Reflexivity and the Separable Quotient Problem

for a Class of Banach Spaces

Marek Wojtowicz

Abstract

Let E be a Banach lattice and let X be its closed subspace such that:X is complemented in E, or the norm of E is order continuous. Then X

is reflexive iff X∗ contains no isomorphic copy of `1 iff for every n ≥ 1,

the nth dual X(n) of X contains no isomorphic copy of `1 iff X has no

quotient isomorphic to c0 and X does not have a subspace isomorphicto `1 (Theorem 2). This is an extension of the results obtained earlierby Lozanovskii, Tzafriri, Meyer-Nieberg, and Diaz and Fernandez. Thetheorem is applied to show that many Banach spaces possess separablequotients isomorphic to one of the following spaces: c0, `1, or a reflexivespace with a Schauder basis.

2000 MS Classification: 46B10, 46B25, 46B42,Key words: reflexivity of Banach spaces, embeddability of `1, Banach lattices,separable quotients of Banach spaces

1. Introduction. In the present paper the letter X denotes a Banach space,and X∗ denotes its topological dual. The letter E stands for a (real) Banachlattice. All spaces, their subspaces, and quotients are of infinite dimension. Weassume that the reader is familiar with the theory of Banach lattices presentede.g. in the monographs [2], [23], [31].

The main purpose of this article is to indicate a large class of Banach spacesin which reflexivity is equivalent to the non-containment of an isomorphic copyof `1. These results are presented in Section 3 and, although their proofs areshort and elementary, they are new and play significent role in the so calledSeparable Quotient Problem; we apply them in Section 4 to show that manyBanach spaces can be mapped continuously onto one of the spaces: c0, `1, ora reflexive one with a basis (possibly unconditional). Section 4 correspondsalso to the paper by Onal and Terzioglu [26], devoted to concrete quotients ofFrechet spaces, and complements partially general results for separable quotientsof Banach spaces obtained earlier in [30] and [32] (see also [18], [19], [24], [36]).

Now we will present a survey of the results upon which this article is based.

In 1967 Lozanovskii gave the following characterization of reflexivity of Ba-nach lattices (see [20]; cf. [3], Theorem 4.3).

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Proposition 1 (Lozanovskii). For a Banach lattice E the following statementsare equivalent.

(a) E is reflexive.(b) E does not have a subspace isomorphic to c0 or `1.(c) E∗ does not have a subspace isomorphic to `1.

The above equivalence (a) ⇔ (b) is by now a well known classical result (thatgeneralizes also the theorem of R. C. James on reflexivity of a Banach spacewith an unconditional basis (see [21], Theorem 1.c.12), and was carried over in1972 by Tzafriri, and in 1991 by Meyer-Nieberg, to some subspaces of Banachlattices (see [33] and [23], Theorem 5.1.14, respectively). On the other hand,the equivalence (a) ⇔ (c) is not cited in textbooks devoted to the theory ofBanach lattices although it is a simple consequence of the previous equivalenceand the well known result of Bessaga and Pe lczynski on copies of c0 in dualsof Banach spaces. The above-mentioned result of Tzafriri and Meyer-Niebergreads as follows:

Proposition 2 (Tzafriri & Meyer-Nieberg). Let X be a closed subspace of aBanach lattice E. If X is complemented in E or if the norm of E is ordercontinuous, then the following two conditions are equivalent.

(a’) X is reflexive.(b’) X does not have subspaces isomorphic to c0 or `1.

This theorem is a generalization of both the result by Lozanovskii and an earlierone obtained (in 1958), by Bessaga and Pe lczynski, concerning the character-ization of reflexivity of closed subspaces of Banach spaces with unconditionalbases ([21], Theorem 1.c.13). Let us recall that every Banach space with anunconditional basis can be viewed as a Banach lattice with an order continuousnorm ([22], p. 2).

The present author conjeted earlier that conditions (a’) and (b’) in Propo-sition 2 should be equivalent to the folllwing one:

(c’) X∗ does not have a subspace isomorphic to `1.

