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Uniform Boundedness and Twisted Sums of Banach Spaces Houston Journal of Mathematics c 2004 University of Houston ­ Volume 30, No. 2, 2004 UNIFORM BOUNDEDNESS AND TWISTED SUMS OF BANACH SPACES FELIX´ CABELLO SANCHEZ´ AND JESUS´ M. F. CASTILLO Communicated by Gilles Pisier Abstract. We construct a Banach space X admitting an uncomplemented copy of l1 so that X/l1 = c0. To do that we study the uniform boundedness principles that arise when one considers exact sequences of Banach spaces; as well as several elements of homological algebra applied to the construction of nontrivial twisted sum of Banach spaces. The combination of both elements allows one to determine the existence of nontrivial twisted sums for almost all combinations of classical Banach spaces. 1. Introduction The objective of this paper is to be able to construct a nontrivial twisted sum of l1 and c0; precisely, to show the existence of a Banach space X admitting an un- complemented copy of l1 so that X/l1 = c0. To do that we need to develop certain uniform boundedness principles that arise when one considers exact sequences of Banach spaces and some homological constructions applied to the construction of twisted sum of Banach spaces. The combination of both approaches will show the existence of nontrivial twisted sums for almost all combinations of classical Banach spaces. This is not the first time that the existence “uniform boundedness principles” for twisted sums of (quasi) Banach spaces has been considered. Doma´nski[12] developed similar principles; while some of these topics have been recently consid- ered in [20]. There, some kind of uniform boundedness principle is used to attack the problem of which spaces admit a nontrivial twisted sum with l2. However, 2000 Mathematics Subject Classification. 46B25, 46A16, 46B08, 46M10. Supported in part by DGICYT project BFM2001-0813. 523 524 F. CABELLO AND J.M.F. CASTILLO those papers are not easy to read, the former because of the lack of an algebraic approach forces the author to develop time after time some ingenious devices of his own to resolve situations; and the latter because of the rather laconic form in which the difficult point of exactly how one can paste together the pieces (or pro- ceed to obtain local information) is treated. Moreover, the approaches of those two papers are almost disjoint: while Kalton and PeÀlczy´nskiuse hard local theory of Banach spaces (behaviour of cotype constants, Maurey’s theorem ...), Doma´nski has a more operator-ideal oriented point of view. In conclusion, that while there is no doubt that some experts are aware of the existence of some principles (see, e.g. [14], where the author mentions that “some localization technique” allows to conclude his argument) we think that a comprehensive theory for such principles has not been yet developed. Moreover, none of the known principles is able to produce a nontrivial twisted sum of l1 and c0. Such a construction can be seen (extremely) implicit in [2]; and also in [3]. In this paper we present a unified, and far simpler, approach to the topic based on the use of the elements of homological algebra and the theory of quasi-linear maps. Let us briefly outline the contents of the paper. Section 2 of the paper in- troduces the background required (some elements of homological algebra and an adaptation to the locally convex setting of the theory of quasi-linear maps). Section 3 contains the uniform boundedness principles. The section 4 contains nontrivial (in both senses of the word) examples of twisted sums of classical Ba- nach spaces, culminating in the promised nontrivial exact sequence 0 l → 1 → X c 0. The last section of the paper characterizes -spaces showing that → 0 → L1 Z is a -space if and only if Ext(R, Z) = 0 for every reflexive space R. L1 2. Background, with some applications In the sequel, the word operator means linear continuous map. For general in- formation about exact sequences the reader can consult [13]. Information about categorical constructions in the Banach space setting can be found in the mono- graph [5] or in Johnson paper [14]. A diagram 0 Y X Z 0 of Banach → → → → spaces and operators is said to be an exact sequence if the kernel of each arrow coincides with the image of the preceding. This means, by the open mapping theorem, that Y is (isomorphic to) a closed subspace of X and the corresponding quotient is (isomorphic to) Z. We shall also say that X is a twisted sum of Y and Z