A space with no unconditional basis that satisfies the Johnson-Lindenstrauss lemma

TL;DR

This work constructs a nontrivial twisted Hilbert space, $Z(T^2)$, with no unconditional basis that nonetheless satisfies the Johnson-Lindenstrauss lemma, advancing beyond the known $T^2$ result by Johnson-Naor. By proving $Z(T^2)$ is log-Hilbertian, the authors establish JL for this space and link it to Mascioni's question by showing $Z(T^2)$ is asymptotically LSD, providing a structured way to approximate finite-dimensional subspaces by ones close to their duals. They further analyze the block-basis behavior via the invariant $D_n$, showing $D_n(Z(T^2))$ grows like $ oot n$, precluding embedding of the Kalton-Peck space $Z_2$ and differentiating $Z(T^2)$ from $Z(T^p)$ for $p eq2$, which in turn cannot be weak Hilbert. Overall, the paper highlights a delicate interplay between centralizers, derived spaces, and asymptotic duality properties in twisted Hilbert spaces, with implications for the geometry of Banach spaces and the Mascioni problem.

Abstract

We give the first example of a nontrivial twisted Hilbert space that satisfies the Johnson-Lindenstrauss lemma. This space has no unconditional basis. We also show that such a space gives a partial answer to a question of Mascioni.

Theorems & Definitions (25)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• proof
• Lemma 1
• proof
• Theorem 2
• proof
• Proposition 1