$(\lambda^+)$-injective Banach spaces

This paper resolves the open case for $\lambda > 2$ in Pelczyński's theorem by constructing $(\lambda^+)$-injective but not $\lambda$-injective Banach spaces via an iterative "zero-sum" subspace technique, while also establishing a new upper bound of $9+6\sqrt{3}$for the Banach-Mazur distance between$L_\infty[0,1]$and$\ell_\infty$.

The Gist

Imagine you are trying to fit a specific shape (a "Banach space") into a larger, flexible container (like a giant, infinite-dimensional room called $\ell_\infty$). Mathematicians have been asking a very specific question for decades: Can we create a shape that is "almost" perfectly flexible, but not quite?

In the world of these mathematical shapes, "injective" means "flexible." If a shape is $\lambda$-injective, it means you can stretch or squeeze it to fit into a larger container with a "stretch factor" of at most $\lambda$.

• If $\lambda = 1$, it's a perfect fit (no stretching needed).
• If $\lambda = 1.5$, you can stretch it by 50%.
• If a shape is $(\lambda+)$-injective, it means you can stretch it by any amount slightly larger than $\lambda$ (like 1.5000001), but you cannot do it with exactly $\lambda$.

The Mystery of the Missing Proof

Back in the 1960s, a famous mathematician named Pełczyński claimed he could prove that for any stretch factor greater than 1 (say, 1.1, 1.5, or 100), there exists a shape that is "almost" that flexible but not exactly that flexible.

However, Pełczyński's proof was lost to history. It was like a treasure map with the final X missing.

• The Old Problem: In a previous paper, the authors (Kania and Lewicki) solved the mystery for small stretch factors (between 1 and 2). They found a way to build these "almost" shapes.
• The New Problem: But what about big stretch factors (anything greater than 2)? The old method hit a wall. It was like trying to build a skyscraper using only one-story bricks; it just wouldn't go high enough.

The Solution: The "Zero-Sum" Machine

In this new paper, the authors built a special mathematical machine called the "Zero-Sum Subspace."

Think of it like this:

1. The Input: You start with a shape that is "almost" flexible (but not quite) and has a stretch factor of, say, 1.5.
2. The Machine: You take $N$ copies of this shape and stack them together. Then, you apply a rule: "The sum of all these copies must equal zero."
   • Imagine you have $N$ people holding ropes. If they all pull in different directions, the total pull is zero. This specific arrangement creates a new, more complex shape.
3. The Magic Effect: This machine doesn't just make the shape bigger; it multiplies its stretch factor by a specific number ($\mu_N$).
   • The authors found a clever way to tune this machine so that the stretch factor gets multiplied by a number slightly less than 2 (like 1.9).
4. The Loop: Here is the genius part: You can run the output of this machine back through the machine again.
   • Run it once: Stretch factor becomes $1.5 \times 1.9$.
   • Run it twice: Stretch factor becomes $1.5 \times 1.9 \times 1.9$.
   • Run it $m$ times: The stretch factor grows larger and larger.

By choosing the right number of copies ($N$) and the right number of loops ($m$), they can hit any target stretch factor you want, whether it's 2.1, 10, or 100. They have finally completed Pełczyński's map for the entire range of numbers greater than 1.

The Second Discovery: Measuring the Distance Between Shapes

The second part of the paper is like a game of "How far apart are these two cities?"

In math, we measure how different two shapes are using something called Banach-Mazur distance.

• If two shapes are identical, the distance is 1.
• If they are very different, the distance is huge.

The authors looked at two famous "cities":

1. $L_\infty[0, 1]$: A space representing all possible functions on a line segment.
2. $\ell_\infty$: A space representing infinite sequences of numbers.

Previous mathematicians had guessed that the distance between these two cities was less than 19.49. The authors of this paper used a new, optimized strategy (a clever combination of stretching and rotating) to prove that the distance is actually less than 19.39.

It's a small number, but in the world of pure math, shaving off 0.1 from a theoretical limit is a massive victory. It's like finding a shortcut that saves you a few seconds on a journey that takes a lifetime.

Summary

• The Goal: Prove that for any stretch factor $\lambda > 1$, there is a shape that is "almost" $\lambda$-flexible but not exactly.
• The Breakthrough: They solved the missing piece for large numbers ($\lambda > 2$) by inventing a "Zero-Sum" machine that can multiply flexibility factors.
• The Result: They completed a 60-year-old theorem, proving the rule works for all numbers greater than 1.
• Bonus: They also found a tighter, more precise measurement for how different two major mathematical spaces are from each other.

In short, they took a broken puzzle, invented a new tool to fit the missing pieces, and finally completed the picture.

Technical Summary

Here is a detailed technical summary of the paper "(λ+)-Injective Banach Spaces" by Tomasz Kania and Grzegorz Lewicki.

1. Problem Statement and Context

The paper addresses a long-standing open problem in Banach space theory concerning the existence of spaces with specific injectivity properties.

• Definitions:

  • A Banach space $X$ is $\lambda$-injective if every bounded linear operator $T: E \to X$ defined on a subspace $E$ of any Banach space $Z$ can be extended to $\tilde{T}: Z \to X$ with $\|\tilde{T}\| \leq \lambda \|T\|$.
  • $X$ is $(\lambda+)$-injective if it is $(\lambda + \varepsilon)$-injective for every $\varepsilon > 0$.
  • The relative projection constant $\lambda(Y, Z)$ is the infimum of the norms of projections from $Z$ onto a subspace $Y$. If this infimum is not attained, $Y$ is $(\lambda(Y, Z)+)$-injective but not $\lambda(Y, Z)$-injective.