The main result of Section 3, included in Theorem 2, says not only that, underthe assumptions of Proposition 2, condition (c’) is true, but that the reflexivityof X is equivalent to the condition that for every positive integer n, the nth dualX(n) of X does not have a subspace isomorphic to `1 as well. As a consequence ofTheorem 2 we obtain that many Banach spaces have ”nice” separable quotients(Theorems 4 and 5).

Below we list the results that will be often used in this paper. The first oneis the classical Bessaga-Pe lczynski theorem ([21], Proposition 2.e.8).

Proposition 3. Let X be a Banach space such that X∗ contains a subspaceisomorphic to c0. Then X contains a complemented subspace isomorphic to `1.

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The next theorem deals with complemented copies of c0 and `1 in Banachlattices with order continuous norms ([6]; Remark 3, p. 150; [23]; Corollary3.4.14).

Proposition 4. Let E be a Banach lattice with order continuous norm, and letW be a closed subspace of E.

(a) If W is isomorphic to c0, then W is complemented in E.(b) If W contains a subspace U that is isomorphic to `1, then U contains a

subspace U1 isomorphic to `1 and complemented in E (and hence, also in W ).

Theorems of Section 4 depend essentially on the following form of the resultof Rosenthal ([29]; Remark 2, p. 181).

Proposition 5. Let X be a Banach space. If (fn) is a basic sequence in the dualspace X∗ with V := [fn] reflexive, then the quotient space X/V⊥ is isomorphicto the reflexive space V ∗, that is, to the space [f∗

n], where f∗n denotes the nth

biorthogonal functional in V ∗. Moreover, if the sequence (fn) is unconditional,then (f∗

n) is unconditional as well.

The following theorem is due to Gonzalez and Onieva (see [12], Corollary2.3; the reader should note that the quoted part (a) of this corollary dependson ([13], Corollary 1) proved by Hagler and Johnson for real Banach spaces):

Proposition 6. Let X be a real Banach space. Then X∗ contains a subspaceisomorphic to `1 if and only if X has a quotient isomorphic to c0 or X containsa subspace isomorphic to `1.

Acknowledgements. The author would like to thank Professors J. Diestel andD. Yost for making him aware of the papers [5] and [1], [7], respectively, as wellas the referee whose remarks improved the quality of this article.

2. Notations and terminology. We follow standard notations and ter-minology; see [21]. Let X be a Banach space. If A ⊂ X∗, then A⊥ denotesthe annihilator of A in X . The term ”operator” means ”linear and (norm) con-tinuous operator”. A Banach space Y is said to be a quotient of X if X canbe mapped continuously onto Y . An operator P on X is called a projectionif P 2 = P ; a subspace Y of X is said to be complemented (in X) if Y = PXfor some projection P . If the weak- and weak*-convergence of sequences in X∗

coincide, then X is referred to as a Grothendieck space (`∞ is a nonreflexiveexample of such a space; for other examples see [4], [23], pp. 348-360).

Let E = (E, ‖ ‖) be a Banach lattice, and let E+ denote its positive cone.The norm ‖ ‖ is said to be order continuous if the condition xα ↓ 0 (in E) followsthat ‖xα‖ → 0. E is called a KB-space if every monotone and norm-boundedsequence in E+ is convergent. Every KB-space has an order continuous norm.For every 1 ≤ p < ∞ the (real) space Lp(µ) is a KB-space; c0 is not a KB-spacebut its norm is order continuous; no C(K)-lattice (with K compact (infinite)

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Hausdorff) has an order continuous norm. For other properties of Banach lat-tices with order continuous norms we refer the reader to the monographs ofMeyer-Nieberg [23], and Wnuk [35]. The class X of Banach lattices with ordercontinuous norms contains the class of all (real) Orlicz spaces generated by Orliczfunctions fulfilling the so called ∆2-condition (see e.g. [34]), and, by the remarkfollowing Proposition 2, the class of all Banach spaces with unconditional basescan be regarded as a subclass of X . E is said to be an AM-space [AL-space,resp.] if ‖sup{x, y}‖ = max{‖x‖, ‖y‖} [respectivey, ‖x + y‖ = ‖x‖ + ‖y‖ ] forall x, y ∈ E+; every AM-space is (order isomorphic to) a closed sublattice of aC(K)-lattice, and every AL-space can be represented as an L1(µ)-lattice (see[23]).

