- Email: [email protected]
The behaviour of quasi-linear maps on C(K)-spaces
J. Math. Anal. Appl. 475 (2019) 1714–1719
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
The behaviour of quasi-linear maps on C(K)-spaces ✩ Félix Cabello Sánchez a , Jesús M.F. Castillo a,∗ , Alberto Salguero-Alarcón b a b
Instituto de Matemáticas, Universidad de Extremadura, Avenida de Elvas, 06071-Badajoz, Spain Departamento de Matemáticas, Universidad de Extremadura, Avenida de Elvas, 06071-Badajoz, Spain
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 29 November 2018 Available online 23 March 2019 Submitted by R.M. Aron
In this paper we combine topological and functional analysis methods to prove that a non-locally trivial quasi-linear map defined on a C(K) must be nontrivial on a subspace isomorphic to c0 . We conclude the paper with a few examples showing that the result is optimal, and providing an application to the existence of nontrivial twisted sums of 1 and c0 . © 2019 Elsevier Inc. All rights reserved.
Keywords: Banach spaces of continuous functions Quasi-linear maps
1. Introduction and preliminaries This paper treats the behaviour of quasi-linear maps defined on C(K)-spaces. More precisely, we will prove that a quasi-linear map Ω : C(K) → Y is either uniformly trivial on finite dimensional subspaces or nontrivial on a certain copy of c0 . This behaviour of quasi-linear maps on C(K)-spaces has consequences about the existence and properties of exact sequences 0 −−−−→ Y −−−−→ X −−−−→ C(K) −−−−→ 0
(1.1)
via the correspondence between exact sequences and quasi-linear maps we describe below. Recall that an exact sequence of Banach (rsp. quasi-Banach) spaces is a diagram 0 −−−−→ Y −−−−→ X −−−−→ Z −−−−→ 0
(1.2)
formed by Banach (resp. quasi-Banach) spaces and linear continuous operators in which the kernel of each arrow coincides with the image of the preceding one. The middle space X is usually called a twisted sum of ✩ This research has been supported in part by project MTM2016-76958-C2-1-P and Project IB16056 de la Junta de Extremadura. The third author is supported by a grant associated to Project IB16056. * Corresponding author. E-mail addresses: [email protected] (F. Cabello Sánchez), [email protected] (J.M.F. Castillo), [email protected] (A. Salguero-Alarcón).
https://doi.org/10.1016/j.jmaa.2019.03.040 0022-247X/© 2019 Elsevier Inc. All rights reserved.
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Y and Z. By the open mapping theorem, Y must be isomorphic to a subspace of X and Z to the quotient X/Y . It was the discovery of Kalton [6,8] that exact sequences of quasi-Banach spaces (1.2) correspond to a certain type of non-linear maps Z → Y called quasi-linear maps, which are homogeneous maps for which there is a constant Q such that Ω(x + y) − Ω(x) − Ω(y) ≤ Q (x + y) for every x, y ∈ X. The least constant Q that satisfies the previous inequality is called the quasi-linearity constant of Ω. Be warned of a surprising fact [1]: the space X in an exact sequence (1.2) is not necessarily a Banach space when Y, Z are Banach spaces. Kalton and Roberts [9] however proved in a deep theorem that, in combination with [1], means that every quasi-linear map Ω defined on an L∞ -space enjoys the additional property that there is a constant K such that for every n ∈ N and every x1 , . . . , xn ∈ X one has n n n xi − Ω(xi ) ≤ K xi , Ω i=1
i=1
(1.3)
i=1