• The Gap:

  • Known Results: A theorem by Pełczyński (unpublished) claimed that for every $\lambda > 1$, there exists a Banach space that is $(\lambda+)$-injective but not $\lambda$-injective.
  • Previous Work: In a companion paper (2023), the authors proved this for the range $\lambda \in (1, 2]$ using hyperplane kernels and singular functionals on $\ell_\infty$.
  • The Open Case: The range $\lambda \in (2, \infty)$ remained unresolved. Standard hyperplane techniques fail here because the projection constant of any hyperplane in a Banach space is at most 2.

2. Methodology

The authors develop a novel, explicit construction to extend the result from the range $(1, 2]$ to all $\lambda > 2$. The core of their methodology relies on a specific subspace construction called the "zero-sum" subspace.

The Zero-Sum Construction

Given a Banach space $X$ and an integer $N \geq 2$, they define:

1. The Space: $X_\infty^N = X^N$ equipped with the $\ell_\infty$-norm.
2. The Subspace: $\Sigma_N(X) = \{ (x_1, \dots, x_N) \in X_\infty^N : \sum_{i=1}^N x_i = 0 \}$.
3. The Centring Projection: $S_X^N(x_1, \dots, x_N) = (x_i - \frac{1}{N}\sum_{j=1}^N x_j)_{i=1}^N$.

Key Mechanism (Lemma 3.2)

The central technical result establishes how the projection constant scales under this construction:

• If $Y \subset Z$ is a closed subspace with relative projection constant $\lambda(Y, Z) = \alpha$ (where the infimum is not attained), then for the zero-sum subspace $W = \Sigma_N(Y) \subset Z_\infty^N$:
  $$\lambda(W, Z_\infty^N) = \mu_N \cdot \alpha$$
  where $\mu_N = 2 - \frac{2}{N}$.
• Crucial Property: The non-attainment of the infimum is preserved. If the infimum was not attained for $Y$, it is not attained for $W$.
• Scaling Factor: Since $\mu_N < 2$, this operation allows the authors to multiply the projection constant by a factor strictly less than 2. By iterating this process, they can scale a base constant $\alpha \in (1, 2]$ to any target $\lambda > 2$.

3. Key Contributions and Results

Theorem A: Resolution of Pełczyński's Theorem

The authors prove that for every $\lambda > 1$, there exists a Banach space that is $(\lambda+)$-injective but not $\lambda$-injective.

• Construction for $\lambda > 2$:
  1. Start with a base space $Y_0 \subset \ell_\infty$ having projection constant $\alpha \in (1, 2]$ (guaranteed by their companion paper).
  2. Choose integers $m$ and $N$ such that $\mu_N^m \cdot \alpha = \lambda$.
  3. Iteratively apply the zero-sum construction $m$ times: $Y_k = \Sigma_N(Y_{k-1}) \subset (Z_{k-1})_\infty^N$.
  4. The final space $Y_m$ is a subspace of a finite $\ell_\infty$-sum of $\ell_\infty$, which is isometrically isomorphic to $\ell_\infty$. Thus, the example is realized as a subspace of $\ell_\infty$.
• Significance: This completes Pełczyński's theorem for all $\lambda > 1$, providing a unified, explicit construction that bridges the gap left by the hyperplane method.

Theorem B: Banach–Mazur Distance Bound

The paper provides a new quantitative bound on the Banach–Mazur distance between specific Banach spaces.

• Context: The Banach–Mazur distance $d_{BM}(X, Y)$ measures how close two isomorphic spaces are to being isometric.
• Theorem: If $X$ and $Y$ are:
  1. Isometrically square ($X \cong X \oplus_\infty X$ and $Y \cong Y \oplus_\infty Y$).
  2. Each is isometric to a 1-complemented subspace of the other.
     Then, $d_{BM}(X, Y) \leq 9 + 6\sqrt{3} \approx 19.39$.
• Corollary: Applying this to $L_\infty[0, 1]$ and $\ell_\infty$ (which satisfy the conditions), the authors improve the previous upper bound of $(3+\sqrt{2})^2 \approx 19.49$ established by Korpalski and Plebanek.
• Methodology: The proof involves constructing an explicit one-parameter family of isomorphisms $W_a$ between the spaces and optimizing the parameter $a$ to minimize the product of the operator norm and its inverse.

4. Significance and Impact

1. Completeness of Theory: The paper resolves a "forgotten" theorem of Pełczyński, establishing a complete classification of the existence of $(\lambda+)$-injective but non-$\lambda$-injective spaces for all $\lambda > 1$.
2. New Construction Technique: The "zero-sum" subspace $\Sigma_N(Y)$ serves as a powerful new tool in Banach space theory. It allows for the precise manipulation of projection constants while preserving the non-attainment property, overcoming the limitations of hyperplane-based methods.
3. Explicit Examples: Unlike many existence proofs in functional analysis that rely on Zorn's lemma or non-constructive arguments, this paper provides a fully explicit construction of the required spaces as subspaces of $\ell_\infty$.
4. Quantitative Improvements: The result on Banach–Mazur distances provides a sharper bound for the relationship between $L_\infty[0, 1]$ and $\ell_\infty$, two fundamental spaces in analysis. The optimization of the isomorphism parameters demonstrates a refined understanding of the geometric structure of these spaces.

In summary, Kania and Lewicki have not only solved a specific open problem regarding injectivity moduli but have also introduced a versatile construction method and improved quantitative estimates in the geometry of Banach spaces.