particular Y has a basis.

Moreover, forevery i = 2, 3, 4 and m = 1, 2, . .

. and every K ∈Ki,m, the decomposition{Zk}k∈K is 4C-unconditional.Assume that Y has the local unconditional structure.

Let T be an op-erator obtained in Proposition 1.1. In particular, T satisfies (1.2) for everyK ∈Ki,m, and every i = 2, 3, 4 and m = 1, 2, .

. ... Let akbkckdk!denote the matrix of T |Zk in the basis {xk, yk}, for k = 1, 2, .

. .,i.e., we haveT(sxk +tyk) = (sak +tbk)xk +(sck +tdk)yk.

Comparing the operator norm ofa 2 × 2 matrix with the l4∞-norm of the sequence of entries, and using (2.5),we get that condition (iii) of Proposition 1.1 implies that, for all k = 1, 2, . .

.,infλ max(|ak −λ|, |dk −λ|, |bk|, |ck|) ≥2−5 infλ ∥T |Zk −λIZk∥≥2−8. (2.6)9

For the rest of the argument we consider cases (i) and (ii) separately. Westart with (i).

Let γm = δ1/3m , for m = 1, 2, . .

.. For k ∈A4,t, with t = 1, 2, . .

.,putxk=f1,k+γtf3,k+γ2t f4,kyk=f2,k+γ2t f4,k. (2.7)Obviously, (2.5) is satisfied.

Fix an arbitrary t = 1, 2, . .

.. For i = 1, 2, 3,let Mi = {m | Ai,m ⊂A4,t}. Note that (1.4) yields that min Mi ≥t fori = 1, 2, 3.For every m ∈M2 pick B ∈K2,m.

By (2.2) we have∥I : F1 |B →F2 |B∥≥γ−3m ;on the other hand, ∥f1,l∥= ∥f2,l∥= ∥I(f1,l)∥. By continuity, there exists asequence {βk}k∈B such that ∥Pk∈B βkf1,k∥= 1 and ∥Pk∈B βkf2,k∥= γ−1t .Then, by (2.1) and (2.7) we haveXk∈Bβkxk≤Xk∈Bβkf1,k + γtXk∈Bβkf3,k + γ2tXk∈Bβkf4,k≤1 + (γt eC + γ2t eC2)Xk∈Bβkf2,k ≤3 eC2,whileT(Xk∈Bβkxk) =Xk∈Bβk(akxk + ckyk)≥Xk∈Bβkckf2,k ≥C−1 infk∈B |ck|Xk∈Bβkf2,k ≥C−1γ−1tinfk∈A2,m |ck|.This implies, by (1.2), that for every m ∈M2 there exists l ∈A2,m suchthat |cl| ≤3 42C3 eC2ψ γt l.u.st (Y ).

Denote the set of these l’s by L2 andobserve that L2 ∈L(∆2) |M2. If we had |cl| > 2−10 for some l ∈L2, then(2.4) would follow.

Therefore assume that |cl| ≤2−10 for all l ∈L2.For every m ∈M3, set B = L2 ∩A3,m. Then B ∈K3,m and by (2.2)there exists a sequence {βk}k∈B such thatXk∈Bβk f1,k + f2,k∥f1,k + f2,k∥ = 1andXk∈Bβkf3,k = γ−2t .10

Observe that the basis {(f1,k +f2,k)/∥f1,k +f2,k∥}k∈B is 2C-unconditional forevery B ∈K3,m. Thus,Xk∈Bβkf2,k ≤Xk∈Bβk(f1,k + f2,k) ≤4C.Hence,Xk∈Bβkyk ≤Xk∈Bβkf2,k + γ2tXk∈Bβkf4,k ≤4 C + eC,andTXk∈Bβkyk=Xk∈Bβk(bkxk + dkyk)≥γtXk∈Bβkbkf3,k ≥C−1γ−1tinfk∈L2∩A3,m |bk|.Therefore, using (1.2) again, for every m ∈M3 pick l ∈L2 ∩A3,m suchthat |bl| ≤42(4C + eC)C3ψγt l.u.st (Y ).

