construction of an injective representation of $C_0(X)$

Question

Let X be a locally compact noncompact Hausdorff space and consider the C$^*$-Algebra $C_0(X)$ of continuous functions vanishing at infinity. I want to construct an injective *-represenatation of $C_0(X)$ to give an example of Gelfand-Naimark. I want to construct the representation with the GNS-construction. My beginning:
First of all i need the states of $C_0(X)$ and if i consider http://en.wikipedia.org/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem i would say that the positive linear functionals are $\phi(\mu)(\cdot)=\int_X \cdot d\mu$ with $\mu:X\to\mathbb{C}$ is a positive regular borel-measure (therefore $\mu$ has only real values) and its total variation has to be 1 because it has to be state. The next step of the GNS-construction is to adjoin a unit to $C_0(X)$ ( i write $C_0(X)^1$ for the unitalisized C$^*$-algebra) and extend $\phi(\mu)$ to $C_0(X)^1$. I obtain $\tilde{\phi(\mu)}((f,\lambda))=\int_X f d\mu +\lambda\|\phi(\mu)\|$, $\tilde{\phi(\mu)}:C_0(X)^1\to \mathbb{C}$. Because $\phi(\mu)$ is a state, i have $\tilde{\phi(\mu)}((f,\lambda))=\int_X f d\mu +\lambda 1$.
Now, i have lots of problems to continue the constraction: Every state give me a scalar product $\tilde{\phi(\mu)}((g,\eta)^*(f,\lambda)):=<(f,\lambda),(g,\eta)>_{\tilde{\phi(\mu)}}$ (im not sure if it is correct and how to write out $<(f,\lambda),(g,\eta)>=...$.
The next step is to define $N_{\tilde{\phi(\mu)}}:=\{(f,\lambda) \in C_0(X)^1: <(f,\lambda),(f,\lambda)>=0\}$. The $<\cdot,\cdot>_{\tilde{\phi(\mu)}}$ has to be a scalar product on the quotient $C_0(X)/N_{\tilde{\phi(\mu)}}$ and i have to take the Completion of $C_0(X)/N_{\tilde{\phi(\mu)}}$. The GNS-construction is not finished with this step. But the things will be more complicated in every step. For example, which space is $C_0(X)/N_{\tilde{\phi(\mu)}}$ ? Could you help me to construct an injective *-representation of $C_0(X)$ ant to coninue here maybe? I'm not sure if i do this in a correct way. Regards

Edit: If you need some definitions, i will add them.

Answer 1

I don't think you can get too far with your approach, because you want to deal with the set of all measures on $X$, and there is nothing explicit about it.

So you want to use fewer states.

1) Following on what Phoenix87 said, here is an example of a faithful representation (denomination way more common in the literature than "injective"). It is based on using the pure states, which are nothing but the point evaluations.

Let $H$ be a Hilbert space with basis indexed by $X$, i.e. $\{e_x\}_{x\in X}$. Then you map $f$ to the diagonal operator $\{f(x)\}_{x\in X}$. Namely, $$ \pi(f)e_x=f(x)e_x. $$

2) Another way it to use a single state, but such that it is faithul. You can get such a state if you can construct a probability measure $\mu$ on $X$ such that $\int_Xf\,d\mu=0$ implies $f=0$ for any $f\in C_0(X)$. This is for example the case with $\mathbb R$ and the Lebesgue measure, but I don't think you can have something like this for arbitrary $X$.

If you do have such $\mu$, then you can do GNS for the state $f\longmapsto\int_Xf\,d\mu$. If you push this through, you'll find that your representation is the map $f\longmapsto M_f$, where $M_f$ is the multiplication operator by $f$ on $L^2(X,\mu)$.

Answer 2

If I do understand your question correctly, you want a representation $\pi$ of $C_0(X)$ on some Hilbert space, and you want $\pi$ to be injective. As injectivity in this sense is equivalent to "isometric", I'll give you three constructions of isometric representations of $C^*$-algebras that I know of.

Construction 1 This is a universal construction, known as the universal representation $\pi_u$. Roughly speaking, for any state $\omega$ in the state space of your $C^*$-algebra, you consider its GNS representation, and then you take the direct sum over them. So $$\pi_u := \bigoplus_{\omega\in\mathcal S(A)}\pi_\omega$$ where $\mathcal S(A)$ is the state space of $A$ and $\pi_\omega$ is the GNS representation associated to the state $\omega$. This representation is clearly isometric, but the "drawback" is that the Hilbert space of the universal representation is huge (non-separable already for $M_2$).

Construction 2 Since $\pi_u$ is universal because of pure states, one can consider a "smaller" canonical representation by restricting to the extreme points of $\mathcal S(A)$. So another injective representation could be $$\pi := \bigoplus_{\omega\in\partial\mathcal S(A)}\pi_\omega.$$

Construction 3 This applies just to separable $C^*$-algebras. Find a dense sequence $a_1,a_2,\ldots$ in the algebra. There are states $\omega_k$ with the property that $\omega_k(a_k^*a_k) = \Vert a_k\Vert^2$, and therefore it is enough to construct $$\rho :=\bigoplus_{k\in\mathbb N}\pi_{\omega_k}$$ which is an isometric representation of $A$ on a separable Hilbert space.