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ON THE METRIC ENTROPY OF THE BANACH–MAZUR COMPACTUM

Part of: Selfadjoint operator algebras Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  28 May 2014

Gilles Pisier
Gilles Pisier*
Affiliation:
Texas A&M University, College Station, TX 77843, U.S.A. email pisier@math.tamu.edu Université Paris VI, Inst. Math. Jussieu, Équipe d’Analyse Fonctionnelle, Case 186, 75252 Paris Cedex 05, France
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Abstract

We study the metric entropy of the metric space ${\mathcal{B}}_{n}$ 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 $N({\mathcal{B}}_{n},d,1+{\it\varepsilon})$ is the smallest number of “balls” of “radius” $1+{\it\varepsilon}$ that cover ${\mathcal{B}}_{n}$, then for any ${\it\varepsilon}>0$ we have

$$\begin{eqnarray}0<\liminf _{n\rightarrow \infty }n^{-1}\log \log N({\mathcal{B}}_{n},d,1+{\it\varepsilon})\leqslant \limsup _{n\rightarrow \infty }n^{-1}\log \log N({\mathcal{B}}_{n},d,1+{\it\varepsilon})<\infty .\end{eqnarray}$$
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 with $N\times N$ 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”.

MSC classification

Primary: 46B07: Local theory of Banach spaces 46L07: Operator spaces and completely bounded maps
Type
Research Article
Information
Mathematika , Volume 61 , Issue 1 , January 2015 , pp. 179 - 198
Copyright
Copyright © University College London 2014 

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