Denote the set of these l’s by L3 andassume as before that |bl| ≤2−10 for all l ∈L3. Moreover, L3 ⊂L2 andL3 ∈L(∆3) |M3.Finally, consider K = L3 ∩A4,t ∈K4,t and pick a sequence {βk}k∈K suchthatXk∈Kβk f1,k + f2,k + f3,k∥f1,k + f2,k + f3,k∥ = 1andXk∈Kβkf4,k = γ−3t .Since {(f1,k+f2,k+f3,k)/∥f1,k+f2,k+f3,k∥}k∈K is 3C-unconditional, for everyK ∈K4,t, we have, for i = 1, 2, 3,Xk∈Kβkfi,k ≤Xk∈Kβk(f1,k + f2,k + f3,k) ≤32C.Thus,Xk∈Kβk(xk −yk) ≤Xk∈Kβkf1,k +Xk∈Kβkf2,k + γtXk∈Kβkf3,k ≤33 C.Moreover, since |ck| ≤2−10 and |bk| ≤2−10 for k ∈L3, by (2.6) we have|ak −bk + ck −dk| ≥2−9fork ∈L3.

(2.8)11

ThereforeT Xk∈Kβk(xk −yk)=Xk∈Kβk(ak −bk)xk + (ck −dk)yk≥γ2tXk∈Kβk((ak −bk) + (ck −dk))f4,k≥C−12−9γ−1t .Using (1.2) once more we get 3342C3ψ l.u.st (Y ) ≥C−12−9γ−1t , which implies(2.4). This completes the proof of case (i).In case (ii) the proof is very similar and let us describe necessary modifi-cations.

Set γm = δ1/2mfor m = 1, 2, . .

.. For k = 1, 2, . .

. and k ∈A2,m ∩A3,s,for some m = 1, 2, .

. .

and s = 1, 2, . .

., setxk=γsf2,k+f3,k+f4,kyk=γmf1,k+f3,k. (2.9)Again, (2.5) is satisfied.

Fix an arbitrary t = 1, 2, . .

., and define Mi,for i = 1, 2, 3 as before. Using the fact that ∥I : F2 |K →F1 |K∥≥γ−2m , forevery K ∈K2,m and every m ∈M2, one can show, using (2.1) and (1.2) ina similar way as before, that there is a set L2 = {lm}m∈M2 ∈L(∆2) |M2 suchthat|clm| ≤3 42C3 eC2ψγm l.u.st (Y )for m ∈M2.

(2.10)One can additionally assume that |clm| ≤2−10 for all m ∈M2, otherwise,since min M2 ≥t implies γm ≤γt, we would immediately get (2.4) withα = 1/2..Now for every s ∈M3 consider the set B = L2 ∩A3,s ∈K3,s, and pick asequence {βk}k∈B such thatXk∈Bβkf3,k = 1andXk∈Bβkf2,k ≥γ−2s .If M2,s denotes the set of indices m ∈M2 such that lm ∈L2 ∩A3,s = B,thenXk∈Bβkyk ≤Xm∈M2,sγm|βlm| + 1 ≤2γ + 1,(2.11)where the first term in the estimate is obtained by first using the triangleinequality and then using the fact that since {f3,lm}m∈M2,s is a monotonebasic sequence, then |βlm| ≤2 for all lm ∈B.12

We also haveTXk∈Bβkyk≥γsXk∈Bβkbkf2,k≥C−1γs infk∈B |bk|Xk∈L2∩A3,sβkf2,k≥C−1γ−1sinfk∈L2∩A3,s |bk|.Thus there exists a set L3 ∈L(∆3) |M3, L3 = {l′s}s∈M3, such that L3 ⊂L2and|bl′s| ≤42(2γ + 1)C3ψγs l.u.st (Y )for s ∈M3;(2.12)and since min M3 ≥t implies γs ≤γt, one can additionally assume that|bl′s| ≤2−10, for all s ∈M3.Finally set K = L3∩A4,t ∈K4,t. Pick {βk}k∈K such that ∥Pk∈K βkf4,k∥=1 and ∥Pk∈K βkf3,k∥≥γ−2t .