and, consequently, exact sequences of quasi-Banach spaces 0 −−−−→ Y −−−−→ X −−−−→ Z −−−−→ 0 in which Z is an L∞ -space and Y is a Banach space also have X a Banach space. Thus, we will freely speak about quasi-linear maps Ω : C(K) → Y from a space of continuous functions to a Banach space with the understanding that the associated exact sequence 0 → Y → X → C(K) → 0 is a sequence of Banach spaces. We will write 0 → Y → X → Z → 0 ≡ Ω to say that the exact sequence and the quasi-linear map correspond one to each other. j An exact sequence 0 → Y → X → Z → 0 ≡ Ω is said to be trivial, or to split, if the injection j admits a left inverse; i.e., there is a linear continuous projection P : X → Y along j. In terms of Ω, this means that there exists a linear map L : Z → Y such that Ω − L = supx≤1 Ωx − Lx < +∞. The sequence is said to be λ-trivial if Ω − L ≤ λ. Definition 1. We will say that a quasi-linear map Ω : X → Y is locally trivial if there is a constant λ such that whenever F is a finite dimensional subspace of X then the restriction Ω|F : F → Y is λ-trivial. This notion was introduced by Kalton [7] who proved that an exact sequence 0 → Y → X → Z → 0 of Banach spaces is locally trivial if and only if its dual exact sequence 0 → Z ∗ → X ∗ → Y ∗ → 0 is trivial. It is also shown in [7] that a locally trivial sequence 0 → Y → X → Z → 0 in which Y is complemented in its bidual must be trivial. 2. A dichotomy for C(K) spaces Our aim is to prove the following dichotomy: Theorem 2.1. A quasi-linear map Ω : C(K) → Y either is locally trivial or admits a subspace isomorphic to c0 on which its restriction is not locally trivial. Before embarking in the proof we need a technical lemma: Lemma 2.2. Let Ω be a quasi-linear map on X with quasi-linearity constant Q. Let x1 , . . . , xm be points so that Ω|[x1 ,...,xm ] is λ-trivial, and let δ be the Banach-Mazur distance between [x1 , . . . , xm ] and m 1 . If
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y1 , . . . , ym are norm one points so that yi − xi ≤ ε/mδ for all i = 1, . . . , m, then Ω|[y1 ,...,ym ] is λ + 2+ε 1−ε Q-trivial. Proof. Let L be a linear map on [x1 , . . . , xm ] so that Ω − L ≤ λ. Set L the linear map L yi = Lxi + Ω(yi − xi ) on [y1 , . . . , ym ]. If we call (∗) = Ω( λi yi ) − L ( λi yi ), then (∗) = Ω( λi yi ) − L( λi xi ) − λi Ω(yi − xi ) ≤ Ω( λi yi ) − Ω( λi xi ) − λi Ω(yi − xi ) + Ω( λi xi ) − L( λi xi ) ≤ Ω( λi yi ) − Ω( λi xi ) − Ω( λi (yi − xi )) + Ω( λi (yi − xi )) − λi Ω(yi − xi ) +λ λi xi ≤ Q λi yi + λi (yi − xi ) + (m − 1) |λi |yi − xi + λ λi yi ≤ Q λi yi + m |λi |yi − xi + λ λi yi ε ≤ Q λi yi + m |λi | + λ λi yi mδ 1 ≤ Q λi yi + λi yi + λ λi yi 1−ε 2+ε Q λi yi 2 ≤ λ+ 1−ε We pass to the proof of the theorem. Proof. We make the proof for the Cantor set Δ. Let Ω : C(Δ) → Y be a quasi-linear map. Split Δ = Δ− ∪Δ+ in right and left halves Δ− = Δ ∩ [0, 1/3] and Δ+ = Δ ∩ [2/3, 1]. Now, let J+ = {f ∈ C(Δ) : f|Δ− = 0} = C(Δ+ ) and J− = {f ∈ C(Δ) : f|Δ+ = 0} = C(Δ− ). Claim. If Ω+ = Ω|J+ ≡ 0 and Ω− = Ω|J− ≡ 0 then Ω ≡ 0. Proof. If there are linear maps L+ : C(Δ+ ) → Y and L− : C(Δ− ) → Y so that Ω+ − L+ ≤ M and Ω− − L− ≤ N , set the linear map L : C(Δ) −→ Y
,
Lf = L+ (f 1Δ+ ) + L− (f 1Δ− )
Since f = f 1Δ+ + f 1Δ− for every f ∈ C(Δ) one has Ωf − Lf ≤ Ω(f ) − Ω(f 1Δ+ ) − Ω(f 1Δ− )+ + Ω(f 1Δ+ ) − L+ (f 1Δ+ ) + Ω(f 1Δ− ) − L− (f 1Δ− ) ≤ ≤ Q f 1Δ+ + f 1Δ− + M f 1Δ+ + N f 1Δ− ≤ ≤ (M + N + 2Q)f 2 So, if Ω is non-trivial then either Ω+ o Ω− is non-trivial. For given ξ ∈ / Δ we will denote Δξ− = Δ ∩ [0, ξ] and Δξ+ = Δ ∩ [ξ, 1]. Lemma 2.3. Let Ω : C(Δ) → Y be non-locally trivial. Then there is ξ ∈ [0, 1] so that • If ξ ∈ / Δ then Δ = Δξ+ ∪ Δξ− with Δξ+ ∩ Δξ− = ∅ and both Ω|C(Δξ+ ) and Ω|C(Δξ− ) are non-locally trivial.
