Why study operator spaces?
I'm currently enrolled in an operator spaces course and I'm finding it difficult to understand why we study them in the first place. Functional analysis is motivated well enough for me and even though I don't have a firm grasp on them, I guess I can see why $C^{\ast}$-algebras are studied as well (purely from a quantum mechanics point of view). However I fail to see what the motivation is for studying operator spaces or why they're useful/important. What is their motivation? What led people to be interested in them and what is so special about completely bounded/positive maps that we study them?
In short the list of reasons is:
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The modern trend in math today is to develop non-commutative analogues of well known theories. Vaguely speaking operator spaces are normed spaces over "non-commutative" scalars, in fact over matricies.
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There were several long standing problems, that were solved via methods of operator space theory. As soon as a problem is embedded in its natural environment the solution comes in a natural way. For example, thanks to D. P. Blecher, we have a criterion for a Banach algebra $A$ to be isomorphic as an algebra to a closed subalgebra of $\mathcal{B}(H)$: multiplication in the Banach algebra $A$ must be completely bounded for some embedding (as a normed space) of $A$ into some space of bounded operators.
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Mathematicians not only prove theorems but also guess good definitions. Whether a definition is good or not will become evident only after its usage in theorems. It turns out that some notions are better to define withing the scope of operator space theory. For example a criterion of amenability of the Fourier algebra $A(G)$ for a locally compact group $G$ was very unnatural. But if we consider not a simple amenability but an operator amenability then we get a nice characterization in the sense of B. Johnson: Fourier algebra is operator ameanable iff its group is ameanable.
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Operator spaces possess some unexpected interesting properties. For example, in this theory we have a tensor product which both injective and projective and what is more it is not commutative! Also, thanks to W. F. Steinspring, we have quite explicit description of maps between operator spaces. For classical normed spaces this problem is hopeless.
The most I've written here is a a copy-paste from the book Quantum Functional Analysis: Non-coordinate approach by A. Ya. Helemskii.