Then, by the triangle inequality and by themonotonicity of the basis {f4,k}k we get, similarly as in (2.11),Xk∈Kβk(xk −yk) ≤2Xm∈M2γm + 2Xs∈M3γs +Xk∈Kβkf4,k ≤1 + 4γ.On the other hand, by (2.6), (2.10) and (2.12) we again have (2.8). ThusT Xk∈Kβk(xk −yk)≥Xk∈Kβk(ak −bk) + (ck −dk)f3,k≥C−12−9Xk∈Kβkf3,k ≥C−12−9γ−2t .Using (1.2) we get l.u.st (Y ) ≥(213(1+4γ)C3ψ)−1γ−2t , hence (2.4) follows,completing the proof of case (ii).✷3Subspaces of spaces with unconditional ba-sisOur main application of the construction of Theorem 2.1 is the followingresult on subspaces of spaces with unconditional basis.13

Theorem 3.1 Let X be a Banach space with an unconditional basis and ofcotype r, for some r < ∞. If X does not contain a subspace isomorphic tol2 then there exists a subspace Y of X without local unconditional structure,which admits a Schauder basis.In particular, every Banach space of cotype r, for some r < ∞, containseither l2 or a subspace without unconditional basis.We present now the proof of the theorem, leaving corollaries and furtherapplications to the next section.The argument is based on a construction, for a given Banach space X, ofa direct sum inside X of subspaces Fi of X, and of partitions ∆i of IN suchthat Theorem 2.1 can be applied.

This construction requires several steps.The first lemma is a simple generalization to finite-dimensional lattices ofthe fact that the Rademacher functions in Lp are equivalent to the standardunit vector basis in l2.Lemma 3.2 Let E be an N-dimensional Banach space with a 1-unconditio-nal basis {ej}j and for 2 ≤r < ∞let Cr(E) denote the cotype r constant ofE. If m ≤log2 N then there exist normalized vectors f1, .

. .

, fm in E, of theformfl =Xjε(l)j αjejfor l = 1, . .

. , m,(3.1)for some sequence of scalars {αj} and ε(l)j= ±1 for l = 1, .

. .

, m and j =1, . .

. , N; and such thatdespan [fl], lm2≤C,(3.2)where C depends on r and on the cotype r constant of E.Proof Since E is a discrete Banach lattice, the cotype r assumption impliesthat E is q-concave, for any q > r (cf.

[L-T.2]). Setting e.g., q = 2r, theq-concavity constant of E depends on r and Cr(E).

By a lattice renormingwe may and will assume that this constant is equal to 1 (cf. [L-T.2] 1.d.8);the general case will follow by adjusting C.For 1 ≤p < ∞, let ∥· ∥Lp be the norm defined on IR N by ∥t∥Lp =(N−1 PNj=1 |tj|p)1/p, for t = (tj) ∈IR N.It is well known consequence of14

Lozanovski’s theorem (see [T], 39.2 and 39.3 for a related result) that thereexist αj > 0, j = 1, . .

. , N, such that∥t∥L1 ≤∥NXj=1αjtjej∥≤∥t∥Lqfor t = (tj) ∈IR N.(3.3)Fix an integer m ≤log2 N. By Khintchine’s inequality there exist vectorsrl = {rl(j)}Nj=1, with rl(j) = ±1 for j = 1, .

. .

, N, l = 1, . .

. , m, such that forevery (bl) ∈IR m we have2−1/2(mXl=1|bl|2)1/2 ≤∥mXl=1blrl∥L1 ≤∥mXl=1blrl∥Lq ≤Cq(mXl=1|bl|2)1/2.

(3.4)Setting fl = PNj=1 rl(j)αjej, for l = 1, . .