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• If ξ ∈ Δ then Δ = Δξ+ ∪ Δξ− with Δξ+ ∩ Δξ− = {ξ}; and if we call Jξ+ = {f ∈ Δ : f|Δξ− = 0} and Jξ− = {f ∈ Δ : f|Δξ+ = 0} then ker δξ = Jξ+ ∪ Jξ− and both Ω|Jξ+ and Ω|Jξ− are non-locally trivial. Proof. Split Δ = Δ+ ∪ Δ− . If both Ω+ and Ω− are non-trivial, then we are done: ξ = 1/2 works and the intersection is empty. Assume, on the contrary, that one of them is trivial, say, Ω+ . Pick λ1 so that Ω+ is λ1 -trivial but not (λ1 − 1)-trivial. Iterate the argument. If both Ω−− and Ω−+ (the restrictions of Ω to Δ−− and Δ−+ ) are non-trivial, then we are done and Δ− splits in two pieces with empty intersection Δ−− and Δ−+ ∪ Δ+ , so that Ω is non-trivial when restricted to them. Otherwise, we continue. If the decomposition method does not stop, let ξ ∈ {−1, 1}N be the element that represents the non-trivial choices; i.e., the choices of the pieces on which the restriction of Ω was non-trivial. We can assume without loss of generality that −ξ represents the trivial choices and Ω is λn -trivial but not (λn − 1)-trivial on Δξ(1),...,ξ(n−1),−ξ(n) , and also non-trivial on Δξ(1),...,ξ(n) . Now, either supn λn < +∞ or supn λn = +∞. Claim. If supn λn = λ < +∞ then Ω is locally trivial. Proof. Pick functions f1 , . . . , fm . For ε > 0 small enough each of the functions fj is at distance ε of a function of the form fj (ξ)χ + gj with gj supported in a union A = Δ−ξ(1) ∪ Δξ(1),−ξ(2) ∪ · · · ∪ Δξ(1),ξ(2),...,ξ(n−1),−ξ(n) and χ is the characteristic function of Δ \ A. Since Ω is λ-trivial on A and is also 1-trivial on the onedimensional subspace generated by χ, Ω is μ-trivial (with μ = λ + 1 + 2Q) on the finite-dimensional subspace [χ + gj ] generated by them. Therefore, it must be μ + Q(1 + ε) -trivial on [f1 , . . . , fm ]. 2 It just remains that supn λn = +∞. Set the decomposition Δξ− = Δ−ξ(1) ∪ Δξ(1),−ξ(2) ∪ Δξ(1),ξ(2),−ξ(3) ∪ . . . Δξ+ = Δξ(1),ξ(2) ∪ Δξ(1),ξ(2),ξ(3) ∪ . . . with ξ ∈ Δξ+ ∩ Δξ− and Ω is non-trivial on both. We are ready to conclude the proof of the lemma. The different pieces Δξ(1),ξ(2),...,ξ(n−1),−ξ(n) on which Ω is not (λn −1)-trivial are disjoint. So when one considers a finite dimensional subspace of C(Δξ(1),ξ(2),...,−ξ(n) ) n where Ω is not (λn − 1)-trivial, it is contained in a subspace m ∞ of C(Δξ(1),ξ(2),...,−ξ(n) ) on which Ω is not mn (λn − 1)-trivial. The subspaces ∞ are thus disjointly supported, so their closed span yields a copy of c0 where Ω cannot be locally trivial. 2 Now, if C(K) is isomorphic to C(Δ), pick Ω : C(K) → Y non-locally trivial and α : C(Δ) → C(K) an isomorphism. Then Ω = Ω α is not locally trivial, so there is an embedding j : c0 → C(Δ) so that Ωj is not locally trivial. Then Ω αj is not locally trivial and αj : c0 → C(K) is the embedding one needs. In general, let Ω : C(K) → Y be non-locally trivial. Find for each n = 1, 2, ... a finite dimensional subspace Fn ⊂ C(K) so that Ω|Fn is not n-trivial. Let X = [∪n Fn ]. This is a separable subspace of C(K) on which Ω is not locally trivial. Let A be the separable subalgebra that X generates in C(K). This A is isometric to C(H) for some compact H, necessarily metrizable, and the restriction of Ω to C(H) cannot be locally trivial as well. If H is countable, then C(H) = C(α) for some countable ordinal α and the result is clear. If H is uncountable then C(H) = C(Δ) by Milutin’s theorem and the result is also clear. 2 Corollary 2.4. A quasi-linear map defined on a C(K)-space is locally trivial if and only if every restriction to every copy of c0 is locally trivial.