or an extension of Y by Z. Two exact sequences 0 Y X Z 0 → → → → and 0 Y X Z 0 are said to be equivalent if there is an operator → → 1 → → UNIFORM BOUNDEDNESS PRINCIPLES 525 T : X X making the diagram → 1 0 Y X Z 0 → → → → T k ↓ k 0 Y X Z 0 → → 1 → → commutative. The “three-lemma” (see [13, lemma 1.1]) and the open mapping theorem imply that T must be an isomorphism. The exact sequence 0 Y → → X Z 0 is said to split if it is equivalent to the trivial sequence 0 Y → → → → Y Z Z 0. This already implies that the twisted sum X is isomorphic ⊕ → → to the direct sum Y Z (the converse is not true). Given two Banach spaces Y ⊕ and Z we denote by Ext(Z, Y ) the set of exact sequences 0 Y X Z 0 → → → → modulo equivalence. Thus, Ext(Z, Y ) = 0 means that every exact sequence 0 → Y X Z 0 splits. → → → The pull-back square. Let A : U Z and B : V Z be two operators. The → → pull-back of A, B is the space PB = (u, v): Au = Bv U V endowed { } { } ⊂ × with the relative product topology, together with the restrictions of the canonical projections of U V onto, respectively, U and V . If 0 Y X Z 0 is × → → → → an exact sequence with quotient map q and T : V Z is an operator and PB → denotes the pull-back of the couple q, T then the diagram { } 0 Y X Z 0 → → → → T k ↑ ↑ 0 Y PB V 0 → → → → is commutative with exact rows. The three-lemma implies that the operator PB X is onto if and only T is. It is well-known that the pull-back sequence → splits if and only if the operator T can be lifted to X; i.e., there exists an operator τ : V X such that q τ = T . → ◦ 2.1. Application: nontrivial twisted sums of c0 and l . Consider a non- ∞ trivial sequence 0 c JL l (I) 0, where I has the power of continuum → 0 → 2 → 2 → (see [15]). The pull-back diagram 0 c JL l (I) 0 → 0 → 2 → 2 → q k ↑ ↑ 0 c0 PB l 0 → → → ∞ → where q : l l2(I) is a quotient map gives a nontrivial exact sequence as we ∞ → now show. The space PB cannot be isomorphic to c0 l since JL2 cannot be a ⊕ ∞ quotient of c0 l : if Q is such a quotient map, since no nonseparable subspace ⊕ ∞ of JL2 is isomorphic to a subspace of l ([15]), then by Rosenthal’s theorem [24] ∞ 526 F. CABELLO AND J.M.F. CASTILLO Q l is weakly compact and thus its range is separable, and the same occurs to | ∞ Q . |c0 2.2. Application: nontrivial twisted sums of c0 and L1(µ). Start with a nontrivial exact sequence 0 c0 JL c0(I) 0 (see again [15]), and → → ∞ → → observe that for some finite measure µ there exists a quotient map from L1(µ) onto c0(I) (see [24]). The space L1(µ) is WCG since µ is finite. The pull-back diagram 0 c0 JL c0(I) 0 → → ∞ → → q k ↑ ↑ 0 c PB L (µ) 0 → 0 → → 1 → produces a non WCG space PB that cannot therefore be isomorphic to the prod- uct c L (µ). 0 ⊕ 1 The push-out square. Let A : K X and B : K M be two operators. The → → push-out of A, B is the space PO = (M X)/∆, where ∆ = (Ak, Bk): k K { } ⊕ { ∈ } endowed with the quotient topology, together with the restrictions of the quotient map to, respectively, M and X. If 0 Y X Z 0 is an exact sequence → → → → with injection j and T : Y M is an operator and PO denotes the push-out of → the couple j, T then the diagram { } 0 Y X Z 0 → → → → T ↓ ↓ k 0 M PO Z 0 → → → → is commutative with exact rows. Note that the operator X PO is an isomorphic → embedding if and only if T is. The push-out sequence splits if and only if the operator T can be extended to X; i.e., there exists an operator τ : X M such → that τ j = T . ◦ Zero-linear maps and extensions. The by now classical theory of Kalton and Peck [16, 19] describes short exact sequences of quasi-Banach spaces in terms of the so-called quasi-linear maps. The theory of quasi-linear maps can be easily adapted to the locally convex (Banach) setting as follows (see [4]). A map F : Z Y acting between Banach spaces is said to be zero-linear if it is homogeneous → and satisfies that, for some constant K and all points x Z, one has i ∈ n n n F xi F (xi) K xi . ­ À ! − ­ ≤ À k k! ­ i=1 i=1 ­ i=1 ­ X X ­ X The infimum of the­ constants K as above is­ denoted Z(F ) and referred to as the ­ ­ zero-linearity constant of F . As in the quasi-linear case, zero-linear maps give rise UNIFORM BOUNDEDNESS PRINCIPLES 527 to twisted sums: given a zero-linear map F : Z Y , it is possible to construct → a twisted sum, which we shall denote by Y Z , endowing the product space ⊕F Y Z with the quasi-norm (y, z) = y F z + z which is always equivalent × k kF k − k k k to a norm that makes Y Z into a Banach space.
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