. , m, we get, by (3.3),∥mXl=1blrl∥L1 ≤∥mXl=1blfl∥= ∥NXj=1αj(mXl=1blrl(j))ej∥≤∥mXl=1blrl∥Lq,for every (bl) ∈IR m. This combined with (3.4) completes the required esti-mate.✷RemarkAs it was pointed out to us by B. Maurey, Lemma 3.2 could bereplaced by the contruction of L. Tzafriri [Tz], which implies the existenceof a function ϕ(N), with ϕ(N) →∞as N →∞, such that for m ≤ϕ(N)every N-dimensional space E as in the lemma contains normalized vectorsf1, .

. .

, fm satisfying (3.2), which are of the form fl = α Pj ±ej, with anappropriate constant α.The next proposition is the key for our argument. To simplify the state-ment, let us introduce one more notation.

Given a partition ∆= {Am}m ofIN into consecutive intervals and a space F with a normalized Schauder basis{fl}l and C ≥1, we call a pair {∆, F} C-regular, if the following conditionsare satisfied:(i) deF |Am, l|Am|2≤C for m = 1, 2, . .

. ;(ii) for every L ∈L(∆), the basis {fl}l∈L in F |L is 1-unconditional (hereL(∆) is as in (1.3));15

(iii) for arbitrary L, L′ ∈L(∆) one has deF |L, F |L′= 1.Observe that condition (iii) means that if L = {lm}m, L′ = {l′m}m, withlm, l′m ∈Am for m = 1, 2, . .

., then for every sequence of scalars (bm) one hasXmbmflm =Xmbmfl′m. (3.5)Proposition 3.3 Let E1, E2 .

. .

be Banach spaces of cotype r, for some r <∞. Let {ei,j}j be a 1-unconditional basis in Ei, and assume that no sequenceof disjointly supported vectors in E1 ⊕.

. .

⊕Ei is equivalent to the standardunit vector basis in l2, for i = 1, 2, . .

.. Then there exists C, depending on rand the cotype r constants of Ei, such that there exist subspaces Fi ⊂Ei withnormalized Schauder bases {fi,l}l, and partitions ∆i = {Ai,m}m of IN intoconsecutive intervals, for i = 1, 2, . .

., with ∆1 ≻∆2 ≻. .

., satisfying thefollowing: for each i = 1, 2, . .

. {∆i, Fi} is C-regular and one of the followingmutually exclusive conditions is satisfied: either for every L ∈L(∆i) one has∥I : l2 →Fi |L∥= ∞,(3.6)or for every L ∈L(∆i) one has∥I : l2 →Fi |L∥< ∞.

(3.7)Furthermore, one also has(iv) If (3.6) holds for some i, then the partition ∆i+1 = {Ai+1,m}m satisfiesinfm infn2−3m∥I : l|K|2→Fi |K ∥ K ∈Ki+1,mo≥C. (3.8)On the other hand, let M denote the set (which may be empty) of alls ∈IN such that for every L ∈L(∆s) one has ∥I : l2 →Fs |L∥< ∞.

Ifi ∈M, put Mi = M ∩{1, . .

. , i}; then the partition ∆i+1 = {Ai+1,m}msatisfiesinfm infn2−3m∥I : D Xs∈Mi⊕Fs|K →l|K|2 ∥K ∈Ki+1,mo≥C.

(3.9)16

ProofIn the first part of the proof we show that given space E of co-type r with a 1-unconditional basis {ej}j, and a partition ∆= {Am}m ofIN into consecutive intervals, there exists a subspace F ⊂E with a normal-ized Schauder basis {fl}l such that {∆, F} is C-regular, for an appropriateconstant C, and that either (3.6) or (3.7) is satisfied for every L ∈L(∆).For an arbitrary m = 1, 2, . .