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3. Consequences First of all, let us show that the result just obtained is the only possible one. We have proved that a quasi-linear map Ω : C(K) → ♦ is locally trivial if and only if every restriction to every copy of c0 is locally trivial. It is however false that a quasi-linear map Ω : C(K) → ♦ must be trivial when every restriction to every copy of c0 is trivial, and a simple example like 0 −−−−→ c0 −−−−→ ∞ −−−−→ ∞ /c0 −−−−→ 0 proves it: by Sobczyk’s theorem any restriction to a separable subspace of ∞/c0 must be trivial. The word “locally” is therefore not superfluous. It is also false that a locally trivial quasi-linear map Ω : C(K) → ♦ must have some trivial restriction to some copy of c0 , as it is proved by the example in [3] of a non-trivial locally trivial sequence 0 −−−−→ C[0, 1] −−−−→ ♦ −−−−→ c0 −−−−→ 0 having strictly singular quotient map (from now on called strictly singular sequences). The quotient map is not an isomorphism when restricted to any infinite dimensional subspace and consequently the associated quasi-linear map is not trivial when restricted to any infinite dimensional subspace. Such quasi-linear maps will be also called strictly singular in what follows. To conclude this analysis, observe that strictly singular but not locally trivial maps Ω : c0 → Y do exist: any exact sequence 0 → K → 1 → c0 → 0 works (it is not locally trivial because a L∞ -script space cannot be a complemented subspace of a L1 -space); there also exist strictly singular sequences in which the subspace is complemented in its bidual: pick any notrivial sequence 0 → 2 → ♦ → c0 → 0 and apply the method in [4] to obtain a strictly singular sequence 0 −−−−→ ∞ (Γ, 2 ) −−−−→ ♣ −−−−→ c0 −−−−→ 0 Such sequences cannot locally split. Now we pass to applications. Exact sequences 0 −−−−→ 1 −−−−→ ♠ −−−−→ c0 −−−−→ 0
(3.1)
are somewhat a mystery, even though their existence has been shown in [2,3]. Both papers contain different although complementary methods to show the existence of non-trivial sequences 0 −−−−→ L1 (0, 1) −−−−→ ♣ −−−−→ C[0, 1] −−−−→ 0
(3.2)
Since L1 and 1 (resp. C[0, 1] and c0 ) have the same local structure, one can apply the local theory of exact sequences [2] to pass from a non-trivial twisting of L1 and C[0, 1] to a non-trivial twisting of 1 and c0 . The final sequence, however, remains invisible since it has emerged via an abstract existence result. Theorem 2.1 yields a much more explicit way to get such sequences. Let us call Ω the exact sequence (3.2), which is not locally trivial because L1 is complemented in its bidual, and therefore there is an injection j : c0 → C[0, 1] so that Ωj, namely the lower sequence in the commutative diagram 0 −−−−→ L1 (0, 1) −−−−→
♣ −−−−→ C[0, 1] −−−−→ 0 ≡ Ω ⏐j ⏐ ⏐ ⏐
0 −−−−→ L1 (0, 1) −−−−→ ♥ −−−−→ is not locally trivial. This implies that its dual sequence
c0
−−−−→ 0 ≡ Ωj
(3.3)
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0 −−−−→ 1 −−−−→ ♥∗ −−−−→ L1 (0, 1)∗ −−−−→ 0 ≡ j∗ Ω∗ is not locally trivial. Since L1 (0, 1)∗ is a C(K)-space, Theorem 2.1 applies again to obtain an injection ı : c0 → L1 (0, 1)∗ such that the restriction (Ωj)∗ ı, namely, the lower sequence in the commutative diagram 0 −−−−→ 1 −−−−→
♥∗ −−−−→ L1 (0, 1)∗ −−−−→ 0 ≡ (Ωj)∗ ⏐ı ⏐ ⏐ ⏐
0 −−−−→ 1 −−−−→ ♦ −−−−→
c0
(3.4)
−−−−→ 0 ≡ (Ωj)∗ ı
is not locally trivial. In particular, it is not trivial. One can show that the middle space ♠ in an exact sequence (3.1) cannot be isomorphic to 1 ⊕ c0 [5]. However, it is open to decide whether strictly singular sequences (3.1) exist. References [1] F. Cabello Sánchez, J.M.F. Castillo, Duality and twisted sums of Banach spaces, J. Funct. Anal. 175 (2000) 1–16. [2] F. Cabello Sánchez, J.M.F. Castillo, Uniform boundedness and twisted sums of Banach spaces, Houston J. Math. 30 (2004) 523–536. [3] F. Cabello Sánchez, J.M.F. Castillo, N.J. Kalton, D.T. Yost, Twisted sums with C(K) spaces, Trans. Amer. Math. Soc. 355 (2003) 4523–4541. [4] J.M.F. Castillo, Y. Moreno, Strictly singular quasi-linear maps, Nonlinear Anal. – TMA 49 (2002) 897–904. [5] J.M.F. Castillo, M. Simões, Positions in 1 , Banach J. Math. 9 (2015) 395–404. [6] N.J. Kalton, The three-space problem for locally bounded F-spaces, Compos. Math. 37 (1978) 243–276. [7] N.J. Kalton, Locally complemented subspaces and Lp for p < 1, Math. Nachr. 115 (1984) 71–97. [8] N.J. Kalton, N.T. Peck, Twisted sums of sequence spaces and the three-space problem, Trans. Amer. Math. Soc. 255 (1979) 1–30. [9] N.J. Kalton, W. Roberts, Uniformly exhaustive submeasures and nearly additive set functions, Trans. Amer. Math. Soc. 278 (1983) 803–816.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
The behaviour of quasi-linear maps on C(K)-spaces ✩ Félix Cabello Sánchez a , Jesús M.F. Castillo a,∗ , Alberto Salguero-Alarcón b a b
Instituto de Matemáticas, Universidad de Extremadura, Avenida de Elvas, 06071-Badajoz, Spain Departamento de Matemáticas, Universidad de Extremadura, Avenida de Elvas, 06071-Badajoz, Spain
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 29 November 2018 Available online 23 March 2019 Submitted by R.M. Aron
In this paper we combine topological and functional analysis methods to prove that a non-locally trivial quasi-linear map defined on a C(K) must be nontrivial on a subspace isomorphic to c0 . We conclude the paper with a few examples showing that the result is optimal, and providing an application to the existence of nontrivial twisted sums of 1 and c0 . © 2019 Elsevier Inc. All rights reserved.
