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The scale of Schatten-von Neumann classes is noncommutatitve analog of classical $\ell_p$-spaces. A lot of researchers devoted their lives to study Banach geometric structure of these spaces. Different geometric properties where invented along the way. Just to name a few

  • The cotype and type of Banach space
  • The Radon-Nykodym property
  • The approximation property
  • The Dunford-Pettis property
  • The property of being an $\mathscr{L}_p$-space.

I would like to know if there exists any survey on noncommutative analogs of these properties. Of course I'm interested in those properties that posessed by Schatten-von Neumann classes. I doubt that such survey exists, so references to specific properties are welcomed too.

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    $\begingroup$ You can start with Pisier, Gilles; Xu, Quanhua Non-commutative Lp-spaces. Handbook of the geometry of Banach spaces, Vol. 2, 1459–1517, North-Holland, Amsterdam, 2003. $\endgroup$ Jul 18, 2015 at 23:45
  • $\begingroup$ @BillJohnson, thank you! This paper mostly deals with the case $1<p<\infty$. As for the edge cases, do you know a non-commutative analog of Dunford-Pettis property, a kind of property shared by bounded and nuclear operators? $\endgroup$
    – Norbert
    Jul 19, 2015 at 0:21
  • $\begingroup$ No, I have not seen anything about a non commutative Dunford Pettis property. $\endgroup$ Jul 20, 2015 at 13:05
  • $\begingroup$ I don't know if this is relevant to your explicit questions, but there is a long series of articles by P.G. Dodds with various collaborators on non commutative analogues of classical function spaces. These might be of interest and are easily traceable on mathscinet. $\endgroup$
    – priel
    Aug 25, 2015 at 4:38

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Indeed the $\mathscr{L}_p$-spaces have been generalised to the non-commutative setting

M. Junge, N.J Nielsen, Z.-J. Ruan, Q. Xu, $\mathscr{COL}_p$-spaces—the local structure of non-commutative $L_p$-spaces, Advances in Mathematics 187, (2004), 257–319.

There is also operator approximation property (try to google that name); this notion originates from the classical paper by Haagerup:

U. Haagerup, An example of a nonnuclear C*-algebra, which has the metric approximation property, Invent. Math. 50 (1978/79), no. 3, 279–293.

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