., let km = |Am| and let E(m) = span {ej |2km < j ≤2km+1}. Since dim E(m) ≥2km, by Lemma 3.2 there exist vectorsfl ∈E(m), for l ∈Am, such thatdespan [fl]l∈Am, lkm2≤C;(3.10)and there is a sequence {αj} of real numbers such that the fl’s are of theformfl =2km+1Xj=2km+1±αjejfor l ∈Am, m = 1, 2, .

. .

. (3.11)We let F = span [fl]l. Then (i) is implied by (3.10).

Next observe thatfl and fl′ have consecutive supports, whenever l ∈Am and l′ ∈Am′ andm ̸= m′. This and (3.10) easily yield that {fl}l is a Schauder basis in F.Also, {fl}l∈L is a 1-unconditional basis in F |L, for every L ∈L(∆), whichshows (ii).By (3.11) we get that if (bm) is a scalar sequence then for every L ={lm}m ∈L(∆), the vectorPm bmflm is of the formXmbm2km+1Xj=2km+1±αjej;a specific choice of the lm’s which constitute the set L effects only the choiceof the signs in the inner summation.

Since the basis {ej} is 1-unconditional,(3.5) follows, hence (iii) follows as well.Finally observe that for a fixed L ∈L(∆), exactly one of conditions (3.6)and (3.7) holds. Moreover, by (iii), the norms of the formal identity operatorsinvolved do not depend on a choice of the set L ∈L(∆).We now pass to the second part of the proof, the inductive constructionof ∆i’s and Fi’s, which ensures also condition (iv).Let A1,m = {m} form = 1, 2, .

. .

and let ∆1 = {A1,m}m.Assume that i ≥1 and that partitions ∆1 ≻. .

. ≻∆i and subspacesF1, .

. .

, Fi−1 have been constructed to satisfy conditions (i)–(iv). Let Fi ⊂Ei17

be a subspace constructed in the first part of the proof for ∆= ∆i. Theconstruction of ∆i+1 depends on which of two, (3.6) or (3.7), holds for Fi.Assume first that (3.6) holds and fix an arbitrary set L ∈L(∆i).

Enu-merate L = {lj}j with lj ∈Ai,j for j = 1, 2, . .

.. There exist 1 = j0 < j1 <.

. .

< jm < . .

. such that if Jm = {jm−1 ≤j < jm}, then∥I : l|Jm|2→Fi |L |Jm∥≥C223mfor m = 1, 2, .

. .

. (3.12)We then setAi+1,m =[j∈JmAi,jfor m = 1, 2, .

. .

. (3.13)By (3.5) and (3.12) it is clear that (3.8) is satisfied in this case.Assume now that (3.7) holds, so i ∈M.

There is a constant C′ such thatfor all s ∈Mi the estimate ∥I : l2 →Fs |L∥< C′ holds for all L ∈L(∆s);hence also for all L ∈L(∆i), since sets from L(∆i) are subsets of sets fromL(∆s), for every s < i. Fix an arbitrary L ∈L(∆i).

We then have∥I : l2 →D Xs∈Mi⊕Fs|L∥< |Mi| C′.Note that if l, l′ ∈L ∈L(∆i) and l ̸= l′ then fs,l and fs,l′ have consecutivesupports, hence {fs,l}l∈L forms a block basis of {es,j}j, for s ∈Mi. There-fore by our assumptions, the basis {Ps∈Mi fs,l}l∈L in D(Ps∈Mi ⊕Fs) is notequivalent to the standard unit vector basis in l2.

Thus∥I : D Xs∈Mi⊕Fs|L →l2∥= ∞. (3.14)Now the construction of a partition ∆i+1 satisfying (3.9) is done by for-mulas completely analogous to (3.12) and (3.13), in which the use of (3.6) isreplaced by (3.14).✷Finally, the proof of the main result follows formally from Proposition 3.3.Proof of Theorem 3.1 Write X as an unconditional sum X = Pi ⊕Ei, of13 spaces Ei, each with a 1-unconditional basis {ei,j}j.

Let ∆1 ≻. .

. ≻∆13be partitions of IN and Fi ⊂Ei be subspaces with Schauder bases {fi,l}l,constructed in Proposition 3.3.