Keywords: Banach spaces of continuous functions Quasi-linear maps
1. Introduction and preliminaries This paper treats the behaviour of quasi-linear maps defined on C(K)-spaces. More precisely, we will prove that a quasi-linear map Ω : C(K) → Y is either uniformly trivial on finite dimensional subspaces or nontrivial on a certain copy of c0 . This behaviour of quasi-linear maps on C(K)-spaces has consequences about the existence and properties of exact sequences 0 −−−−→ Y −−−−→ X −−−−→ C(K) −−−−→ 0
(1.1)
via the correspondence between exact sequences and quasi-linear maps we describe below. Recall that an exact sequence of Banach (rsp. quasi-Banach) spaces is a diagram 0 −−−−→ Y −−−−→ X −−−−→ Z −−−−→ 0
(1.2)
formed by Banach (resp. quasi-Banach) spaces and linear continuous operators in which the kernel of each arrow coincides with the image of the preceding one. The middle space X is usually called a twisted sum of ✩ This research has been supported in part by project MTM2016-76958-C2-1-P and Project IB16056 de la Junta de Extremadura. The third author is supported by a grant associated to Project IB16056. * Corresponding author. E-mail addresses: [email protected] (F. Cabello Sánchez), [email protected] (J.M.F. Castillo), [email protected] (A. Salguero-Alarcón).
https://doi.org/10.1016/j.jmaa.2019.03.040 0022-247X/© 2019 Elsevier Inc. All rights reserved.
F. Cabello Sánchez et al. / J. Math. Anal. Appl. 475 (2019) 1714–1719
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Y and Z. By the open mapping theorem, Y must be isomorphic to a subspace of X and Z to the quotient X/Y . It was the discovery of Kalton [6,8] that exact sequences of quasi-Banach spaces (1.2) correspond to a certain type of non-linear maps Z → Y called quasi-linear maps, which are homogeneous maps for which there is a constant Q such that Ω(x + y) − Ω(x) − Ω(y) ≤ Q (x + y) for every x, y ∈ X. The least constant Q that satisfies the previous inequality is called the quasi-linearity constant of Ω. Be warned of a surprising fact [1]: the space X in an exact sequence (1.2) is not necessarily a Banach space when Y, Z are Banach spaces. Kalton and Roberts [9] however proved in a deep theorem that, in combination with [1], means that every quasi-linear map Ω defined on an L∞ -space enjoys the additional property that there is a constant K such that for every n ∈ N and every x1 , . . . , xn ∈ X one has n n n xi − Ω(xi ) ≤ K xi , Ω i=1
i=1
(1.3)
i=1
and, consequently, exact sequences of quasi-Banach spaces 0 −−−−→ Y −−−−→ X −−−−→ Z −−−−→ 0 in which Z is an L∞ -space and Y is a Banach space also have X a Banach space. Thus, we will freely speak about quasi-linear maps Ω : C(K) → Y from a space of continuous functions to a Banach space with the understanding that the associated exact sequence 0 → Y → X → C(K) → 0 is a sequence of Banach spaces. We will write 0 → Y → X → Z → 0 ≡ Ω to say that the exact sequence and the quasi-linear map correspond one to each other. j An exact sequence 0 → Y → X → Z → 0 ≡ Ω is said to be trivial, or to split, if the injection j admits a left