[Paper Review] Dual Banach spaces which contain an isometric copy of $L_1$

S. J. Dilworth, Maria Girardi+1 more|ArXiv.org|Apr 27, 2000

Advanced Banach Space Theory7 references20 citations

TL;DR

This paper establishes a precise duality between a Banach space containing asymptotically isometric copies of $\ell_1$ and its dual space containing an isometric copy of $L_1$. The key result shows that $X$ contains asymptotically isometric copies of $\ell_1$ if and only if $X^*$ contains $L_1$ isometrically, resolving a long-standing duality question in Banach space theory with implications for fixed point theory and renorming of $\ell_1$.

ABSTRACT

A Banach space contains asymptotically isometric copies of $\ell_1$ if and only if its dual space contains an isometric copy of $L_1$.

Motivation & Objective

• Establish a precise duality between the presence of asymptotically isometric copies of $\ell_1$ in a Banach space $X$ and the existence of an isometric copy of $L_1$ in its dual $X^*$.
• Resolve the open question of whether asymptotic isometric embeddability of $\ell_1$ in $X$ implies isometric embeddability of $L_1$ in $X^*$, confirming the equivalence.
• Investigate the structural implications of such isometric embeddings for renormings of $\ell_1$ and the existence of fixed-point-free isometries on weakly compact convex sets in dual spaces.
• Provide a characterization of when a dual space contains $L_1$ isometrically through quotient and perturbation structures of $\ell_1$-type sequences.
• Extend and generalize prior results by Peczynski and Hagler on duality between $\ell_1$-like structures and $L_1$-like structures in dual spaces.

Proposed method

• The paper defines asymptotically isometric copies of $\ell_1$ via sequences $ (x_n) $ in $ X $ satisfying $ (1 - \varepsilon_n) \sum |a_n| \leq \left\| \sum a_n x_n \right\| \leq \sum |a_n| $ for a null sequence $ (\varepsilon_n) $, with equivalent formulations in terms of tail estimates and perturbations.
• Uses the concept of $ (1+\varepsilon_n) $-perturbations of isometric copies of $\ell_1$, where $ \varepsilon_n \to 0 $, to characterize asymptotic isometric embeddability.
• Applies duality theory: shows that $ X^* $ contains $ L_1 $ isometrically if and only if $ X $ admits a quotient space isometric to $ \ell_1 $, using quotient mappings and annihilators.
• Constructs explicit renormings of $ \ell_1 $ via $ \| (a_n) \|_{1}^{\prime} = \inf \left\{ \left[ \| (a_n + b_n) \|_1^2 + \| (\gamma_n^{-1} b_n) \|_2^2 \right]^{1/2} \right\} $ for $ \| (\gamma_n) \|_2 < \varepsilon $, and proves such spaces do not contain asymptotically isometric copies of $ \ell_1 $.
• Applies the duality between isometric quotient mappings and isometric embeddings of duals, showing $ T $ is an isometric quotient iff $ T^* $ is an isometric embedding.
• Uses the fact that if $ X^* $ contains $ L_1 $ isometrically, then $ X $ must contain a sequence satisfying the asymptotic isometric condition, and vice versa, via duality and quotient space constructions.

Experimental results

Research questions

• RQ1 Does the presence of asymptotically isometric copies of $\ell_1$ in a Banach space $X$ imply that its dual $X^*$ contains an isometric copy of $L_1$?
• RQ2 Is the converse true: if $X^*$ contains $L_1$ isometrically, does $X$ necessarily contain asymptotically isometric copies of $\ell_1$?
• RQ3 Can the duality between $\ell_1$-type structures and $L_1$-type structures in dual spaces be made isometric rather than just isomorphic or almost isometric?
• RQ4 Are there renormings of $\ell_1$ that avoid containing asymptotically isometric copies of $\ell_1$, and what structural conditions prevent this?
• RQ5 Does the existence of an isometric copy of $L_1$ in $X^*$ imply the existence of a fixed-point-free isometry on a weakly compact convex subset of $X^*$?

Key findings

• The main result establishes a complete duality: a Banach space $X$ contains asymptotically isometric copies of $\ell_1$ if and only if its dual $X^*$ contains an isometric copy of $L_1$.
• An explicit renorming of $\ell_1$ is constructed via $ \| (a_n) \|_{1}^{\prime} = \inf \left\{ \left[ \| (a_n + b_n) \|_1^2 + \| (\gamma_n^{-1} b_n) \|_2^2 \right]^{1/2} \right\} $ with $ \| (\gamma_n) \|_2 < \varepsilon $, which does not contain asymptotically isometric copies of $\ell_1$.
• The dual of this renormed $\ell_1$ space is isometric to $ (\ell_\infty, \| \cdot \|_\infty') $ with $ \| (c_n) \|_\infty' = \left[ \| (c_n) \|_\infty^2 + \| (\gamma_n c_n) \|_2^2 \right]^{1/2} $, which does not contain an isometric copy of $L_1$.
• Since the dual space does not contain $L_1$ isometrically, by Theorem 2, the renormed $\ell_1$ space does not contain asymptotically isometric copies of $\ell_1$, confirming the sharpness of the duality.
• An isometric quotient mapping from $X$ onto $\ell_1$ exists if and only if $X^*$ contains $L_1$ isometrically, linking quotient structures to dual space embeddings.
• Alspach’s example of a fixed-point-free isometry on a weakly compact convex set in $L_1$ implies that if $X^*$ contains $L_1$ isometrically, then $X^*$ supports such an isometry, yielding a new existence result for fixed-point-free isometries.