3. Reflexivity and the space `1. From the Rosenthal-Dor `1-theorem([21], Theorem 2.e.5) we obtain the following characterization of reflexivity.

Proposition 7. A Banach space X is reflexive if and only if X is weakly se-quentially complete and X contains no subspace isomorphic to `1. In particular,a weakly sequentially complete Banach space is reflexive if and only if it containsno subspace isomorphic to `1.

In this section we show that except for weakly sequentially complete Banachspaces there is a large class of spaces the reflexivity of which can be characterizedby means of the non-containment of only the space `1 as well. To shorten thetext we say that a Banach space X is of class R`1 provided that X is reflexiveor X contains a subspace isomorphic to `1. The first theorem of this sectionapplies to the structure of the class R`1 and will be applied to concrete cases.

Theorem 1. Let X be a Banach space. If X is of class R`1 or X contains acopy of c0 then:

(a) for every positive integer n, the nth conjugate X(n) space is of class R`1

also, and

(b) X is reflexive if and only if one of the following equivalent conditions issatisfied:

(i) X∗ does not contain a subspace isomorphic to `1;

(ii) for some integer n ≥ 0, the space X(n) does not contain a subspaceisomorphic to `1;

(iii) for every integer n ≥ 0, the space X(n) does not contain a subspaceisomorphic to `1.

Moreover, if X is a real Banach lattice, then these conditions are equivalentto:

(iv) X has no quotient isomorphic to c0 and X does not contain a subspaceisomorphic to `1.

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(From the result of Janicka [16] it follows that the above condition (i) is alsoequivalent to the condition that X∗∗ has the so called weak Radon-Nikodymproperty (see [25] and the references therein), but the study of this connectionis laying beyond the scope of this paper).

Proof. It is enough to consider the case X nonreflexive.If X contains a subspace isomorphic to c0 then, by the Bessaga-Pe lczynski

theorem (see Proposition 3), the space X∗ contains a subspace isomorphic to `1.Now let X contain a subspace isomorphic to `1.

Claim. Then X∗ contains a subspace isomorphic to `1 also. By the result ofPe lczynski ([28], Proposition 3.3) we have that X∗ contains a subspace isomor-phic to `1(Γ) with card(Γ) ≥ 2ℵ0 . An alternative (and elementary) proof of theClaim has been communicated to the present author by D. Yost: If X contains`1 then `∞ is a quotient of X∗, whence `1 is a subspace of a quotient of X∗;by the lifting property, `1 is a (complemented) subspace of a subspace of X .(Similarly, by the lifting property of `1 one can obtain an elementary proof ofthe above c0-case)

Part (a) is now obvious. In part (b), observe first that X is reflexive iffX∗ is reflexive iff (by part (a)) condition (i) is true iff (by part (a) again)condition (iii) is true. The implication (iii) ⇒ (ii) is obvious, and from theabove Claim we obtain that (ii) ⇒ (i). The equivalence (i) ⇔ (iv) follows fromProposition 6. 2

The theorem below not only complements the Tzafriri and Meyer-Niebergresult (Proposition 2) in the spirit of Proposition 1, but it extends the resultof Diaz and Fernandez ([7], Theorem 3.1) asserting that a Banach lattice E isreflexive iff it does not contain any isomorphic copy of `1 and any complementedcopy of c0, as well (see the equivalence (i) ⇔ (v)). The equivalence (i) ⇔ (iv)′

for X = E is given in ([7], Theorem 3.3).

Theorem 2. Let X be a closed subspace of a Banach lattice E. If X is com-plemented in E or if the norm of E is order continuous, then the followingconditions are equivalent.

(i) X is reflexive.

(ii) X∗ does not contain a subspace isomorphic to `1.

(iii) For every (for some) positive integer n, the space X(n) does not contain asubspace isomorphic to `1.

