In non-commutative geometry, Gelfand duality is the construction of multiplicative linear functionals of a commutative C\*-algebra, which can be viewed as the space of all its irreducible complex representations. 


When encountered with a non-commutative C\*-algebra, we can speak of the space of pure states, which is the generalization of multiplicative linear functionals in commutative cases. The GNS construction can then be viewed as a map from the space of states to the space of all representations, and also from the space of pure states to the space of all irreducible representations.


My question is, is the "GNS map" surjective, or injective? What should be viewed as the non-commutative topological space, the space of all irreducible representations, or the space of pure states? And why?



Gelfand duality in NCG
On surjectivity: No, not every representation comes from a state; only the cyclic ones. Every nondegenerate representation of a C\*-algebra is a direct sum of cyclic representations (Zorn), and every cyclic representation comes from a GNS construction. But yes, every irreducible representation (which is also cyclic) comes from a GNS construction.


On injectivity: To even consider this question you probably want to identify representations if they are unitarily equivalent. But no, because for example two unit cyclic vectors in the Hilbert space of a representation will often yield different vector states, but these vector states will yield the same representation.


Which space? That would depend on your applications, and I don't have any absolute answers to that. Instead I'll tell you what comes to mind in the hope that it will help orient you. (Then I'll wait with you to be enlightened by another answer.)


The space of (unitary) equivalence classes of irreducible representations is often called the *spectrum* of a C\*-algebra; a closely related space is the set of kernels of irreducible representations, called the *primitive ideal space*. The primitive ideal space is given the hull kernel topology and is a quotient of the spectrum. I recommend [the book](http://books.google.com/books?id=KV_cC5uYA7EC&dq=morita+raeburn+williams&client=firefox-a&source=gbs_navlinks_s) by Raeburn and Williams; see Appendix A to learn about these spaces, and see the rest of the book for how they're used. 


As for the space of pure states, I'm more accustomed to the idea of studying the space of all states, in which the pure states are the extreme points. There is a nice characterization of the state spaces of C\*-algebras in E. Alfsen, H. Hanche-Olsen and F.W. Shultz: State Spaces of C∗-Algebras, Acta Math. 144 (1980) 267–305, which came up at [this other question](https://mathoverflow.net/questions/4848/gelfand-naimark-from-the-category-theoretic-point-of-view). However, there is yet another option which I learned a little about from [Pedersen's book](http://books.google.com/books?id=yBCoAAAAIAAJ&q=gert+pedersen+automorphism+groups&dq=gert+pedersen+automorphism+groups&ei=ACk_S9-nO4m2NfGZ1bEB&client=firefox-a&cd=1), where the quasi-state space Q is used. A quasi-state is a positive functional of norm at most 1. (Other than 0, the extreme points of Q are also the pure states.) A C\*-algebra can be studied as affine functions on Q; see section 3.10 of Pedersen for details.