inverse; i.e., there is a linear continuous projection P : X → Y along j. In terms of Ω, this means that there exists a linear map L : Z → Y such that Ω − L = supx≤1 Ωx − Lx < +∞. The sequence is said to be λ-trivial if Ω − L ≤ λ. Definition 1. We will say that a quasi-linear map Ω : X → Y is locally trivial if there is a constant λ such that whenever F is a finite dimensional subspace of X then the restriction Ω|F : F → Y is λ-trivial. This notion was introduced by Kalton [7] who proved that an exact sequence 0 → Y → X → Z → 0 of Banach spaces is locally trivial if and only if its dual exact sequence 0 → Z ∗ → X ∗ → Y ∗ → 0 is trivial. It is also shown in [7] that a locally trivial sequence 0 → Y → X → Z → 0 in which Y is complemented in its bidual must be trivial. 2. A dichotomy for C(K) spaces Our aim is to prove the following dichotomy: Theorem 2.1. A quasi-linear map Ω : C(K) → Y either is locally trivial or admits a subspace isomorphic to c0 on which its restriction is not locally trivial. Before embarking in the proof we need a technical lemma: Lemma 2.2. Let Ω be a quasi-linear map on X with quasi-linearity constant Q. Let x1 , . . . , xm be points so that Ω|[x1 ,...,xm ] is λ-trivial, and let δ be the Banach-Mazur distance between [x1 , . . . , xm ] and m 1 . If
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y1 , . . . , ym are norm one points so that yi − xi ≤ ε/mδ for all i = 1, . . . , m, then Ω|[y1 ,...,ym ] is λ + 2+ε 1−ε Q-trivial. Proof. Let L be a linear map on [x1 , . . . , xm ] so that Ω − L ≤ λ. Set L the linear map L yi = Lxi + Ω(yi − xi ) on [y1 , . . . , ym ]. If we call (∗) = Ω( λi yi ) − L ( λi yi ), then (∗) = Ω( λi yi ) − L( λi xi ) − λi Ω(yi − xi ) ≤ Ω( λi yi ) − Ω( λi xi ) − λi Ω(yi − xi ) + Ω( λi xi ) − L( λi xi ) ≤ Ω( λi yi ) − Ω( λi xi ) − Ω( λi (yi − xi )) + Ω( λi (yi − xi )) − λi Ω(yi − xi ) +λ λi xi ≤ Q λi yi + λi (yi − xi ) + (m − 1) |λi |yi − xi + λ λi yi ≤ Q λi yi + m |λi |yi − xi + λ λi yi ε ≤ Q λi yi + m |λi | + λ λi yi mδ 1 ≤ Q λi yi + λi yi + λ λi yi 1−ε 2+ε Q λi yi 2 ≤ λ+ 1−ε We pass to the proof of the theorem. Proof. We make the proof for the Cantor set Δ. Let Ω : C(Δ) → Y be a quasi-linear map. Split Δ = Δ− ∪Δ+ in right and left halves Δ− = Δ ∩ [0, 1/3] and Δ+ = Δ ∩ [2/3, 1]. Now, let J+ = {f ∈ C(Δ) : f|Δ− = 0} = C(Δ+ ) and J− = {f ∈ C(Δ) : f|Δ+ = 0} = C(Δ− ). Claim. If Ω+ = Ω|J+ ≡ 0 and Ω− = Ω|J− ≡ 0 then Ω ≡ 0. Proof. If there are linear maps L+ : C(Δ+ ) → Y and L− : C(Δ− ) → Y so that Ω+ − L+ ≤ M and Ω− − L− ≤ N , set the linear map L : C(Δ) −→ Y
,
Lf = L+ (f 1Δ+ ) + L− (f 1Δ− )
Since f = f 1Δ+ + f 1Δ− for every f ∈ C(Δ) one has Ωf − Lf ≤ Ω(f ) − Ω(f 1Δ+ ) − Ω(f 1Δ− )+ + Ω(f 1Δ+ ) − L+ (f 1Δ+ ) + Ω(f 1Δ− ) − L− (f 1Δ− ) ≤ ≤ Q f 1Δ+ + f 1Δ− + M f 1Δ+ + N f 1Δ− ≤ ≤ (M + N + 2Q)f 2 So, if Ω is non-trivial then either Ω+ o Ω− is non-trivial. For given ξ ∈ / Δ we will denote Δξ− = Δ ∩ [0, ξ] and Δξ+ = Δ ∩ [ξ, 1]. Lemma 2.3. Let Ω : C(Δ) → Y be non-locally trivial. Then there is ξ ∈ [0, 1] so that • If ξ ∈ / Δ then Δ = Δξ+ ∪ Δξ− with Δξ+ ∩ Δξ− = ∅ and both Ω|C(Δξ+ ) and Ω|C(Δξ− ) are non-locally trivial.
