[Paper Review] The solution to the Maurey extension problem for Banach spaces with the Gordon Lewis property and related structures

Peter G. Casazza, N. J. Nielsen|ArXiv.org|Oct 14, 1999

Advanced Banach Space Theory13 references3 citations

TL;DR

This paper resolves the Maurey extension problem for Banach spaces with the Gordon-Lewis property and related structures by proving that if every bounded operator from a subspace of X into a cotype 2 space extends to X, then the space of bounded operators from ℓ∞ to X* equals the space of absolutely p-summing operators. The result establishes that such spaces are of type 2 when they also have the Gaussian average property, solving the extension problem for Banach lattices and subspaces of finite cotype Banach lattices.

ABSTRACT

The main result of this paper states that if a Banach space X has the property that every bounded operator from an arbitrary subspace of X into an arbitrary Banach space of cotype 2 extends to a bounded operator on X, then $B(\ell_{\infty},X^*)=Π_2(\ell_{\infty},X^*)$. If in addition X has the Gaussian average property, then it is of type 2. This implies that the same conclusion holds if X has the Gordon-Lewis property (in particular X could be a Banach lattice) or if X is isomorphic to a subspace of a Banach lattice of finite cotype, thus solving the Maurey extension property for these classes of spaces. The paper also contains a detailed study of the property of extending operators with values in $\ell_p$-spaces, $1\le p

Motivation & Objective

• To resolve the Maurey extension problem for Banach spaces with the Gordon-Lewis property and related classes.
• To characterize the conditions under which bounded operators from subspaces of X into cotype 2 spaces extend to bounded operators on X.
• To establish the implication that such extension properties imply that X is of type 2 when combined with the Gaussian average property.
• To analyze the extension of operators with values in ℓp-spaces for 1 ≤ p < ∞.
• To extend the solution of the Maurey problem to Banach lattices and subspaces of Banach lattices of finite cotype.

Proposed method

• Use of the Gordon-Lewis property as a structural condition to control operator extensions.
• Application of the Gaussian average property to deduce type 2 behavior in Banach spaces.
• Analysis of the equality B(ℓ∞, X*) = Π₂(ℓ∞, X*) as a key characterization of the extension property.
• Use of duality and factorization techniques to relate operator ideals and extension properties.
• Study of operator extension behavior into ℓp-spaces via interpolation and type/cotype theory.
• Leveraging known results on Banach lattices and finite cotype to extend the solution to broader classes of spaces.

Experimental results

Research questions

• RQ1 Under what conditions does every bounded operator from a subspace of X into a cotype 2 space extend to a bounded operator on X?
• RQ2 What is the relationship between the Maurey extension property and the equality B(ℓ∞, X*) = Π₂(ℓ∞, X*)?
• RQ3 How does the Gaussian average property interact with the extension property to imply type 2 behavior?
• RQ4 To what extent does the Gordon-Lewis property ensure the Maurey extension property?
• RQ5 What is the role of ℓp-space valued operators in characterizing the extension property?

Key findings

• If every bounded operator from a subspace of X into a cotype 2 space extends to X, then B(ℓ∞, X*) = Π₂(ℓ∞, X*).
• If X has the Gaussian average property and satisfies the extension condition, then X is of type 2.
• The Maurey extension problem is solved for Banach spaces with the Gordon-Lewis property, including all Banach lattices.
• The result extends to any Banach space isomorphic to a subspace of a Banach lattice of finite cotype.
• The paper provides a detailed analysis of operator extension into ℓp-spaces for 1 ≤ p < ∞.
• The equality B(ℓ∞, X*) = Π₂(ℓ∞, X*) serves as a central characterization for the extension property in the studied classes.