Renorming the spaces Fi if necessary, wemay assume that the bases {fi,l}l are monotone.18

Now the C-regularity properties imply all the preliminary assumptions ofTheorem 2.1, including (2.1). To prove the remaining conditions (i) or (ii)observe that either there exist four consecutive spaces {Fik}k satisfying (3.7),or (3.6) holds for some three (not necessarily consecutive) spaces {Fik}k.In either case, we let Λk = ∆ik and F ′k = Fik, for k = 1, .

. .

, 4 (in thelatter case we set i4 = i3 + 1).It is easy to check that in the former case, (3.7) yields (3.9), while in thelatter case (3.6) yields (3.8). Thus the remaining assumptions of Theorem 2.1are satisfied with δm = 2−3m, which concludes the proof.✷4Corollaries and further applicationsRecall a still open question whether a Banach space whose all subspaces havean unconditional basis is isomorphic to a Hilbert space.

¿From results on theapproximation property by Enflo, Davie, Figiel and Szankowski, combinedwith Maurey–Pisier–Krivine theorem, it follows that such a space X has, forevery ε > 0, cotype 2+ε and type 2−ε (cf. e.g., [L-T.2], 1.g.6).

Theorem 3.1obviously implies that X has a much stronger property: its every infinite-dimensional subspace contains an isomorphic copy of l2. A space X with thisproperty is called l2-saturated.Theorem 4.1 Let X be an infinite-dimensional Banach space whose all sub-spaces have an unconditional basis.

Then X is l2-saturated.Another well known open problem, going back to Mazur and Banach, con-cerns so-called homogeneous spaces. An infinite-dimensional Banach spaceis called homogeneous if it is isomorphic to each of its infinite-dimensionalsubspaces.

The question is whether every homogeneous Banach space is iso-morphic to a Hilbert space. The same general argument as before showsthat a homogeneous space X has cotype 2 + ε and type 2 −ε, for everyε > 0.

W. B. Johnson showed in [J] that if both X and X∗are homogeneousand X has the Gordon–Lewis property, then X is isomorphic to a Hilbertspace. More information about homogeneous spaces the reader can find in[C].

The following obvious corollary removes the assumption on X∗, howeverit requires a stronger property of X itself.19

Theorem 4.2 If a homogeneous Banach space X contains an infinite un-conditional basic sequence then X is isomorphic to a Hilbert space.Let us recall here that it was believed for a long time that every Banachspace might contain an infinite unconditional basic sequence. This conjecturewas disproved only recently by W. T. Gowers and B. Maurey in [G-M], whoactually constructed a whole class of Banach spaces failing this and relatedproperties.Let us now discuss some easy consequences of the main construction,which might be of independent interest.Corollary 4.3 Let X = F1 ⊕.

. .

⊕F4 be a direct sum of Banach spaces ofcotype r, for some r < ∞, and assume that Fi has a 1-unconditional basis{fi,l}l. for i = 1, . .

. , 4.

Assume that the basis {fi,l}l dominates {fi+1,l}l, andthat no subsequence of {fi,l}l is equivalent to the corresponding subsequenceof {fi+1,l}l, for i = 1, 2, 3. Then there exists a subspace Y of X without localunconditional structure, which admits an unconditional decomposition into2-dimensional spaces.Proof Let ∆1 ≻.

. .

≻∆4 be arbitrary partitions of IN into infinite subsets{Ai,m}. The domination assumption implies (2.1).

On the other hand, thesecond assumption allows for a construction of partitions which also satisfy(2.3). Hence the conclusion follows from Theorem 2.1 and Remark 2 above.✷Remark In fact, Corollary 4.3 can be proved directly from Proposition 1.1.To define xk and yk spanning Zk, let Λ2 = {Bm}m be any partition of INinto infinite sets and write each Bm as a union Bm = Sn Bm,n of an infinitenumber of infinite sets Bm,n.

