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I have recently read a paper by Talagrand: Embedding subspaces of $L_{1}$ into $l^{N}_{1}$, and part of the chapter: Finite dimensional subspaces of $L_{p}$, written by Jonson and Schechtman. The theory of finite dimensional subspaces of $L_{p}$ is beautiful and fruitful, which has been well studied.

It is quite natural to expect similar results for noncommutative Lp spaces (or, more specifically, the Schatten p-classes, denoted by $S_{p}$). However, it seems that the theory of finite dimensional subspaces of Schatten p-classes is not well developed.

So, I am pretty interested in this topic. Are there any related references to embedding finite dimensional subspaces of $S_{p}$ into $S^{N}_{p}$? And what is the difficulty of such investigations in noncommutative settings?

PS: $S_{p}$ (resp. $S^{N}_{p}$) is the space of all infinite (resp. $N\times N$) matrices equipped with the Schatten-p norm $\|A\|_{p}=(Tr(|A|^{p}))^{1/p}$ with $|A|=(A^{*}A)^{1/2}$.

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See

Schechtman, Gideon (IL-WEIZ) Three observations regarding Schatten p classes. (English summary) J. Operator Theory 75 (2016), no. 1, 139–149. 46B28 (47B10)

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  • $\begingroup$ Thank you very much for your kind suggestions, Prof. Johnson. In Section 3 of ”Three observations...” , Schechtman shown that only if $n\geq D^{\frac{-2p}{|p-2|k}}$, $l_{p}^{k}$ can be linearly embedded into $C_{p}^{n}$ with distortion at most $D$. This result is quite related to my question. However, I prefer to look for investigations of the following specific problem: Given $\varepsilon>0$ and $n$-dimensional subspace $X\subseteq C_{p}$, what is the smallest $N(\varepsilon,n)$ such that there exists a subspace $Y\subseteq C_{p}^{N(\varepsilon,n)}$ such that $d_{B-M}(X, Y)<1+\varepsilon$? $\endgroup$
    – Sijie Luo
    Commented May 16, 2022 at 6:26

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