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• If ξ ∈ Δ then Δ = Δξ+ ∪ Δξ− with Δξ+ ∩ Δξ− = {ξ}; and if we call Jξ+ = {f ∈ Δ : f|Δξ− = 0} and Jξ− = {f ∈ Δ : f|Δξ+ = 0} then ker δξ = Jξ+ ∪ Jξ− and both Ω|Jξ+ and Ω|Jξ− are non-locally trivial. Proof. Split Δ = Δ+ ∪ Δ− . If both Ω+ and Ω− are non-trivial, then we are done: ξ = 1/2 works and the intersection is empty. Assume, on the contrary, that one of them is trivial, say, Ω+ . Pick λ1 so that Ω+ is λ1 -trivial but not (λ1 − 1)-trivial. Iterate the argument. If both Ω−− and Ω−+ (the restrictions of Ω to Δ−− and Δ−+ ) are non-trivial, then we are done and Δ− splits in two pieces with empty intersection Δ−− and Δ−+ ∪ Δ+ , so that Ω is non-trivial when restricted to them. Otherwise, we continue. If the decomposition method does not stop, let ξ ∈ {−1, 1}N be the element that represents the non-trivial choices; i.e., the choices of the pieces on which the restriction of Ω was non-trivial. We can assume without loss of generality that −ξ represents the trivial choices and Ω is λn -trivial but not (λn − 1)-trivial on Δξ(1),...,ξ(n−1),−ξ(n) , and also non-trivial on Δξ(1),...,ξ(n) . Now, either supn λn < +∞ or supn λn = +∞. Claim. If supn λn = λ < +∞ then Ω is locally trivial. Proof. Pick functions f1 , . . . , fm . For ε > 0 small enough each of the functions fj is at distance ε of a function of the form fj (ξ)χ + gj with gj supported in a union A = Δ−ξ(1) ∪ Δξ(1),−ξ(2) ∪ · · · ∪ Δξ(1),ξ(2),...,ξ(n−1),−ξ(n) and χ is the characteristic function of Δ \ A. Since Ω is λ-trivial on A and is also 1-trivial on the onedimensional subspace generated by χ, Ω is μ-trivial (with μ = λ + 1 + 2Q) on the finite-dimensional subspace [χ + gj ] generated by them. Therefore, it must be μ + Q(1 + ε) -trivial on [f1 , . . . , fm ]. 2 It just remains that supn λn = +∞. Set the decomposition Δξ− = Δ−ξ(1) ∪ Δξ(1),−ξ(2) ∪ Δξ(1),ξ(2),−ξ(3) ∪ . . . Δξ+ = Δξ(1),ξ(2) ∪ Δξ(1),ξ(2),ξ(3) ∪ . . . with ξ ∈ Δξ+ ∩ Δξ− and Ω is non-trivial on both. We are ready to conclude the proof of the lemma. The different pieces Δξ(1),ξ(2),...,ξ(n−1),−ξ(n) on which Ω is not (λn −1)-trivial are disjoint. So when one considers a finite dimensional subspace of C(Δξ(1),ξ(2),...,−ξ(n) ) n where Ω is not (λn − 1)-trivial, it is contained in a subspace m ∞ of C(Δξ(1),ξ(2),...,−ξ(n) ) on which Ω is not mn (λn − 1)-trivial. The subspaces ∞ are thus disjointly supported, so their closed span yields a copy of c0 where Ω cannot be locally trivial. 2 Now, if C(K) is isomorphic to C(Δ), pick Ω : C(K) → Y non-locally trivial and α : C(Δ) → C(K) an isomorphism. Then Ω = Ω α is not locally trivial, so there is an embedding j : c0 → C(Δ) so that Ωj is not locally trivial. Then Ω αj is not locally trivial and αj : c0 → C(K) is the embedding one needs. In general, let Ω : C(K) → Y be non-locally trivial. Find for each n = 1, 2, ... a finite dimensional subspace Fn ⊂ C(K) so that Ω|Fn is not n-trivial. Let X = [∪n Fn ]. This is a separable subspace of C(K) on which Ω is not locally trivial. Let A be the separable subalgebra that X generates in C(K). This A is isometric to C(H) for some compact H, necessarily metrizable, and the restriction of Ω to C(H) cannot be locally trivial as well. If H is countable, then C(H) = C(α) for some countable ordinal α and the result is clear. If H is uncountable then C(H) = C(Δ) by Milutin’s theorem and the result is also clear. 2 Corollary 2.4. A quasi-linear map defined on a C(K)-space is locally trivial if and only if every restriction to every copy of c0 is locally trivial.
