On the metric entropy of the Banach-Mazur compactum
Gilles Pisier

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 of all n-dimensional Banach spaces (the so-called Banach-Mazur compactum) equipped with the Banach-Mazur (multiplicative) "distance" . We are interested either in estimates independent of the dimension or in asymptotic estimates when the dimension tends to . For instance, we prove that, if is the smallest number of "balls" of "radius" that cover , then for any we have We also prove an analogous result for the metric entropy of the set of n-dimensional operator spaces equipped with the distance naturally associated to -matrices with operator entries. In that case is arbitrary but our estimates are valid…
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