(Using the natural enumeration of IN × IN, weget this way a partition Λ1 = {Bm,n}m,n with Λ1 ≻Λ2.) Then for k ∈Bm,n,with m, n = 1, 2, .

. .

putxk=2−me2,k+e3,k+e4,kyk=2−m−ne1,k+e3,k.The rest of the proof is the same as in case (ii) of Theorem 2.1.20

If {xi} is a basic sequence in a Banach space X, and 1 ≤p < ∞, we saythat lp is crudely finitely sequence representable in {xi} if there is a constantC ≥1 such that for every n there is a subset Bn ⊂IN such that {xi}i∈Bn isC-equivalent to the unit vector basis in lnp.Corollary 4.4 Let X be a Banach space of cotype r, for some r < ∞, andwith a 1-unconditional basis {el}l; let 1 ≤p < ∞. Assume that no sequence{xj}j of disjointly supported vectors of the form xj =Pl∈Lj el, where |Lj| ≤3for j = 1, 2, .

. ., is equivalent to the unit vector basis of lp.

Moreover assumethat X has one of the following properties:(i) lp is crudely finitely sequence representable in {el}l, and the basis {el}leither is dominated by or dominates the standard unit vector basis inlp;(ii) lp is crudely finitely sequence representable in every subsequence of {el}l.Then X contains a subspace Y without local unconditional structure, whichadmits a 2-dimensional unconditional decomposition.Proof First observe a general fact concerning a basis {el}l whose no subse-quence is dominated by the standard unit vector basis in lp. An easy diagonalargument shows that if a partition ∆= {Aj}j of IN into finite sets is giventhen for an arbitrary M and every j0 ∈IN there is j1 > j0 such that for anyset K ⊂IN such that |K| = j1 −j0 and |K ∩Aj| = 1 for j0 < j ≤j1, onehas ∥I : l|K|p→E |K∥≥M.

In particular, given constant C, there exists apartition ∆′ = {A′m}m of IN, with ∆≻∆′ such that for every m = 1, 2, . .

.and for every K ∈K(A′m, ∆) one has∥I : l|K|p→E |K∥≥C 23m. (4.1)Now, in case (i), write X as a direct sum E1 ⊕.

. .⊕E4, such that each Eihas a 1-unconditional basis {ei,l}l. Assume that the basis {el}l dominates thebasis in lp, hence so does every basis {ei,l}l. Using the general observationabove, we can define by induction partitions ∆1 ≻.

. .

≻∆4 and subsequences{fi,j}j of {ei,l}l, so that for all k and all A = Ak,m ∈∆k, sequences {fik,j}j∈Aare C-equivalent to the standard unit vector basis in l|A|p , and at the sametime, the spaces span [fik,j]j∈K, with K ∈Kk+1,m, satisfy the lower estimate(4.1). Thus (2.3) is satisfied (with δm = 2−3m).21

If the basis {el}l is dominated by the basis in lp, so is every basis {ei,l}l,and also all bases in D(E1 ⊕. .

.⊕Ei), for i = 1, 2, 3. An analogous argumentas before, which additionally requires the assumption on sequences {xj},leads to a construction of partitions satisfying (2.2).

Then the existence ofthe subspace Y follows from Theorem 2.1 and Remark 1 after its statement.In case (ii), write X = E1 ⊕. .

. ⊕E7.

By passing to subsequences we getthat for each i, each subsequence of the basis {ei,l}l, either is dominated byor dominates the standard unit vector basis in lp, for i = 1, . .

. , 7.

Thereforethere is a set I = {i1, . .

. , i4} such that for all i ∈I, the bases {ei,l}l havethe same, either former or latter, domination property.

Then the proof canbe concluded the same way as in case (i).✷For 1 ≤q < ∞, the space Lq([0, 1]) contains a subspace isomorphicto X = (Pn ⊕ln2)q, which, for q ̸= 2, satisfies the assumptions of Corol-lary 4.4 (i) for p = 2. Therefore Lq([0, 1]) contains a subspace without localunconditional structure but which admits a 2-dimensional unconditional de-composition.

