A $\mathrm{C}^\*$-algebra $\mathcal{A}\subset B(\mathsf{H})$ is a norm-closed, self-adjoint subalgebra of bounded operators on a Hilbert space. If we then take a unital self-adjoint (possibly closed) subspace (not subalgebra) we have an *operator system* $\mathcal{O}\subset \mathcal{A}$.


(I am aware there are abstract definitions of both $\mathrm{C}^\*$-algebras and operator systems).


My question is:



> 
> *Why are operator systems studied?*
> 
> 
> 


I presume that they are not studied for their own sake. I have tried to find the answer to this question by skimming papers, but to no avail. All I can distill is that perhaps they can help study $\mathrm{C}^\*$-algebras, either their $\mathrm{C}^\*$-envelopes, or perhaps their ambient $\mathrm{C}^\*$-algebras $\mathcal{A}\supset \mathcal{O}$. Perhaps they are studied to learn about completely positive maps, the appropriate morphisms of operator systems.


I would have thought they are not coming from a physics motivation: from my point of view the algebra structure for observables makes more sense than the vector structure. I have asked this question in that context [here](https://physics.stackexchange.com/questions/728432/operator-systems-vs-operator-algebras). If the motivation is well addressed in some paper, I would be happy to be pointed to such a reference.


There is one MO user who I think I could email this question to directly but that would be rather presumptive on my part.



Why operator systems?
Collating some comments, it appears that one reason why operator systems are studied is because they are *useful*:


* Uri Bader mentions [a paper of Kalantar & Kennedy](https://arxiv.org/pdf/1405.4359.pdf) which makes extensive use of Hamana's theory for injective envelopes for operator systems equipped with a group action.
* მამუკა ჯიბლაძე explains that conditional complete C\*-algebras may be recovered from their underling injective operator system, and so the underlying operator system may be used to study the algebra.
* t.c. points to [a paper of Connes & van Suijlekom](https://arxiv.org/pdf/2004.14115.pdf), where an operator systems approach to noncommutative geometry is explored, and t.c. suggests that this approach has physical applications in mind.


Of a slightly different bent, Narutaka OZAWA alludes to the question "*what are the appropriate morphisms for a category of C$^\*$-algebras?*", and suggests that a usual choice is too rigid, something which is not the case for the category of operator systems.