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3. Consequences First of all, let us show that the result just obtained is the only possible one. We have proved that a quasi-linear map Ω : C(K) → ♦ is locally trivial if and only if every restriction to every copy of c0 is locally trivial. It is however false that a quasi-linear map Ω : C(K) → ♦ must be trivial when every restriction to every copy of c0 is trivial, and a simple example like 0 −−−−→ c0 −−−−→ ∞ −−−−→ ∞ /c0 −−−−→ 0 proves it: by Sobczyk’s theorem any restriction to a separable subspace of ∞/c0 must be trivial. The word “locally” is therefore not superfluous. It is also false that a locally trivial quasi-linear map Ω : C(K) → ♦ must have some trivial restriction to some copy of c0 , as it is proved by the example in [3] of a non-trivial locally trivial sequence 0 −−−−→ C[0, 1] −−−−→ ♦ −−−−→ c0 −−−−→ 0 having strictly singular quotient map (from now on called strictly singular sequences). The quotient map is not an isomorphism when restricted to any infinite dimensional subspace and consequently the associated quasi-linear map is not trivial when restricted to any infinite dimensional subspace. Such quasi-linear maps will be also called strictly singular in what follows. To conclude this analysis, observe that strictly singular but not locally trivial maps Ω : c0 → Y do exist: any exact sequence 0 → K → 1 → c0 → 0 works (it is not locally trivial because a L∞ -script space cannot be a complemented subspace of a L1 -space); there also exist strictly singular sequences in which the subspace is complemented in its bidual: pick any notrivial sequence 0 → 2 → ♦ → c0 → 0 and apply the method in [4] to obtain a strictly singular sequence 0 −−−−→ ∞ (Γ, 2 ) −−−−→ ♣ −−−−→ c0 −−−−→ 0 Such sequences cannot locally split. Now we pass to applications. Exact sequences 0 −−−−→ 1 −−−−→ ♠ −−−−→ c0 −−−−→ 0
(3.1)
are somewhat a mystery, even though their existence has been shown in [2,3]. Both papers contain different although complementary methods to show the existence of non-trivial sequences 0 −−−−→ L1 (0, 1) −−−−→ ♣ −−−−→ C[0, 1] −−−−→ 0
(3.2)
Since L1 and 1 (resp. C[0, 1] and c0 ) have the same local structure, one can apply the local theory of exact sequences [2] to pass from a non-trivial twisting of L1 and C[0, 1] to a non-trivial twisting of 1 and c0 . The final sequence, however, remains invisible since it has emerged via an abstract existence result. Theorem 2.1 yields a much more explicit way to get such sequences. Let us call Ω the exact sequence (3.2), which is not locally trivial because L1 is complemented in its bidual, and therefore there is an injection j : c0 → C[0, 1] so that Ωj, namely the lower sequence in the commutative diagram 0 −−−−→ L1 (0, 1) −−−−→
♣ −−−−→ C[0, 1] −−−−→ 0 ≡ Ω ⏐j ⏐ ⏐ ⏐
0 −−−−→ L1 (0, 1) −−−−→ ♥ −−−−→ is not locally trivial. This implies that its dual sequence
c0
−−−−→ 0 ≡ Ωj
(3.3)
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0 −−−−→ 1 −−−−→ ♥∗ −−−−→ L1 (0, 1)∗ −−−−→ 0 ≡ j∗ Ω∗ is not locally trivial. Since L1 (0, 1)∗ is a C(K)-space, Theorem 2.1 applies again to obtain an injection ı : c0 → L1 (0, 1)∗ such that the restriction (Ωj)∗ ı, namely, the lower sequence in the commutative diagram 0 −−−−→ 1 −−−−→
♥∗ −−−−→ L1 (0, 1)∗ −−−−→ 0 ≡ (Ωj)∗ ⏐ı ⏐ ⏐ ⏐
0 −−−−→ 1 −−−−→ ♦ −−−−→
c0
(3.4)
−−−−→ 0 ≡ (Ωj)∗ ı
is not locally trivial. In particular, it is not trivial. One can show that the middle space ♠ in an exact sequence (3.1) cannot be isomorphic to 1 ⊕ c0 [5]. However, it is open to decide whether strictly singular sequences (3.1) exist. References [1] F. Cabello Sánchez, J.M.F. Castillo, Duality and twisted sums of Banach spaces, J. Funct. Anal. 175 (2000) 1–16. [2] F. Cabello Sánchez, J.M.F. Castillo, Uniform boundedness and twisted sums of Banach spaces, Houston J. Math. 30 (2004) 523–536. [3] F. Cabello Sánchez, J.M.F. Castillo, N.J. Kalton, D.T. Yost, Twisted sums with C(K) spaces, Trans. Amer. Math. Soc. 355 (2003) 4523–4541. [4] J.M.F. Castillo, Y. Moreno, Strictly singular quasi-linear maps, Nonlinear Anal. – TMA 49 (2002) 897–904. [5] J.M.F. Castillo, M. Simões, Positions in 1 , Banach J. Math. 9 (2015) 395–404. [6] N.J. Kalton, The three-space problem for locally bounded F-spaces, Compos. Math. 37 (1978) 243–276. [7] N.J. Kalton, Locally complemented subspaces and Lp for p < 1, Math. Nachr. 115 (1984) 71–97. [8] N.J. Kalton, N.T. Peck, Twisted sums of sequence spaces and the three-space problem, Trans. Amer. Math. Soc. 255 (1979) 1–30. [9] N.J. Kalton, W. Roberts, Uniformly exhaustive submeasures and nearly additive set functions, Trans. Amer. Math. Soc. 278 (1983) 803–816.