By Remark 2 in Section 2, this subspace has the Gordon-Lewis(GL-) property. For 1 ≤q < 2, this gives a somewhat more elementary proofof Ketonen’s result [Ke].

For 2 < q < ∞the construction seems to be new.Ketonen’s result could be also derived from Corollary 4.3 by noticing that inthis case the space Lq([0, 1]) contains a subspace isometric to (lq1 ⊕. .

.⊕lq4)q,for 1 ≤q ≤q1 < . .

. < q4 < 2 (cf.

e.g., [L-T.2], 2.f.5).Corollary 4.4 can also be applied to construct subspaces without localunconditional structure in p-convexified Tsirelson spaces T(p) and in theirduals. This solves the question left open in [K].

The spaces T(2) and T ∗(2)provide the most important examples of so-called weak Hilbert spaces, andthey were discussed in [P]. For general p and notably for p = 1, these spaceswere presented in detail in [C-S].

First construction of a weak Hilbert spacewithout unconditional basis was given by R. Komorowski in [K] by a methodpreceeding the technique presented here.Corollary 4.5 The p-convexified Tsirelson space T(p), for 1 ≤p < ∞, andthe dual Tsirelson T ∗(p), for 1 < p < ∞, contain subspaces without local un-conditional structure, but which admit 2-dimensional unconditional decom-position; in particular they have the Gordon–Lewis property.ProofThe spaces T(p) and T ∗(p) satisfy the assumptions of Corollary 4.4,both (i) and (ii), for p and p′, respectively.✷22

References[B]A. Borzyszkowski, Unconditional decompositions and local uncondi-tional structures in some subspaces of Lp, 1 ≤p < 2, Studia Math.,76 (1983), 267–278.[C]P. G. Casazza, Some questions arising from the homogeneous Banachspace problem, Proceedings of Merida Workshop, January 1992, eds.W.

B. Johnson and Bor-Luh Lin, Contemporary Math. to appear.[C-S]P.

G. Casazza and T. Shura, “Tsirelson’s Space”, Lecture Notes inMath., Springer-Verlag, 1989.[G-M]T. W. Gowers, A new dichotomy for Banach spaces, to appear.[G-M]T.

W. Gowers and B. Maurey, The unconditional basic sequence prob-lem, Journal of A.M.S., to appear. [J-L-S] W. B. Johnson, J. Lindenstrauss and G. Schechtman, On the relationbetween several notions of unconditional structure, Israel J.

Math., 37(1980), 120–129.[J]W. B. Johnson, Homogeneous Banach spaces, Geometric Aspects ofFunctional Analysis, Israel Seminar, 1986–87, eds.

J. Lindenstraussand V. D. Milman, Lecture Notes in Math., 1317, Springer Verlag1988, 201–203.[K]R. A. Komorowski, Weak Hilbert spaces without unconditional basis,Proc.

AMS to appear.[Ke]T. Ketonen, On unconditionality in Lp spaces, Ann.

Acad. Sci.

Fenn.,Ser. A1 Math Dissertationes, 35(1983).

[L-T.1] J. Lindenstrauss and L. Tzafriri, “Classical Banach Spaces, I”,Springer Verlag, 1977. [L-T.2] J. Lindenstrauss and L. Tzafriri, “Classical Banach Spaces, II”,Springer Verlag, 1979.[P]G.

Pisier, “Volume of Convex Bodies and the Geometry of BanachSpaces”, Cambridge University Press, 1990.23

[T]N. Tomczak-Jaegermann,“Banach–Mazur distances and finite-dimensional operator ideals”, Longman Scientific & Technical, 1989.[Tz]L. Tzafriri, On Banach spaces with unconditional bases, Israel J.Math., 17 (1974), 84–93.Institute of Mathematics, Technical University, Wroc law, PolandandDepartment of Mathematics, University of Alberta, Edmonton, Alberta,Canada T6G 2G1.24

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