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Can one explain some philosophy behind "quantum functional analysis" (or "quantized functional analysis") which was initiated and developed by such researchers as: Ruan Z.-J., Pisier J., Effros E.G., Haagerup U., et al....

The main notion of this subject is a quantum space: $(E, \{\|\cdot\|_n\})$ -- some normed space, equipped with "quantum norms" $\|\cdot\|_n, \, n\in \mathbb{N}$ (in turn, these quantum norms $\{\|\cdot\|_n\}$ are not the norms on $E$ but on matrices $M_n(E)$). It appears that classical bounded operators in $(E, \|\cdot\|_1)$ may have good ("quantum brothers") properties in whole quantum space, and as far as I understood they are called "completely bounded" operators. So quantum functional analysis is a theory about "such" operators, about matrices of operators (which are themselves operators) and from the book A.J. Helemsky I've read that it is a beautiful theory, "much simpler" than the classical functional analysis.

Also there should be a relationship with quantum mechanics, since there is a notion of "quantization" of normed spaces (but I don't know anything more than that).

I was wondering, can one here post some beautiful results of this theory? (for example, problem of Halmos about polynomially contracting operators was negatively resolved by Pisier using techniques from this theory). Among these beautiful results those ones which speak about classical analysis are extremely beautiful. Are there many extremely beautiful results?

Fedor Goncharov
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Okay, I'll take this one. First let me say that the English term is "completely bounded" (or "complete isometry", etc.).

About the term "quantum". The general principle is that analyzing some aspect of a physical system typically involves very different kinds of mathematical structures, depending on whether the system is classical or quantum. Thus if you study a classical system using topological techniques, or measure theory, or Lie groups, or graphs, the analogous analysis of a quantum system would probably involve C*-algebras, or von Neumann algebras, or quantum groups, or operator systems. (For instance, here's a paper of mine that talks about how operator systems arise in quantum error correction in the same way that graphs arise in classical error correction.)

There is no simple rule for translating classical structures into quantum structures, but we have a pretty large dictionary and one can identify broad themes. My take is that operator spaces fit into this scheme as the quantum analog of relations on a set. See Section 2 of the paper linked above. I'm not aware of any really compelling direct connections with real-world quantum mechanics in this instance, but the word "quantum" may be appropriate just because the subject fits into the general scheme of mathematical quantization which does include many direct links.

Now, as to the "beautiful results" you ask for. Well, there's a lot of good stuff so the selection would be idiosyncratic. One general principle is that when you're studying things related to Hilbert space (single operators, C*-algebras, etc.) it is often the case that you get much nicer results if you assume the "complete" (at all matrix levels) version of your hypothesis. For example, if two C*-algebras are linearly isometric, are they isomorphic as C*-algebras? Not in general, but yes if they are completely linearly isometric, i.e., linearly isometric at all matrix levels. (I think this is folklore.)

The result of Pisier that you refer to was actually a counterexample, showing that a polynomially bounded operator need not be similar to a contraction. The positive result is due to Paulsen and says that any completely polynomially bounded operator is similar to a contraction.

Here's another nice result, this one due to Zhong-Jin Ruan, one of the giants of the subject. B. Johnson proved that a locally compact group $G$ is amenable if and only if the Banach algebra $L^1(G)$ is amenable. Ruan showed that this happens if and only if the Fourier algebra $A(G)$ is completely amenable.

A good place to learn more is the book Operator Spaces by Effros and Ruan.

References:

G. Pisier, A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc. 10 (1997), 351–369.

V. Paulsen, Every completely polynomially bounded operator is similar to a contraction, J. Funct. Anal. 55 (1984), 1–17.

Z-J. Ruan, The operator amenability of A(G), Amer. J. Math. 117 (1995), 1449–1474.

Nik Weaver
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    I have to say I prefer "completely amenable" to the horrible-but-standard "operator amenable", although the former does make me wonder if I should go ahead and define a notion of something being utterly amenable ... – Yemon Choi Aug 31 '18 at 16:09
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    I once heard a talk by David Blecher where he started by explaining the meaning of "completely bounded", "completely positive", and so on. Then he put up a lemma and said "The proof of this is completely obvious." I think I was the only one who laughed, though. – Nik Weaver Aug 31 '18 at 16:20
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    Thanks for remark about 'completely bounded operator' etc., and the example with amenable groups is very good. Thank you! – Fedor Goncharov Aug 31 '18 at 16:30
  • @NikWeaver I looked at your paper and it made me wonder: Is there a good quantum analog of arbitrary $n$-ary relations, rather than just binary relations? – James Hanson Apr 11 '19 at 15:12
  • @JamesHanson: good question. I suppose the answer is no, but I haven't really thought about it. Did you have any ideas? – Nik Weaver Apr 11 '19 at 15:18
  • What about subspaces of the space of multilinear maps? – James Hanson Apr 11 '19 at 15:25
  • Possibly. It seems like one coordinate is different from the others, but maybe that's okay. I guess I'm not sure what criteria to use to decide whether that's a "good" definition. If it is fruitful in some way, maybe. – Nik Weaver Apr 11 '19 at 16:22
  • Aren't the two coordinates different already with matrices? But also if you think of them as tensors on a Hilbert space then raising and lowering indices is fine, right? – James Hanson Apr 11 '19 at 16:36
  • True, but for two coordinates you have the adjoint operation which reverses order, so in some sense the definition is symmetrical. For the second comment: you may be right, and if you feel like working this out I'd be interested in seeing the results. – Nik Weaver Apr 11 '19 at 16:46
  • I was also wondering. Why do you say the infinite analog needs to be weak* closed? – James Hanson Apr 11 '19 at 17:29
  • The general principle is that if $\mathcal{M} \subseteq B(H)$ is a von Neumann algebra then a weak* closed bimodule over $\mathcal{M}'$ is a "quantum relation" on $\mathcal{M}$. We need weak* to prove the following theorem: if $\mathcal{M} \cong l^\infty(X)$ then the quantum relations on $\mathcal{M}$ naturally correspond to the relations on $X$. – Nik Weaver Apr 11 '19 at 18:18