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 $|A|=(A^{\*}A)^{1/2}$.



Embedding finite dimensional subspaces of Schatten p-classes
See


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