(iv) X does not contain subspaces isomorphic to c0 or `1.

(v) X has no quotient isomorphic to c0 and X does not contain a subspaceisomorphic to `1.

If the norm of E is order continuous, then condition (iv) can be replaced by

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(iv)’ X does not contain complemented subspaces isomorphic to c0 or `1.

Proof. By Proposition 2, we may apply part (b) of Theorem 1. Part (iv)’ followsfrom (iv) and Proposition 4. 2

In particular, the above theorem can be applied to complemented subspacesof (real) C(K)-spaces, or Orlicz spaces, and to subspaces of (real) Banach spaceswith unconditional bases, or Orlicz spaces generated by Orlicz functions fulfillingthe ∆2-condition.

From the equivalence (iv) ⇔ (v) in Theorem 2 we immediately obtain thefollowing corollary (compare it with the classical Sobczyk theorem [21], Theorem2.f.5, and the result of Diaz and Fernandez cited above):

Corollary 1. Let X be a closed subspace of a Banach lattice E, and assumethat X is complemented in E or that the norm of E is order continuous. If Xdoes not contain a subspace isomorphic to `1, then the following conditions areequivalent.

(i) X contains a subspace isomorphic to c0.(ii) X has a quotient isomorphic to c0.

Since every weakly sequentially complete Banach space is of class R`1 (seeProposition 7), from Theorem 1 we obtain the following

Corollary 2. Let X be a weakly sequentially complete Banach space. ThenX is reflexive if and only if for every (for some) integer n ≥ 0 the space X(n)

contains no subspace isomorphic to `1.

The next result deals with the characterization of reflexivity of quotients ofBanach lattices that have duals with order continuous norms. Recall that (see[23]; Theorems 2.4.14, 2.5.6, Proposition 2.3.11)

(*) if E is a Banach lattice, then its conjugate space E∗ has order continuousnorm iff E∗ is a KB-space iff E∗ is weakly sequentially complete iff E hasno complemented subspace (or, a closed sublattice) isomorphic to `1.

Theorem 3. Let E be a Banach lattice such that E∗ has order continuousnorm (see (∗) above). If X is a quotient space of E, then the following statementsare equivalent.

(i) X is reflexive (and thus, X∗ fulfils the equivalent conditions of Theorem 2).

(ii) X∗ contains no [complemented] subspace isomorphic to `1.

(iii) X [and X∗∗ also] contains no subspace isomorphic to c0.

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Proof. The equivalence (i) ⇔ (ii) follows from Proposition 7; the bracketedcase follows from Proposition 4 (b). The implication (i) ⇒ (iii) is obvious, and(iii) ⇒ (i) follows from the result due to Kuhn, Niculescu and Meyer-Nieberg([23], Theorem 3.4.18) asserting that every operator E → X is weakly compactwhenever E∗ has order continuous norm and X contains no copy of c0. Thebracketed equivalence of (ii) and (iii) follows from Proposition 3. 2

Remark 1. The equivalence (i) ⇔ (iii) in the above Theorem 3 is a genere-lization of the result that can be obtained from the theorem of Pe lczynski ([27],Theorem 1).

4. Applications to the Separable Quotient Problem. Now we willapply the results of Section 3 to the Separable Quotient Problem by providingsome affirmative answers to the question of

(SQ) whether every infinite-dimensional Banach space possesses a separableand infinite-dimensional quotient?

This problem was studied by a number of authors. In the survey article [24]Mujica presented both general results on this subject and its state in 1996 (forthe locally convex case see [32] and the references therein; (SQ) for Banach lat-tices was studied in [36]). By the use of the results obtained in previous section,we shall show that many Banach spaces possess ”nice” separable quotients. Forthis purpose we introduce the following notion.

Definition 1. We say that a Banach space X possesses a perfect quotientif X can be mapped continuously onto one of the spaces: c0, `1, or a reflexiveBanach space Y with an unconditional basis (yn); if (yn) is not necessarilyunconditional, then we say that X has an almost perfect quotient.

