The behaviour of quasi-linear maps on C(K) -spaces

📝 Abstract

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 $c_0 $. 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 $\ell_1$ and $c_0 $.

💡 Analysis

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 $c_0 $. 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 $\ell_1$ and $c_0 $.

📄 Content

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 copy of c 0 . This behaviour of quasi-linear maps on C(K)-spaces has consequences about the existence and properties of exact sequences (1.1) 0 —→ Y —→ X —→ C(K) —→ 0 via the correspondence between exact sequences and quasi-linear maps we describe below. Recall that an exact sequence of Banach spaces is a diagram (1.2) 0 —→ Y —→ X —→ Z —→ 0 formed by 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 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 like 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

for every x, y ∈ X. The least constant Q that satisfies the previous inequality is called the quasilinearity 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 quasi-linear maps Ω defined on an L ∞ -space enjoy the additional property that there is a constant K such that for every n ∈ N and every x 1 , . . . , x n ∈ X one has

and, consequently, exact sequences of quasi-Banach spaces

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.

An exact sequence 0 → Y ȷ → X → Z → 0 ≡ Ω is said to be trivial, or to split, if the injection ȷ admits a left inverse; i.e., there is a linear continuous projection P : X → Y along ȷ. In terms of Ω, this means that there exists a linear map L : Z → Y such that ∥Ω-L∥ = sup ∥x∥<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 if F is a finite dimensional subspace of X then the restriction

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.

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 c 0 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 x 1 , . . . , x m be points so that Ω |[x 1 ,…,xm] is λ-trivial, and let δ be the Banach-Mazur distance between [x 1 , …, x m ] and ℓ m 1 . If y 1 , . . . , y m are norm one points satisfying ∥y i -

We pass to the proof of the theorem.

Proof. We make the proof for the Cantor set ∆. Let Ω :

Claim.

Proof. If there are linear maps

Proof. Split ∆ = ∆ + ∪ ∆ -. 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 λ 1 so that Ω + is λ 1 -trivial but not λ 1 -1-trivial. Iterate the argument. If both Ω –and Ω -+ (the restrictions of Ω to ∆ –and ∆ -+ ) are non-trivial, then we are done and ∆ -splits in two pieces with empty intersection ∆ –and ∆ -+ ∪ ∆ + , 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 λ n -trivial but not (λ n -1)-trivial on ∆ ξ(1),…,ξ(n-1),-ξ(n) , and also non-trivial on ∆ ξ(1),…,ξ(n) . Now, either sup n λ n < +∞ or sup n λ n = +∞.

Claim. If sup n λ n = λ < +∞ then Ω is locally trivial.

Proof. Pick functions f 1 , . . . , f m . For ε > 0 small enough, each of the functions f j is at distance ε of a function of the form f j (ξ)χ + g j with g j supported in a union

and χ is the characteristic function of ∆ \ A. Since Ω is λ-trivial on C(A) and is also 1-trivial on the one-dimensional

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