On the metric entropy of the Banach-Mazur compactum

TL;DR

This paper investigates the asymptotic metric entropy of the space of n-dimensional Banach spaces and operator spaces, providing bounds that are independent of dimension or matrix size, using advanced techniques involving quantum expanders.

Contribution

It establishes new asymptotic bounds for the metric entropy of Banach-Mazur compactum and operator spaces, extending known results and employing quantum expander estimates.

Findings

Asymptotic bounds for metric entropy of Banach spaces

Dimension-independent estimates for operator spaces

Application of quantum expanders in entropy estimation

Abstract

We study the metric entropy of the metric space $B_n$ of all n-dimensional Banach spaces (the so-called Banach-Mazur compactum) equipped with the Banach-Mazur (multiplicative) "distance" $d$. We are interested either in estimates independent of the dimension or in asymptotic estimates when the dimension tends to $\infty$. For instance, we prove that, if $N({B_n},d, 1+\varepsilon)$ is the smallest number of "balls" of "radius" $1+\varepsilon$ that cover $ B_n$, then for any $\varepsilon>0$ we have $$0<\liminf_{n\to \infty}n^{-1} \log\log N(B_n,d,1+\varepsilon)\le \limsup_{n\to \infty}n^{-1} \log\log N(B_n,d,1+\varepsilon)<\infty.$$ We also prove an analogous result for the metric entropy of the set of n-dimensional operator spaces equipped with the distance $d_N$ naturally associated to $N\times N$-matrices with operator entries. In that case $N$ is arbitrary but our estimates are valid independently of $N$. In the Banach space case (i.e.N=1) the above upper bound is part of the folklore, and the lower bound is at least partially known (but apparently has not appeared in print). While we follow the same approach in both cases, the matricial case requires more delicate ingredients, namely estimates (from our previous work) on certain $n$-tuples of $N\times N$ unitary matrices known as "quantum expanders".