The word ”perfect” in the above definition has been borrowed from thetheory of Banach lattices: c0 and `1 are the examples of the so called perfect (=order reflexive) Banach lattices, which means that the order continuous bidual ofthe underlined space coincides with the given one (see [2], p. 59); by the remarkfollowing Proposition 2, every reflexive Banach space with an unconditionalbasis can be regarded as a perfect Banach lattice also.

The study of Banach spaces that have quotients with bases is justified bythe deep theorem of Johnson and Rosenthal ([17], Theorem IV.1) which assertsthat every separable Banach space possesses a quotient with a basis. The readershould also note that from Proposition 4 and James’s characterization of reflex-ivity of Banach spaces with unconditional bases [21] it follows that a Banachspace X has a perfect quotient iff X has a quotient with an unconditional basis.In this section we indicate concrete classes of Banach spaces possessing perfector almost perfect quotients.

We shall now list (useful for our purposes) cases for X to have a perfect oran almost perfect quotient.

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Lemma 1. Let X be a Banach space.

(a) X possesses a perfect quotient in the following cases:

(i) X has a subspace isomorphic to `1 (then X has `2 as a quotient);

(ii) X has a subspace isomorphic to `∞ (then X has `2 as a quotient);

(iii) X has a subspace isomorphic to c0 (then X has `2 or c0 as a quotient);

(iv) X∗ has a subspace isomorphic to `1 (then X has `2 or c0 as a quotient);

(v) X∗ has a subspace isomorphic to c0 (then X has `1 as a quotient).

(b) X possesses an almost perfect quotient in the following cases:

(vi) X is weakly sequentially complete (then X has `2 or another reflexivespace with a basis as a quotient);

(vii) X∗ contains a subspace V that is weakly sequentially complete (thenX has c0, `2, or another reflexive space with a basis as a quotient);

(viii) X is a Grothendieck space (then X has `2, or another reflexive spacewith a basis as a quotient).

Proof. (i) By the result of Pe lczynski ([28], Proposition 3.3; cf. [8], Theorem 2),the space X∗ contains an isomorphic copy of L1[0, 1], and hence a copy of `2;now we apply Proposition 5.(ii) Since every isomorphic copy of `∞ is complemented in X , and `∞ has `2 asa quotient ([21], p. 111); alternatively, we apply the above part (i).(iii) and (iv): In case (iii), X∗ contains a complemented copy of `1 (by Propo-sition 3); in both cases we apply Proposition 6 and the result of part (i).(v) By Proposition 3, the space X contains a complemented copy of `1.(vi) By Proposition 7, X is reflexive (then we apply Proposition 5) or X containsan isomorphic copy of `1 (then we apply the above part (i)).(vii) By Proposition 7, X∗ contains a reflexive subspace (then we apply Propo-sition 5) or an isomorphic copy of `1 (then we apply the above part (iv)).(viii) If X a Grothendieck space, then X∗ is weakly sequentially complete; nowwe apply part (vii) keeping in mind that every operator from X into c0 is weaklycompact ([23], Proposition 5.3.10). 2

Let X be a closed subspace of a Banach lattice E. We see that if thespaces X and E fulfil the assumptions of Theorem 2 then, by Lemma 1 (iv)and Proposition 5, X has an almost perfect quotient. The two theorems belowdescribe this situation in more detail (a particular case was discussed in [36],Theorem 2).

Theorem 4. Let X be a closed subspace of a Banach lattice E with ordercontinuous norm. Then, for every integer k ≥ 0, the even conjugate spaceX(2k) has an almost perfect quotient, and the odd conjugate space X(2k+1) hasa perfect quotient.

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Theorem 5. Let E be a Banach lattice, and let X be a Banach space.

(a) If X is isomorphic to a complemented subspace of E then, for every integern ≥ 0, the nth conjugate space X(n) has a perfect quotient.

(b) If E∗ has order continuous norm and X is a quotient of E then, for everyinteger k ≥ 0, the even conjugate space X(2k) has a perfect quotient, andthe odd conjugate space X(2k+1) has an almost perfect quotient.

Remark 2. The assumption in Theorem 5 (b) regarding order continuity ofthe norm of E∗ is essential. Indeed, the Banach lattice E := `1 can be mappedcontinuously onto the dual G∗ of the separable and reflexive Banach space G(constructed by Gowers and Maurey in [11], cf. [1]) not containing any uncon-ditional basic sequence. We thus have that the quotient space X := G∗ of Ecannot possess a perfect quotient; obviously, E∗ = `∞ has no order continuousnorm.

Proof of Theorem 4. If X is not reflexive then, by Theorem 2 (iv)’, the space Xhas a complemented subspace isomorphic to c0 or `1. It follows that for everypositive integer n the space X(n) has a complemented subspace isomorphic to`1 or `∞; thus, by Lemma 1 (ii), X(n) has `1 or `2 as a quotient.

If X is reflexive then, by Proposition 5, for every n ≥ 0 the space X(n) has analmost perfect quotient. Moreover, by the result of Figiel, Johnson and Tzafriri([9], Theorem 4.1), the space X contains an unconditional basic sequence. Fromthe second part of Proposition 5 it now follows that, for every positive integerk, the space X(2k+1)(= X∗) has a perfect quotient. 2

Proof of Theorem 5. Part (a). Since the nth conjugate space X(n) is isomorphicto a complemented subspace of the Banach lattice E(n), we may assume thatn = 0 (i.e., X(n) = X). If X is nonreflexive then, by Theorem 2 (v), we mayonly consider the case X contains a subspace isomorphic to `1; by Lemma 1 (a)(i), then X has `2 as a quotient.

Now consider the case X reflexive. By the theorem of Johnson and Tzafriri([2], Theorem 14.16), each of the spaces X and X∗ is isomorphic to a comple-mented subspace of a KB-space (and hence, we may assume that E has ordercontinuous norm). By Figiel-Johnson-Tzafriri’s theorem quoted in the proof ofTheorem 4, the both spaces, X and X∗, contain unconditional basic sequencesand therefore, by Proposition 5, they possess perfect (reflexive) quotients.

Part (b). Since X∗ embeds continuously into E∗, from Theorem 4 it followsthat it is enough to consider only the case k = 0. If X is reflexive then, by thearguments similar to these of part (a), the space X has a perfect quotient. Thecase X nonreflexive follows from Theorem 3 and Lemma 1 (iii). 2

Since the dual of every AM-space (being an AL-space) has order continuousnorm, from Theorem 5 (b) we immediately obtain:

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Corollary 3. Let X be a quotient of an AM-space. Then, for every integern ≥ 0, the space X(n) has an almost perfect quotient, and a perfect quotient forn even.

5. Final remarks. The reader should note that our results have referenceto the class of Banach spaces that are reflexive, or contain c0 or `1 (this classdoes not exhaust of all Banach spaces: see [21], pp. 25-26, or [10]). One mayobviously set the question whether the conclusions of Section 4 remain valid if re-flexivity is replaced by a weaker condition; more precisely, can a (nonseparable)Banach space X be mapped continuously onto a separable Banach space when-ever X contains a reflexive subspace?, or, in the context of the result by Rosen-thal presented in Proposition 5, if X contains a reflexive subspace, does it followthat X∗ contains a reflexive subspace as well? Although the first question seemsto be open, the second one has a surprising negative answer: for every p ∈ (1,∞)there exists a (separable) Banach space Yp such that Y ∗

p is isomorphic to `1 andevery infinite-dimensional subspace of Yp contains a subspace isomorphic to `p;a class of such spaces was constructed in 1980 by Bourgain and Delbaen [5] andstudied in 2000 by Haydon [14]. One should also note that, by the main resultof Herman and Whitley [15] and Proposition 5, the above two questions havepositive answers for the class of quasireflexive (and nonreflexive) spaces.

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INSTITUTE OF MATHEMATICS, UNIVERSITY OF ZIELONA GORA,PODGORNA 50, 65-246 ZIELONA GORA, POLAND

(INSTYTUT MATEMATYKI, UNIWERSYTET ZIELONOGORSKI)e-mail: M.Wojtowicz@im.uz.zgora.pl

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